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## LIMITS INTRODUCTION WITH SOLVED PROBLEMS

1. Introduction to Limits in Calculus Eight worked limit problems. Illustrates the concepts of limits using graphs. The limits of periodic functions, such as sine functions, are found using graphs and using the calculator. Rating A
2. Limits of Basic Functions Just a few introductory examples that illustrate properties of limits, such as the limit of a constant function, the limit of a first degree equation.
3. Introduction and Basic Definitions 5 worked limit problems. Includes Absolute Value of x over x and sin(1/x). Some more theoretical or thinking problems.
4. Definition of a Limit A few examples related to negative infinity and positive infinity.
5. Basic Limit Problems. 5 introductory limit problems are given for the beginner. Shows graphically that as x approaches 0, the sin(x)/x appoaches 1. However there is a hole at x=0.
6. Some Basic Properties 5 limit problems solved. One problem requires that you know that limit of sin(x)/x = 1 and the limit cos(x) =1 both as x approaches 0. A fairly difficult limit problem is also given that requires rationalization of the denominator and numerator. One solution requires cross-multiplication and simplification.
7. Find Limits of Functions in Calculus Fifteen limit problems with complete solutions with explanations. This section focuses on limits that involve absolute value, rational expressions, the ratio of two polynomials, as well as complex problems that involve the ratio of trigonometric expressions, natural logarithms, exponents and complex radicals. Various methods of solving for limits is illustrated, such as the use of L'Hopitals rule. Rating A
8. Properties of Limits in Calculus Five examples illustrating the basic properties of limits: Setup well for illustrating the use of f(x) and g(x) involving a limit problem when its divided up into two or more functions: the limit of the sum of two functions, the limit of the product of two functions, the limit of the quotient of two functions, the limit of the nth root of a function. Basic formatting of the examples could use improvement. Rating C
9. The Limit Includes five examples with solutions. Provides solutions to some more difficult limit problems for piecewise functions. The limit of the Heaviside function is shown to not exist. Describes the limit function from a graphical standpoint, but more in detail than section 1. Touches on the concept of epsilon.
10. Limits an Introduction An elementary introduction to limits
11. Understanding Epsilon-Delta proofs
12. Finding Delta Given a Specific Epsilon. Four problems with worked solutions
13. Finding Delta Given a Specific Epsilon. Ten problems with worked solutions These problems are relatively easy and serve as a good introduction to the basic solution to delta epsilon problems.
14. Verification of Limits Through the Relationship between Epsilon and Delta. Eight problems with explained solutions
15. Determination of Vertical Asymptote 10 problems with worked solutions
16. Determination of Horizontal Asymptotes using Limits 10 problems with worked solutions.
17. Definition of a Limit and Calculation of Delta for a Given Epsilon and a General Epsilon. Provides example problems with solutions
18. The Definition of the Limit Describes analytically the definition of a limit. Lists nine definitions related to limits. Has six problems with explained solutions for limit proofs.
19. One-Sided Limits Offers an explanation of one sided limits. Includes four multi-part problems. Provides a method for quick graphical analysis of limits.
20. Limit Properties A concise list of nine basic limit properties. Three high level limit problems with complete solutions.
21. Basic Limit Laws
22. Zero over Zero
23. Limits - Formal (e,d) Definitions
24. Limits - Techniques 1 - Finite fixed point (x0), finite limit (y0)
25. Computing Limits
26. Introduction to Limits
27. Infinite Limits
28. Rules for the Determination of Horizontal and Vertical Asymptotes for Rational Functions. Includes ten worked examples that illustrates how to determine if the the function defines a hole or a vertical asymptote. Use of rules that pertain the degree of of the polynomial in the numerator and denominator are listed. Limits are not used for these examples.
29. Definition and Determination of Asymptotes Through the Use of Calculus from Wikipedia. Includes section on oblique asymptotes, Specials cases for determination of horizontal asymptote for different polynomial degrees. Determination of the horizontal and vertical asymptote through transformation techniques. Arctanget asymptote is discussed. Taking the limit of a function as x approaches infinity and negative infinity to determine the horizontal asymptotes.
30. Limits at Infinity 5 problems with complete solutions. Emphasizes the use of factoring to find limits. In this problem set, limits of functions as x approaches positive infinity and negative infinity to find the horizontal asymptotes of a function.
31. Limits at Infinity Provides a few limit word problems with solutions. using exponential decay functions. Also uses limits evaluated at positive and negative infinity to find horizontal asymptotes. Vertical asymptotes, which are determined initially through examination of the denominator, are verified, by verifying that the limit approaches negative infinity and positive infinity as x approaches the value of the asymptote.
32. Limits at Infinity Provides several problems and solutions with detailed explanations through graphs. Emphasizes dominating terms to logically evaluate limits. For example, the growth rate of the numerator function f(x) is compared against the denominator function growth rate, g(x). Provides basic rules for evaluating the limits of polynomial based rational expressions. Discusses the limits of periodic and oscillating functions as they approach infinity. Illustrates the squeeze theorem to find the limit of a function that is periodic as it approaches specific numbers, that is not negative or positive infinity.
33. Limits At Infinity, Part I
34. Limits At Infinity, Part II
35. Limits of Functions as x approaches a Constant 15 problems with complete solutions for limits that often are mistakenly miscalculated at infinity. The use of L'Hopitals rule is used in the solution for some of the problems.
36. Limits to Infinity
37. Limits and Infinity
38. Limits of Functions as X Approaches Infinity 20 problems with complete solutions as for limits approaching infinity
39. Limits of Functions as X Approaches Infinity
40. Precise Limits of Functions as x Approaches a Constant
41. Squeeze Theorem- Provides solution to the limit of x^2*sin(1/x) through use of squeeze theorem. Also provides proof of the squeeze theorem, which is also known as the pinching theorem and the sandwich theorem.
42. Use of Squeezing Theorem to Find Limits Intermediate problem illustrates the setup of a math function for evaluating the limit using the squeeze theorem also known as the sandwich thereon. Requires that you determine the range of one part of the function in order to bound the limit. Proof http://www.analyzemath.com/calculus/limits/squeezing.html
43. The Pinching or Sandwich Theorem
44. The Squeeze Theorem - More Advanced Problems. Features 3 problems with solutions that illustrate the use of the squeeze theorem. Provides an introduction that explains basic procedure for using the squeeze theorem.
45. Limits of Functions using the Squeeze Principle
46. Calculate Limits of Trigonometric Functions Seven examples complete with solutions for finding limits of complex trigonometric functions.
47. Limit Problems with Solutions from Spark Notes
48. Limits and L'Hopitals Rule
49. Limits Test Review Summaries and Cheat Sheets A two page review sheet of the basic limit definitions, properties and theorems you will need to know to excel in limits. Summarizes the common types of continuous functions. Also includes several different procedures used for finding limits, factoring, rationalizing, rational expression combinations, limits of functions to remember, limits of piecewise functions.
50. List of Unsolved Problems about Limits

## CONTINUITY WITH PROBLEMS AND SOLUTIONS

1. Continuity 1 Definitions of Continuity with Respect to Limits. Includes 5 worked example problems. Examples include Jump Discontinuities and problems related to the Intermediate Value Theorem. Recites that the continuity of function at a point is only guaranteed if the limit of the function as x approaches that point is equal to the same value of the function at that point. An example illustrates the use of the Intermediate Value Theorem to find the roots of a polynomial equation.
2. Continuity 2 Illustrates graphs of different types of discontinuities for different functions. Includes jump discontinuities, removable discontinuities, for functions such as 1/x, sine functions, and piecewise functions. A few example problems with complete and graphed explanations and solutions. Expresses the point that the function, f(x), is continuous at if and only if the right- and left-hand limits exist and are both equal to f(c)
3. The Intermediate Value Theorem
4. Continuity contains solved problems that require that the continuity of a specific function be discussed. Coefficients of a functions must be determined such that the piecewise functions is continuous at given point that they are approaching at the same time. Solutions to these problems can be done in two different ways. The solutions presented here takes the right and left hand limits of the functions and equates them.
5. Continuity of Functions of One Variable Lists seven properties of continuous functions which are products, quotients, sums and differences of continuous functions. Also lists functions that are always continuous such as exponentials, sine and cosine trigonometric functions and polynomial functions. Provides 15 problems with detailed solutions.
6. Continuous Functions in Calculus Introduction and definition of continuous functions. Provides examples of continuous and discontinuous functions. Defines three required conditions of continuity. Provides three example continuity problems with complete solutions. Problems emphasize continuity over intervals.
7. Continuity Theorems and Their use in Calculus Provides five theorems used for the evaluation of continuous functions: polynomial, sine, cosine, arctan, e^x, continuity of a rational functions, continuity of composite functions. Also includes a few worked examples to clarify the main points of the five theorems.
8. Continuity Problems and Solutions from Spark Notes: Provides 7 problems with complete explanations. Problems require that you determine if the function is continuous at a specific point or over a specific interval
9. Continuity of Piecewise Defined Functions from the University of Tennessee at Knoxville - Provides 8 piecewise problems with complete solutions. Problems illustrate how to create a piecewise function that will be continuous over an interval or at a point.
10. Differentiation and Continuity
11. The Bisection Method

## LIMITS AND DERIVITIVES WITH PROBLEMS AND SOLUTIONS

1. The Definition of the Derivative Includes worked examples on use of the definition of the derivative to calculate the derivative. Also provides information on alternative notation used instead of the more common prime notation.
2. Definition of the Derivative
3. The Definition of the Derivative
4. Using the Definition to Compute the Derivative
5. Difference Quotient The difference quotient is defined as the slope of the secant line. Four example problems with solutions are given that make use of the definition as it applies to the limit of the difference quotient.
6. Limit Definition of the Derivative - HMC Calculus Tutorial Emphasizes the geometric definition of the derivative through the secant and tangent. Includes some examples.
7. Interpretation of the Derivative Relates the concept of rate of change, such as velocity, water flow and others to the derivative. Provides a method on how to solve for the derivative of a function from its graph.
8. Use Definition to Find Derivative Provides 3 worked problems with solutions using the definition of a derivative to find the derivative of a functions.
9. Derivatives Using the Limit Definition 12 problems with complete solutions focused on using the limit definition to compute the derivative. Also has example problems that illustrate the process of showing the a function is differentiable.
10. Proof of Various Limit Properties Provides a list of 9 limit properties and provides complete proofs for each limit property. Proofs utilize triangle inequality, induction and the other properties of limits for the proofs.
11. Proof of Two Commonly Used Trigonometric Limits A graphical illustration is included to improve understanding. Limits are for sin(x)/x and (cos( h) - 1)/ h
12. Proof of Trigonometric Limits Provides another proof for the trigonometric limits of sin (x) / x and (cos( x) - 1)x
13. Instantaneous Velocity as Defined in the Limit. Desribes instaneous velocity in the simpler of terms with a table and numerical calculations. Also defines instantaneous velocity as a limit of distance over time.
14. Solving for Instaneous Velocity Using the Definition of a Limit. Provides a completely worked example that requires you solve for the instaneous velocity at a given time given a distance function.
15. Applications of Limits to Find Instaneous Velocity. Average velocity is discussed in relation to the slope of secant line. Determining instaneous velocity at a given point in time from a function with a dependent variable of distance and an independent variable of time using the definition of a limit. Requires that you evaluate the limit as the final time approaches the beginning time or as delta time approaches zero. Solves the problem numerically. (s(t2) - s(t1)) /( t2 -t1) as t1 approaches t2 or delta t approaches zero.
16. Derivative - Wikipedia, the free encyclopedia Provides a complete overview of continuity, limits and derivatives. Illustrates the geometric concept of a derivative through an animation of the tangent and secant functions.

## REVIEW OF LIMITS AND CONTINIUITY

1. Review for limits and continuity Includes worked problems, graphing facilities and calculator instructions. Visual explanations of epsilon-delta definition, numerical determination of limits, bisection method, horizontal asymptotes, vertical asymptotes

## DERIVITIVE RULES AND PROCEDURES

1. Differentiation and Continuity
2. Calculus: Derivatives
3. Calculus: Properties of Derivatives
4. Solving for Derivitives of Polynnominals Using Basic Derivitive Rules.
5. Rules of Differentiation of Functions in Calculus Basic rules of differentiation with example and solution for each rule. Includes power rule, sum rule, difference rule, product rule, quotient rule and constant rules of differentiation.
6. Product and Quotient Rule Includes four product and quotient differentiation problems with solutions. Emphasizes a common mistake that students make when using the product and quotient rule. Also gives a related rates word problem that requires the use of the quotient rule to solve. Indicates that the product and quotient rule can be applied to products and quotients that consist of three or more functions.
7. Product Rule for Derivatives Three product rule differentiation problems with solutions. Illustrates the product rule for finding derivatives that are the products of trigonometric functions and the exponential function.
8. The Product Rule
9. Proof of the Derivative Product Rule http://www.math.hmc.edu/calculus/tutorials/prodrule/proof.pdf
10. Differentiation Using the Product Rule 19 product rule differentiation problems with solutions. Illustrates by example use of the triple product rule. As well, lines that are tangent and parallel to a function are determined with the product rule. Product rule derivative problems are determined with products that include exponentials, the exponential, natural logarithms, and trigonometric functions such as sine, cosine and tangent.
11. Quotient Rule for Derivatives Two quotient rule derivative problems with solutions.
12. Proof of the Derivative Quotient Rule http://www.math.hmc.edu/calculus/tutorials/quotient_rule/proof.pdf
13. The Quotient Rule
14. Differentiation Using the Quotient Rule Twenty-one quotient rule differentiation problems with solutions. Problems require the use of the quotient rule for derivatives to solve. Some problems also require that the chain rule also be applied.
15. Differentiation Formulas Ten differentiation problems using the sum and difference rules for taking derivatives. Also includes an in-depth problem related to velocity and direction of change of velocity.
16. Differentiation Using the Chain Rule Twenty chain rule differentiation problems with solutions. Complete explanations are given. Chain rule differentiation problems include functions that include sine, cosine, natural log, arctangent, and exponential functions. Illustrates the chain rule to find a differentiable function and to find the value at a specific point of the function.
17. Use the Chain Rule of Differentiation in Calculus Four chain rule differentiation problems with solutions. Shows solutions that use implement du substitution through the evaluation of du/dx and dy/du.
18. Chain Rule Features two chain rule differentiation problems with solutions with du/dx and dy/du format. Also shows alternate notation often used to represent chain rule differentiation
19. The Chain Rule
20. The Chain Rule
21. Partial Derivatives Examples

## Finding Derivatives of Specific Types of Functions

1. Find Derivatives of Functions in Calculus 11 derivative problems with solutions that are solved with the chain rule, product rule and quotient rule
2. Differentiation of Trigonometry Functions 18 trigonometric derivative problems with solutions that make use of the derivatives for cosine, sine, tangent, cosecant, secant and cotangent. One problem requires that you find the line that is perpendicular to the tangent line of the trigonometric function.
3. Differentiation of Trigonometric Functions 3 trigonometric derivative problems with explained solution.
4. Derivatives of Trigonometric Functions 10 trigonometric derisive problems with solutions. Uses the theorems of limits for limit (sin x)/x = 1 and (cos (x)-1)/x = 0 to determine the derivatives. Also illustrates how to find the savings rate as a specific point in time when the financial savings rate is a trigonometric function.
5. Derivatives of Trigonometric Functions Shows that limit (cos (x)-1)/x = 0 through the use of the fact that limit (sin x)/x = 1. Also has three derivative trigonometric problems with solutions.
6. Differentiation of Inverse Trigonometric Functions 15 inverse trigonometric derivative problems with solutions. Inverse trig functions include arcsin, arccos, arctan, arccotan, arccsc and arcsec. One word problem is also solved. A problem that requires that you take the limit of a function that involves inverse trigonometric problems is also solved.
7. Derivative of Inverse Trigonometric Functions 3 inverse trigonometric derivative problems with solutions.
8. Derivatives of Inverse Trig Functions Provides a proof of the derivative for the arcsin using composite inverse definitions. Also provides a few examples for finding the derivatives of inverse trig functions.
9. Derivative of Inverse Function Example showing how to find the derivative of the inverse of a function, given an inverse. That function is a linear function. Uses another example to show how the properties of inverse functions are used to find the derivatives of the arcsin, also known as the inverse sine function.
10. Derivatives of Exponential and Logarithm Functions Basic formulas for finding depravities of exponentials and logarithms are given. Proofs are presented. Two sample problems are given as well as word problem about exponentials and velocity.
11. Logarithmic Differentiation 8 examples for solving complex differentiations through the use of logarithmic differentiation.
12. Logarithmic Differentiation 3 worked examples for solving complex differentiation through the use of logarithmic differentiation, or what might be referred to also as implicit differentiation
13. Logarithmic Differentiation 7 problems which required logarithmic differentiation are explained and solved. These types of problems are identified by the fact that the variable in the equation is also used as an exponent for variable. In these cases the ordinary rules of differentiation do "not" apply and techniques of logarithmic differentiation and / or implicit differentiation are used. These types of problems are common in solutions for differential equations.
14. Differentiation of Exponential Functions 4 example exponential differentiation problems with solutions. Functions of the form b^x are solved.
15. Find Derivative of y = x^x. Classic example of using logarithmic differentiation to find the derivative of a function that raises the power of a variable to a multiple of the variable itself.
16. Differentiation of Logarithmic Functions 4 examples that show how to take the derivative of a function of the form logb(x) where b is the base of the log. Formula for the derivative of a log of a given base is given. Examples provide solutions.
17. Differentiation of Hyperbolic Functions Formulas with examples are given on the derivatives of hyperbolic functions. Graphs are given of the hyperbolic functions. sinh, cosh, tanh, sech, coth, csch
18. Derivatives of Hyperbolic Functions The derivative of hyperbolic functions are derived with the hyperbolic exponential function definitions. List of derivatives of the the six hyperbolic functions. Graphs are included. Identities of the hyperbolic are given.
19. Differentiating Inverse Functions
20. Derivatives Involving Absolute Value 2 examples of absolute value derivative problems with complete solutions.
21. Differentiating Special Functions A basic review of differentiation of trigonometric, logarithmic, and exponential functions. The derivatives of sine  and cosine are derived through the definition of a limit.
22. Techniques of Differentiation A few review problems with complete solutions. Problems include quotient and product rule. Also examples illustrate the use of the derivative to find the slope of a tangent line and the slope of a line perpendicular to the tangent lines.

Derivitive Calculator

1. Derivite Calculator. This calcualtor will give the derivitve of a function entered. It will also plot the original function entered as well as the derivitive of that function.

DERIVITIVE PROOFS

1. Fact, Formula, Test Review Sheets, Cheat Sheets
2. Proofs of Derivative Applications Facts/Formulas Fermat’s Theorem, Rolles Theorem, Mean Value Theorem

LIMITS DETERMINATION WITH L'HOPITAL'S RULE, LIMITS BY DERIVITIVES

1. L'Hopital's Rule 8 example problems with solutions that use L'Hopitals rules to find the limit of a function.
2. L'Hopital's Rule And The Indeterminate forms 0 / 0 5 limit problems with explanations that illustrate finding the limits of functions with indeterminate form 0/0 with derivitives as defined by L'Hopitals rule.
3. Indeterminate forms of Limits Five limit problems with explained solution that involve complex functions such as exponential, logs, and trigonometric functions. The seven forms of indeterminate limits are listed. L'Hopital's rule requires that the function has one of the indeterminate forms. The problems require the identification of the indeterminate form type and then use of L'Hopitals limit rule, when and only when, the rules indeterminate form requirement are met.
4. L’Hospital’s Rule and Indeterminate Forms 7 limit problems with solutions. These problems are more difficult than most. An introduction to L'Hopitals rule and a list of the seven forms of the indeterminate limits are given. A procedure for converting a function that is a product of functions to a quotient of functions. One problem's solution requires that the natural log of the functions be taken first in order to evaluate form of the indeterminate limit.

TABLES OF DERIVITIVES

1. TABLE OF DERIVITIVES A short table of about twenty functions and their derivatives
2. Table of Derivatives

NEWTONS METHOD

1. Newton’s Method Two Newton Methods problems with explained solution. Includes introduction to Newton's Method and applications where it is used. Newton's method is a successive approximation method that lends itself as a computer software algorithm. A initial approximation to the numerical answer to the equation must be supplied. '
2. Newton's Method to Find Zeros of a Function Three newton methods problems with explained solutions are given. The use of making a rough graph of the function to make an approximation to the numerical solution of the equation, also known as the zeros of the function.
3. The Newton-Raphson Method

LINEAR APPROXIMATIONS

1. The Tangent Line Approximation Explanation of how to find a tangent line at a point on a function
2. Linear Approximation of Functions 3 linear approximation problems with solutions. The problems use derivatives to determine a linear approximation to a function at a point on the function
3. Linear Approximations Two linear approximation problems with solutions. Includes graphs to illustrate the solution.
4. Linear Approximations

TANGENT LINES

1. Solve Tangent Lines Problems in Calculus Two tangent line problems that require that a line parallel to the tangent be found.
2. Slope of the tangent line and the normal line to a function is determined through the use of a derivitive http://www.cliffsnotes.com/study_guide/Tangent-and-Normal-Lines.topicArticleId-39909,articleId-39887.html

RATE OF CHANGE PROBLEMS

1. Basic Related Rate Problems Using Derivitive to Solve. Shows an example and provides several problems with answers only.
2. Solve Rate of Change Problems in Calculus 3 rate of change problems with graphical illustrated solutions. One fluid flow rate of change problem, one velocity and distance rate of change problem, and one electrical rate of change problem that involves resistances that change with time.
3. Solve Rate of Change Problems in Calculus One rate of change problem with solution involving the rate of air flow and rate of volume change of a balloon. An in-depth and animated graphical illustration for determining the tangent line of a function at a given point.
4. Rates of Change of Tangent Lines
5. Rates of Change Two rates of change problems. One problem involves cars traveling in different directions.
6. Tangent Lines and Rates of Change

RELATED RATE PROBLEMS

1. Related Rates Related rate problem with solution. rate at which the radius of the balloon is increasing when the diameter of the balloon is at a given length
2. Related Rates 6 multipart related rate problems with graphically illustrated solutions. Related rate classical ladder problem. Related rates of a distances as a function of angular direction Tank leading related rate problem, rate at which water depth is changing at a specific depth. Related rate of length of shadows changing.

### MINIMUMS, MAXIMUMS, FIRST AND SECOND DERIVITIVE TEST

1. Monotonicity and the Sign of the Derivative
2. 3 example problems that involve finding the minimum and maximum of a function through the use of the first derivative.
3. Critical Points 7 critical ;point problems with solutions. The critical points are found for a number of complex functions. Section also provides a definition for a critical point.
4. Critical Points
5. Plotting Functions By Using the First Derivitive of a Function. The Shape of the Graph. This web page has 5 examples that illustrate how you plot a graph of a given function by finding the critical points, the points where the maximum and minimum occur. The first deritivive of the function is taken and then set to zero and solved to determine the critical points. Then the direction of the slopes of the intervals between the critical points is evaluated using the the first derivitive and test points between each crtical point.
6. Find Critical Numbers of Functions 4 critical number problems with solutions. Definition of critical number (critical point) is given.
7. Derivative, Maximum, Minimum of Quadratic Functions Two maximum minimum problems with solutions for quadratic functions (second degree equations or more commonly known as the parabola).
8. Minimum and Maximum Values 6 maximum and minimum problems with graphical explained solutions. Includes the mathematical definition for maximum and minimum. Text based and graphical definition of global and local minimum and maximums also called absolute and relative minimum and maximums. Extreme Value theorem and Fermet's theorem are discussed.
9. Finding Absolute Extrema 5 absolute extrema problems with solutions.
10. Global Extrema
11. Concavity and Points of Inflection
12. Determine the Concavity of Quadratic Functions 2 concavity examples that illustrate how to determine the concavity of a parabola.
13. Concavity and the Second Derivative Test 2 example problems. Explanation and definition of concavity and inflection points. Use of the second derivative test to determine where a function is concave down and concave up and where the inflection points are.
14. The Shape of a Graph, Part I 5 examples with solutions that make use of the first derivitive to determine if a function is increasing or decreasing over an interval and if a function ever increases or decrease. Critical points are also used in the solutions to the problem.
15. The Shape of a Graph, Part II 3 concavity examples with solutions. Gives a practical working definition of concavity. Also gives the mathematical definition of concavity. Discusses misconceptions related to concave up and concave down.
16. First, Second Derivatives and Graphs Of Functions 2 examples of using the first and second derivatives to graph a function. The examples are organized in steps and include graphs to illustrate the somewhat lengthy process.
17. Graphing Functions using First and Second Derivatives 11 examples for graphing functions using the first and second derivatives. Horizontal and vertical asymptotes are found in the procedure. Problems, which come compete with worked solutions, include the most basic to the most difficult. Problem number 11 is one of the more interesting problems.

MAXIMUM / MINIMUM OPTIMIZATION PROBLEMS(Using first and second derivitve tests)

1. Maximum and Minimum Problems 21 solved maximum and minimum problems. The problem set focuses on solving a variety of word problems that require that you find the minimum or maximum. Section also includes a six step procedure for solving minimum and maximum optimization problems. .
2. Minimum Distance Problem Provides two methods to solve a classical minimum distance problem. This minimum distance problem is often seen on many tests, Knowing how to work this problem is often essential is passing a Calculus course.
3. Maximum Area of Rectangle - Problem with Solution The example illustrates how to find the largest rectangle that can be inscribed in a given triangle. A graphically illustrated solution is presented.
4. Maximum Radius of Circle - Problem with Solution This example illustrates how to find the largest circle that can be inscribed in a right triangle.
5. Maximum Area of Triangle - Problem with Solution This example illustrates how to find the largest triangle that can be inscribed in a circle. As with all minimum and maximum problems the first derivative test is used.
6. Maximum Area of Triangle - Problem with Solution
7. Maximize Volume of a Box This example illustrates how to find the maximum area of a box that can be made from a piece of metal of a given size. This classical problem includes a graphic of the metal sheet as well as a plot of the volume of the box versus the length of the corner cutout on the box.
8. Maximize Power Delivered to Circuits This example is a must for electrical engineers studying calculus. The example illustrates the process of finding the resistor value needed to maximize the power delivered to a load. An electrical schematic is given along with the graph of power versus resistance for the circuit.
9. Use First Derivative to Minimize Area of Pyramid This example illustrates how to find the minimum surface area of a pyramid that has a square base and a fixed volume of 1000 cubic centimeters.
10. Use Derivatives to solve problems: Distance-time Optimization For those wanting to minimize their travel time, this illustrated and animated example shows how to use derivatives to find the fastest route to travel. The solution provides an animated applet that lets the student visualize the results. A plot of the time it takes for traversing the different route options is presented in the applet. The point should be dragged slowly in the Applet for best results. Also be sure to turn on the radio button on the graph. A complete step -by-step analytical solution is also presented.
11. Use Derivatives to solve problems: Area Optimization This example illustrates how to find the dimensions of a rectangle that has a fixed perimeter that will maximize the area of the rectangle. An interactive applet illustrates the function that describes the area as a function of the width and length of the rectangle. A complete-step-by-step analytical solution is presented. Be sure to turn the graph radio button on to use the applet. Also move the point on the applet slowly to improve the response of the applet.
12. Use Derivatives to solve problems: Area Optimization
13. Optimization Problems 6 area optimization and volume optimization problems are presented. They are solved using four different methods, the first derivative method, a variant of the first derivative method, the absolute extream method, and the second derivative method. Problems focus on finding the minimum amount of materials needed to construct geometric objects with given areas or volumes.
14. More Optimization Problems 8 optimization problems are worked through with complete solutions. Most of the problems relate to wire length optimization. These problems can be used for minimizing costs when wires from utility poles must be staked to the ground. Problems have a relatively high level of difficulty. The problem set features a classical wire minimization problem. This problem requires that you determine where a wire must be cut to form a triangle and a square that has the maximum area. Other optimization problems presented relate to the fields of architecture and design. A window must be built that lets the maximum amount of light in, given that only a fixed amount of framing materials is available.
15. Business Applications For those studying business and business calculus, this section features 8 optimization problems with solutions that provide the methods to maximize revenue and profit and minimize costs based on given business models. Profit, cost and profit general math equations are used in these solutions along with the derivative. The profit function, the demand function, the price function, the marginal revenue function and the marginal profit function are all discussed and used to solve these business problems.

MEAN VALUE THEOREM

1. Mean Value Thereon An explanation of the Mean Value Theorem as an extension of Rolle's theorem.
2. Mean Value Definition and Concept This section explains the mean value theorem and has a interactive graphical tutorial to further explain the concept. The mean value theorem is considered one of the most important concepts in calculus. It states that a tangent to a function will have a secant line, which has its end points on the function, that is parallel
3. The Mean Value Theorem 4 problems which require the use of the Mean Value Theorem are solved. The problems presented give insight into the type of problems that the mean value theorem can be used to solve.
4. Mean Value Theorem Problems 2 problems are solved that use the Mean Value Theorem
5. The Mean-Value Theorem
6. Mean Value Theorems for Integrals

IMPLICIT DIFFERENTATION

1. Implicit Differentiation One example that solves for dy/dx using implicit differentiation technique.
2. Implicit Differentiation Six implicit differentiation problems of varying levels of difficulty. Complete with solutions. Two methods to solve the implicit differentiation problems are discussed. Implicit differentiation is applied to complex functions that involve exponentials, natural logs and trigonometric functions.
3. Implicit Differentiation Problems 16 solved implicit differentiation problems of varying degrees of difficulty. An introduction to implicit differentiation is given with an involved example. It is noted that because many equations can not be solved explicitly for y, the derivative can not be found explicitly. This requires the use of implicit differentiation techniques, which makes use of the Chain rule.
4. Implicit Differentiation 4 examples of implicit differentiation problems, one that includes using implicit differentiation to find the tangent line to an ellipse are solved.

ARC LENGTH

1. Arc Length This page, which takes a little while to load up, is worth the wait. A proof is given for the derivation of the arc length formula for a given function over a specific interval. The Mean Value Theorem and Riemnes sums are used in the proof. An example is also given and solved. The example is illustrated with an animated mathematical java applet. The other problems are illustrated with well defined graphics and the mathematical symbols used are clear.
2. Differentials Three solved differential problems are solved. Computation of delta x and delta y for given variations in an independent variable. Differentials are used to estimate change in dimensions, such as in the volume problem that is solved and also estimate error.
3. Higher Order Derivatives Solving for the second and third derivative of a function. Three higher order derivative problems are solved. Implicit differentiation is applied to find second devitrifies and third derivatives of complex functions.

DERIVITITIVE TEST REVIEW SUMMARY AND CHEAT SHEETS

1. Derivative Test Review Summary and Cheat Sheet A four page review of derivatives with the most common definitions, procedures, applications and concepts. Also illustrates the basic terms with small graphs and helpful reminders. Definitely a must before your first midterm in Calculus I.
2. More Problems on the Derivative

## INTEGRATION

1. Area Problem The basics concepts of using Riemann Sums to estimate the area underneath a function. Different ways to define Riemann Sums for higher levels of accuracy. Illustrated with numerous graphs to instruct on finding the area. Summation notation is also defined and the use of the limit of the summation to find the actual area under a curve.
2. The Algebra of Summation Notation 14 summation problems with solutions. Basic summation formulas are given for common functions. Telescomping series, limits and partial fractions are also used to solve the problems. http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/summationdirectory/Summation.html#PROBLEM 1
3. The Fundamental Theorem of Calculus
4. Antiderivatives A proof related to antideritives is given as well as an example.
5. Riemanns Sums A graphical illustrated introduction to Riemanns Sums and the notation used to represent the area of function in the notation for Riemann sums. Rules related to the calculation of common Riemann sum sections.
6. Calculus: Construction of the Riemann Integral
7. Calculus: Antiderivatives and the Riemann Integral
8. The Limit Definition of a Definite Integral 15 solved summation problems using the limits definition of a definite integral
9. Properties of Integrals - Tutorial 3 solved integration problems are given that are solved with basic integration techniques: substitution, integration by parts, the integration of the sum of functions and the integration of the difference of the sum of functions.
10. Evaluate integrals 4 solved integral problems that involve trigonometric functions and exponential functions.
11. Table of Integrals http://www.analyzemath.com/calculus/Integrals/table_integrals.html

## INDEFINITE INTEGRALS

1. Indefinite Integrals 4 solved indefinite integral problems for polynomial functions. Basic properties of indefinite integrals are discussed. Links to proof these properties. http://tutorial.math.lamar.edu/Classes/CalcI/ProofIntProp.aspx#Extras_IntPf_Indef1 http://tutorial.math.lamar.edu/Classes/CalcI/ProofIntProp.aspx#Extras_IntPf_Indef2
2. Computing Indefinite Integrals 4 multipart indefinite integral problems are solved
3. Substitution Rule for Indefinite Integrals
4. More Substitution Rule 5 multi-part indefinite integrals are solved with the substitution rule. Several trigonometric integrals are found using identities and substitutions.

## DEFINITE INTEGRALS

1. Definition of the Definite Integral 7 solved problems that involve definite integrals. The definition of the definite integral and Riemanns sums are used. The Fundamental Theorem of Calculus is discussed. Several properties of integrals are listed and explained.
2. Properties of the Definite Integral
3. Computing Definite Integrals 4 multipart solved definite integral problems. Includes integrals of piece wise functions and absolute value functions.

## Substition

1. Substitution Rule for Definite Integrals 5 multipart definite integral problems are solved through single and multivariable substitution. Odd and even function integration solutions are discussed.
2. The Method of U-Substitution 18 indefinite integrals are solved through substitution techniques. Discussion of anti-derivitive to determine the terms to substitute for U.
3. Computing Integrals by Substitution 4 definite integral problems are solved through substitution techniques. Two methods are give for some of the problems.
4. Substitution
5. Integration Substitution Methods
6. Calculus: U-Substitution
7. Numerical Integration
8. Problems on Techniques of Integration

INTEGRATION BY COMPLETING THE SQUARE

1. Computing Integrals by Completing the Square The square of a function is completed to obtain the form needed for integration that equates to the arccos and arctan. The proof the quadratic formula is given by completing the square.

INTEGRATION OF EXPONENTIAL FUNCTIONS

1. The integration of Exponential Functions12 indefinite integral problems are solved. Two formulas are given for the integral of natural (base e) logarithm and other exponential functions.

INTEGRATION OF LOGARITHM FUNCTIONS

1. Evaluate Integrals Involving Logarithms - Tutorial 3 solved indefinite logarithm integral problems. Substitution is used to solve
2. Calculus: The Natural Logarithm Function

INTEGRATION OF NON-ELEMENTARY FUNCTIONS

INTEGRATION OF RATIONAL FUNCTIONS

INTEGRATION OF TRIGONOMETRIC FUNCTIONS

1. Integration of Trigonometric Integrals 27 trigonometric integration problems are solved. Basic integrals for trigonometric functions are listed along with the trigonometric identities often used to solve trigonometric integral problems
2. Trigonometric Substitution Two common integration problems are solved through the use of trigonometric substitution Guidelines for substitutions are given using right triangle sketches.
3. Trigonometric Substitutions A non-pdf solution of the previous problems
4. Special Trigonometric Integrals Formulas used for the evaluation of trigonometric functions that are of the form sin(mx), cos(nx) and use various sum and difference trigonometric identities.
5. Trigonometric Substitution
6. Rational Expressions of Trigonometric Functions
7. Integrating Powers and Product of Trigonometric Functions
8. More on Product of Sines and Cosines
9. Other Trigonometric Powers

INTEGRATION BY PARTS

1. The Method of Integration by Parts 23 indefinite integral problems are solved with the integration by parts method. The formula for integration by parts is also given.
2. Integration by Parts 2 integration by parts are solved. One involves the arctan function and the other is an exponential sinusoid.
3. The integration of Rational Functions Resulting in Logarithmic or Arctangent functions 23 integration problems with solutions that involve rational functions and result in an answer that involves the logarithmic or arctangent functions.
4. Integration by Parts
5. Integration By Parts
6. Calculus: Integration by parts

PARTIAL FRACTION INTEGRATION

1. Partial Fractions Comprehensive examples are given for finding the integrals through partial fraction decomposition
2. Partial Fractions Decompositions. 3 problems that use the method of partial fraction decomposition to find the integral of different types of the form P(x)/Q(x).
3. The Method of Integration by Partial Fractions 20 problems are solved through partial decomposition techniques.
4. Method of Partial Fractions

POWER SUBSTITUTION INTEGRATION

1. The Method of Integration by Power Substitution 14 integral problems are solved that use the power substitution process, a process used when the integral to be solved contains many variables that are to the nth root, such as x^1/4.

ACCELERATION, VELOCITY AND OTHER PHYSICS PROBLEMS

1. Average Function Value Two solved problems that use the average function value or Mean Value Theorem for Integrals to find the average value of a function over an interval.
2. Work 2 problems are solved applying the integral to solve for work given the force function and the distance an object moves.

AREA INTEGRATION PROBLEMS

1. More on the Area Problem Two solved area integration problems are presented. Both involve the finding the area of a region, one which is bounded by two curves and the other region, a triangle, bounded by three lines. Graphics are included.
2. Find Area Under Curve 2 problems are solved using integrals to find the area between two curves and within a polygonal region.
3. The Area Problem and the Definite Integral Three example area problems are solved using relations related to the Riemann sum. Advanced notation is used here to describe the relationship between sums, limits and integration. The problems here and presentation level are rated at a high difficulty level.
4. Area Between Two Curves 7 solved area integral problems. The section explains how to setup different types of area problems and when to integrate over the x axis, dx, and when to integrated across the y axis, dy.
5. Find The Area of Circle Using Integrals in Calculus Formula for the area of a circle is derived using integrals. Using integrals to prove the formula for an area of a circle is involved and requires splitting the circle into quadrants and the use of trigonometric identities.
6. Find The Area of an Ellipse Using Calculus The formula for the area of the ellipse is derived through integration. The problem is solved through symmetry and the use of trigonometric identities The area of one-quadrant of the ellipse is found first.

VOLUMES

1. Volumes of Solids of Revolution / Method of Rings 4 volumes defined by rotating a function about an axis are found with the method of rings integration, which uses the inner and outer radius. The lengthy solutions include descriptive 2D and 3D graphics to illustrate the shape of the revolved function and the original function
2. Volumes of Solids of Revolution / Method of Cylinders 4 solved problems are presented here that use the method of cylinders, also known as the method of shells, to find that volume of complex functions that are rotated about the axis. 3D graphics are provided that enable the visualization of the shape after it is rotated. Emphasis is on the concept of using different types of cross-sections to simplify the mathematics involved in the integration. s
3. Find The Volume of a Frustum Using Calculus The volume of a frustum is derived by revolving the edge of the frustum about the x-axis and integrating the result.
4. Find The Volume of a Square Pyramid Using Integrals The formula of the volume for a pyramid with a square base is derived through integration. The cross-sections uses are squares that are parallel to the pyramid's base.
5. Find The Volume of a Solid of Revolution 4 problems that require revolving a function about an axis to find the volume. The problem set, which comes with illustrated solutions, shows how the volume can be found by either rotating about the x-axis or the y-axis.
6. Volume, Volume of a Sphere The volume of a sphere is found by three different methods, the method of discs, the methods of washers and the method of cylindrical shells.
7. More Volume Problems 6 advanced volume by cross-sectional area problems are solved with complete explanations and 3D graphics. The volumes solved for include the cap of a sphere, a square pyramid, a torus, a solid that has a disk base and whose cross-sections are equilateral triangles, a cylinder as well as wedge cut from a cylinder. These volume problems cannot be solved through standard volumes of solid of revolutions techniques.

REVIEW SHEETS AND TABLES

1. Table of Integrals
2. Derivatives and Integration Review and Summery, Test Review Sheets, Cheat Sheets Basic properties of Derivatives, Common Derivatives, Properties, Formulas, and Rules of Integrals, Standard Integration Techniques, Integration of Products and Quotients of Trigonometric Functions, Standard Right Triangle Trigonometric Substitutions for Integrals
3. Integral Review and Summery, Test Review Sheets, Cheat Sheets Integral definition, Fundamental Theorem of Calculus, Properties of Integrals, Common Integrals, Standard Integration Techniques, Integration by Parts, Products and Quotients and Trigonometric Functions, Trigonometric Substitutions, Partial Fractions, Application of Integrals, Area Between Curves, Volumes of Revolution, Arc Length Surface Area, Improper Integrals, Discontinuous Integrand, Approximating Definite Integrals, Midpoint Rule, Trapezoid Rule, Simpson's Rule
4. Table of Common Integrals Lists 18 commonly used integrals. Also gives an overview of the different techniques used for integrating specific types of functions and expressions, such as rational expressions.
5. Notes for First Year Calculus A 78 page review of the basic calculus areas covered in the first year of calculus.

INTEGRATION AND INTEGRAL PROOFS, FORMULAS

2. ### Area and Volume Formulas

3. Fundamental Theorem of Calculus

CALCULUS AND RELATIONSHIP TO INFINITY

Tables

## SERIES AND CONVERGENCE

2. ### Taylor Series

Constant of Integration

Summation Notation

## OTHER MATHEMATICAL TABLES

Table of Laplace Transforms

Table of Fourier Transforms

Fourier Transform of Rectangular Functions

Tables of Mathematical Formulas

Series Convergence and Tests

Definition of a Series

Series Tests

Common Series

Geometric Series

P-Series

Telescoping Series

Integral Test

Limit Comparison Test

Alternating Series Test

Absolute Convergence Test

Ratio Test

Root Test

Using Series Tests

Finding the Form of a Series

Finding the Form a Series, No N Powers

Finding the Form a Series, N Only in the Power

Finding the Form a Series, N in the Power and in the Base

Finding the Form a Series, N Factorial in the Series

Power Series Introduction

Taylor Series

Maclaurin Series

Taylor Polynomials and Error Bounds

Comparison Test

Divergence Test

Power Series as Functions

To solve for the integral of 1/(1+x^2)

1) http://www.sosmath.com/calculus/calculus.html

http://archives.math.utk.edu/visual.calculus/

http://math.mit.edu/~djk/calculus_beginners/

http://www.geom.uiuc.edu/~banchoff/Calculus/PartDerivb&w.html

http://web.mit.edu/wwmath/vectorc/scalar/partial.html

http://www.math.umn.edu/~garrett/qy/Secant.html

http://mss.math.vanderbilt.edu/cgi-bin/MSSAgent/~pscrooke/MSS/partialfract.def

http://www.studentxpress.ie/educ/maths/maths3/maths3.html

http://mss.math.vanderbilt.edu/~pscrooke/MSS/numintegration.html

http://mss.math.vanderbilt.edu/~pscrooke/MSS/arclength.html

http://library.thinkquest.org/3616/Calc/S3/AoSoR.html

http://maths.abdn.ac.uk/~igc/tch/ma2001/notes/node18.html

Examples of Sequences http://maths.abdn.ac.uk/~igc/tch/ma2001/notes/node18.html

Substitute x=tan(z), where z is the angle Zeta

then dx = sec2(z) dz

Substitute back into the original integral

so you now find the integral of (sec2(z)*dz)/(1+tan2(z))

since 1 + sec2(z) = tan2(z)

we have the integral of (tan2(z)*dz/(tan2(z)

the integral simplifes to the integral of dz, since the tan2(z) terms cancel

The integral of dz is z a, since

Since z is equal to the the arctan (x), the integral of 1/(1+x^2) is simply the arctan(x)

http://www.jtaylor1142001.net/calcjat/Solutions/INT/IArcTan/IATanSimp.htm

Trigonometric Identities http://www.clarku.edu/~djoyce/trig/identities.html


text:
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# A retailer originally bought 50 equally priced phones for a total of z

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A retailer originally bought 50 equally priced phones for a total of z  [#permalink]

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24 Jan 2019, 02:51
00:00

Difficulty:

45% (medium)

Question Stats:

50% (01:28) correct 50% (01:27) wrong based on 15 sessions

### HideShow timer Statistics

A retailer originally bought 50 equally priced phones for a total of z dollars. If he sold each phone for 25% more than he paid for it, then in terms of z, how much was each phone sold for?

A z/50

B z/40

C 5z/4

D 4z/5

E 62.5z

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Re: A retailer originally bought 50 equally priced phones for a total of z  [#permalink]

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24 Jan 2019, 03:01
Bunuel wrote:
A retailer originally bought 50 equally priced phones for a total of z dollars. If he sold each phone for 25% more than he paid for it, then in terms of z, how much was each phone sold for?

A z/50

B z/40

C 5z/4

D 4z/5

E 62.5z

IMO B

50 mobiles were bought for 5000 dollars, just chose a value
1 mobile will be bought for 100 dollars

Now he sold for each phone for 25% greater value 125

Plug value back for z in the options to get value of each phone

5000/40 = 125
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Re: A retailer originally bought 50 equally priced phones for a total of z  [#permalink]

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24 Jan 2019, 03:02
Bunuel wrote:
A retailer originally bought 50 equally priced phones for a total of z dollars. If he sold each phone for 25% more than he paid for it, then in terms of z, how much was each phone sold for?

A z/50

B z/40

C 5z/4

D 4z/5

E 62.5z

Cost price of each set = z/50

Selling price = z/50 * 125/100

= z/40.

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Re: A retailer originally bought 50 equally priced phones for a total of z  [#permalink]

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24 Jan 2019, 04:07

Solution

Given:
• Original number of phones bought = 50, at a total price of z dollars
• Price of each phone is same
• Each phone is sold at a price of 25% more than the price at which he bought it

To find:
• Selling price of each phone

Approach and Working:
• Cost price of each phone = $$\frac{z}{50}$$
• Implies, selling price of each phone = $$\frac{z}{50}$$ + 25% of $$\frac{z}{50} = 1.25 * \frac{z}{50} = \frac{z}{40}$$

Hence, the correct answer is Option B

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Re: A retailer originally bought 50 equally priced phones for a total of z  [#permalink]

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24 Jan 2019, 04:17
Bunuel wrote:
A retailer originally bought 50 equally priced phones for a total of z dollars. If he sold each phone for 25% more than he paid for it, then in terms of z, how much was each phone sold for?

A z/50

B z/40

C 5z/4

D 4z/5

E 62.5z

original price z/50
sp = 2*1.25/50 ; z/40
IMO B
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Re: A retailer originally bought 50 equally priced phones for a total of z   [#permalink] 24 Jan 2019, 04:17
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# String conservation with springs

I was trying to solve this Problem that asked to find the period of small oscillations for this system.

To do so I used the fact that for a massless pulley with strings around it, the sum of the displacements of the string around the pulley is twice the displacement of the pulley.

This gives the correct answer but it bothered me because the above fact is dependent on the length of the string being constant but springs can change their length.

Now:

1. Can I justify this by saying that for small oscillations the change in string length is approximately zero?
2. If not, then how can I find the period otherwise?


text:
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YSTW2015: Saharon Shelah is telling a joke..:

Damjan Kalajdzievski: How to show Con(ZFC + omega_1=u<a) from Con(ZFC)
Place: Fields Institute (Room 210)
Date: February 8 , 2019 (13:30-15:00)
Speaker: Damjan Kalajdzievski
Title: How to show Con(ZFC + omega_1=u<a) from Con(ZFC)
Abstract: I will outline how to prove the result in the title by joint work with Osvaldo Guzman
settheory.mathtalks.org/damjan

Berkeley conference on inner model theory, July 08–19, 2019
Berkeley conference on

inner model theory

July 08--19, 2019
Organizers: Ralf Schindler (Münster) and John Steel (Berkeley).

This conference will be a sequel to the 1st Conference on the core model induction and hod mice that was held in Münster (FRG), July 19 -- August 06, 2010, to the 2nd Conference on the core model indu
settheory.mathtalks.org/berkel

Logic Fest in the Windy City, Chicago, May 30 – June 2, 2019
The conference will take place at the University of Illinois at Chicago on May 30 - June 2. We will cover topics in set theory, descriptive set theory, model theory, and various applications. The workshop is aimed at graduate students and
settheory.mathtalks.org/logicf

Hossein Lamei Ramandi: $\Sigma^2_2$-absoluteness
Place: Fields Institute (Room 210)
Date: January 25 , 2018 (13:30-15:00)
Speaker: Hossein Lamei Ramandi
Title: $\Sigma^2_2$-absoluteness
Abstract:  We will show there is a $\Sigma^2_2$ sentence $\Phi$ such that both $\Phi$ and $\neg \Phi$ are consistent with $\diamondsuit$. This answers a question due to Woodin.

Benjamin Vejnar – Complexity of the homeomorphism relation between compact spaces
Talk held by Benjamin Vejnar (Charles University, Prague, Czech Republic) at the KGRC seminar on 2019-01-24. Abstract: We study the complexity of the homeomorphism relation of compact metrizable spaces when restricted to some subclasses such as continua, regular continua or regular compacta. [...]
settheory.mathtalks.org/benjam

Marcin Sabok: Measurable Hall’s theorem for actions of Z^n
Place: Fields Institute (Room 210) Date: January 18, 2018 (13:30-15:00) Speaker: Marcin Sabok Title: Measurable Hall's theorem for actions of Z^n Abstract: In the 1920's Tarski asked if it is possible to divide the unit square into finitely many pieces, rearrange them by translations and get a disc of area 1. It turns out that this is[...]
settheory.mathtalks.org/marcin

Moritz Müller – Forcing against bounded arithmetic
Talk held by Moritz Müller (Universitat Politècnica de Catalunya, Barcelona, Spain) at the KGRC seminar on 2019-01-17. Abstract: We study the following problem. Given a nonstandard model of arithmetic we want to expand it by a binary relation that does something prohibitive, e.g. violates the pigeonhole principle in the sense that it is the graph…[...]
settheory.mathtalks.org/moritz

Menachem Magidor: Omitting types in the logic of metric structures
HUJI Logic Seminar 16/Jan/2019, 11-13, Ross 63. Speaker: Menachem Magidor Title: Omitting types in the logic of metric structures Abstract. (joint work with I. Farah) The logic of metric structures was introduced by Ben Yaacov, Berenstein , Henson and Usvyatsov. It is a version of continuous logic which allows fruitful mode[...]
settheory.mathtalks.org/menach

Natasha Dobrinen – Mini-course on Infinitary Ramsey theory
Time and Place: Tuesday, January 8 and Wednesday, January 9  at 10:30am in the KGRC lecture room (both parts) at the KGRC. Part I.    Topological Ramsey spaces and applications to ultrafilters Part II.   Ramsey theory on trees and applications to big Ramsey degrees The Infinite Ramsey Theorem states that given $n,r\ge 1$ and a color[...]
settheory.mathtalks.org/natash

Natasha Dobrinen – Ramsey Theory of the Henson graphs
Abstract: A central question in the theory of ultrahomogeneous relational structures asks, How close of an analogue to the Infinite Ramsey Theorem does it carry? An infinite structure S is ultrahomogeneous if any isomorphism between two finitely generated substructures of S can be extended to an automorphism of S. We say that S has finite [...]
settheory.mathtalks.org/natash

Miguel Moreno: An introduction to generalized descriptive set theory, part 2
BIU Infinite Combinatorics Seminar Date : 31/12/2018 - 13:00 - 15:00 Speaker: Miguel Moreno (BIU) Title : An introduction to generalized descriptive set theory, part 2 Abstract. After introducing the notions of $\kappa$-Borel class, $\kappa$-$\Delta_1^1$ class, $\kappa$-Borel^* class in the previous talk,[...]
settheory.mathtalks.org/miguel

Ur Yaar: The Modal Logic of Forcing
HUJI Set Theory Seminar On Wednesday, December 26, Ur Yaar will talk about the modal logic of forcing. Title: The Modal Logic of Forcing   Abstract: Modal logic is used to study various modalities, i.e. various ways in which statements can be true, the most notable of which are the modalities of necessity and possibility. In set-theory,…[...]
settheory.mathtalks.org/ur-yaa

Matt Foreman: Games on weakly compact cardinals
TAU Forcing Seminar Tuesday, 25/12/18 Speaker: Matt Foreman Title: Games on weakly compact cardinals Abstract: Attached.[...]
settheory.mathtalks.org/matt-f

Assaf Rinot: Hindman’s theorem and uncountable Abelian groups
Colloquium, Hebrew University of Jerusalem Thu, 20/12/2018 - 14:30 to 15:30 Location: Manchester Building (Hall 2), Hebrew University Jerusalem Speaker: Assaf Rinot Title: Hindman’s theorem and uncountable Abelian groups Abstract. In the early 1970’s, Hindman proved a beautiful theorem in additive Ramsey theory asserting that [...]
settheory.mathtalks.org/assaf-

Daniel T. Soukup – New aspects of ladder system uniformization II
Talk held by Daniel Soukup (KGRC) at the KGRC seminar on 2018-12-13. Abstract: We continue the previous lecture and present proofs for some of the new results. We show that $\diamondsuit$ implies that for any Aronszajn-tree $T$, there is a ladder system with a 2-colouring with no $T$-uniformization. However, if $\diamondsuit^[...] settheory.mathtalks.org/daniel Asaf Karagila: On countable unions of countable sets BIU seminar in Set Theory December 17, 2018 Speaker: Asaf Karagila (UEA) Title: On countable unions of countable sets Abstract. How big can countable unions of countable sets be? Assuming the axiom of choice, countable. Not assuming the axiom of choice, it is not hard to arrange situation where there are many incomparable cardinals which…[...] settheory.mathtalks.org/asaf-k Ilijas Farah: On the model theory of C*-algebras HUJI Logic Seminar 12/December/18, 11 am, in Ross 63. Speaker: Ilijas Farah Title: On the model theory of C*-algebras Abstract. Ultrapowers and reduced products play a central role in the Elliott classification program for separable (nuclear, etc.) C*-algebras. Although an ultrapower of a separable C*-algebra A is quite different from the reduced product$\e[...]
settheory.mathtalks.org/ilijas

Clovis Hamel: Stability and Definability in Continuous Logics, Cp-theory and the Tsirelson space.
Place: Fields Institute Library Date: Decembver 7, 2018 (13:30-15:00) Speaker: Clovis Hamel Title: Stability and Definability in Continuous Logics, Cp-theory and the Tsirelson space. Abstract: An old question in Functional Analysis inquired whether there is a Ba[...]
settheory.mathtalks.org/clovis

Stevo Todorcevic – Ramsey degrees of topological spaces
Talk held by Stevo Todorcevic (University of Toronto, Canada) at the KGRC seminar on 2018-12-04. Abstract: This will be an overview of structural Ramsey theory when the objects are topological spaces. Open problems and directions for further research in this area will also be examined.[...]
settheory.mathtalks.org/stevo-

James Cummings: Regular cardinals and compactness
Mathematical logic seminar - Dec 4 2018 Time:     3:30pm - 4:30 pm Room:     Wean Hall 8220 Speaker:         James Cummings Department of Mathematical Sciences CMU Title:     Regular cardinals and compactness Abstract: This talk is a sequel of sorts to last week's talk on singular compactness, but is completely independent of it. I will [...]
settheory.mathtalks.org/james-


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# Fan Appreciation Night, Part 2

Key Terms:

Intersection
Union
Addition Rule

## Create and Analyze

The is known as finding the probability of the intersection of A and B, and may also be written as .  Let’s explore what finding means for mutually inclusive events.  Since mutually inclusive events can occur at the same time, the situation can be represented with a Venn diagram:

To find the probability of the intersection of A or B, , we cannot simply find the , because the two probabilities overlap with the section .  This section would be counted twice.

Since the , the union of A and B, is counted twice, you would need to subtract one of these sections to find the .  So, when events are mutually inclusive, you apply the Addition Rule,

Let’s explore how to find the probability of winning a t-shirt or a hat.


text:
doc-src/preface.tex
 changeset 356 2e6545875982 parent 329 92586a978648 child 5374 6ef3742b6153
equal inserted replaced
355:77150178beb2 356:2e6545875982
     1 \chapter*{Preface}
     1 \chapter*{Preface}
     2 \markboth{Preface}{Preface}   %or Preface ?
     2 \markboth{Preface}{Preface}   %or Preface ?
     3 \addcontentsline{toc}{chapter}{Preface}
     3 %%\addcontentsline{toc}{chapter}{Preface}
     4
     4
     5 Most theorem provers support a fixed logic, such as first-order or
     5 Most theorem provers support a fixed logic, such as first-order or
     6 equational logic.  They bring sophisticated proof procedures to bear upon
     6 equational logic.  They bring sophisticated proof procedures to bear upon
     7 the conjectured formula.  The resolution prover Otter~\cite{wos-bledsoe} is
     7 the conjectured formula.  The resolution prover Otter~\cite{wos-bledsoe} is
     8 an impressive example.
     8 an impressive example.
    46
    46
    47 The diagram below illustrates some of the logics distributed with Isabelle.
    47 The diagram below illustrates some of the logics distributed with Isabelle.
    48 These include first-order logic (intuitionistic and classical), the sequent
    48 These include first-order logic (intuitionistic and classical), the sequent
    49 calculus, higher-order logic, Zermelo-Fraenkel set theory~\cite{suppes72},
    49 calculus, higher-order logic, Zermelo-Fraenkel set theory~\cite{suppes72},
    50 a version of Constructive Type Theory~\cite{nordstrom90}, several modal
    50 a version of Constructive Type Theory~\cite{nordstrom90}, several modal
    51 logics, and a Logic for Computable Functions.  Several experimental logics
    51 logics, and a Logic for Computable Functions~\cite{paulson87}.  Several
    52 are being developed, such as linear logic.
    52 experimental logics are being developed, such as linear logic.
    53
    53
    54 \centerline{\epsfbox{Isa-logics.eps}}
    54 \centerline{\epsfbox{Isa-logics.eps}}
    55
    55
    56
    56
    57 \section*{How to read this book}
    57 \section*{How to read this book}
   112
   112
   113 \item Technical University of Munich\\
   113 \item Technical University of Munich\\
   114         host {\tt ftp.informatik.tu-muenchen.de}\\
   114         host {\tt ftp.informatik.tu-muenchen.de}\\
   115         directory {\tt local/lehrstuhl/nipkow}
   115         directory {\tt local/lehrstuhl/nipkow}
   116 \end{itemize}
   116 \end{itemize}
   117 My electronic mail address is {\tt lcp\at cl.cam.ac.uk}.  Please report any
   117 The electronic distribution list {\tt isabelle-users\at cl.cam.ac.uk}
   118 errors you find in this book and your problems or successes with Isabelle.
   118 provides a forum for discussing problems and applications involving
   119
   119 Isabelle.  To join, send me a message via {\tt lcp\at cl.cam.ac.uk}.

   120 Please notify me of any errors you find in this book.
   120
   121
   121 \section*{Acknowledgements}
   122 \section*{Acknowledgements}
   122 Tobias Nipkow has made immense contributions to Isabelle, including the
   123 Tobias Nipkow has made immense contributions to Isabelle, including the
   123 parser generator, type classes, the simplifier, and several object-logics.
   124 parser generator, type classes, the simplifier, and several object-logics.
   124 He also arranged for several of his students to help.  Carsten Clasohm
   125 He also arranged for several of his students to help.  Carsten Clasohm


text:
# The change in the vertical stress in the soil mass estimated by Boussinesq’s equation when Poisson’s ratio of soil changes from 0.3 to 0.5 will be

This question was previously asked in
Gujarat Engineering Service 2017 Official Paper (Civil Part 1)
View all GPSC Engineering Services Papers >
1. reduction by 30%
2. increase by 50%
3. reduction by 20%
4. no change

## Answer (Detailed Solution Below)

Option 4 : no change
Free
ST 1: Logical reasoning
5280
20 Questions 20 Marks 20 Mins

## Detailed Solution

Explanation

we know, Boussinesq’s equation for vertical stress

$${\sigma _z} = \frac{Q}{{{Z^2}}} \times \frac{3}{{2\pi }} \times {\left[ {\frac{1}{{1 + {{\left( {\frac{r}{Z}} \right)}^2}}}} \right]^{\frac{5}{2}}}$$

Where,

σ= vertical stress in kN/m2

Q = concentrated load in kN

Z = point below the ground level where stress has to be calculated

r = radial distance

when we need to calculate the vertical stress exactly below the concentrated point load then,

From the Equation, $${\sigma _z} = \frac{Q}{{{Z^2}}} \times \frac{3}{{2\pi }} \times {\left[ {\frac{1}{{1 + {{\left( {\frac{r}{Z}} \right)}^2}}}} \right]^{\frac{5}{2}}}$$ we can say that vertical stress is independent of Poisson’s ratio.

∴  There will be no change in the vertical stress.


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# Transistor Common Emitter Amplifier

### The common emitter amplifier configuration is possibly the most widely used form of transistor circuit providing voltage gain.

Transistor Circuit Design Tutorial Includes:
Transistor circuit design     Circuit configurations     Common emitter     Common emitter circuit design     Emitter follower     Common base

The common emitter transistor amplifier circuit is one of the mainstay circuits for use within electronic design.

The common emitter circuit configuration is used as a basic switch for logic circuits, as an analogue amplifier and in many other applications.

The common emitter circuit configuration provides voltage gain combined with a moderate current gain, as well as a medium input and a medium output impedance. As such the common emitter configuration is a good all round circuit for use in many applications.

## Common emitter transistor amplifier basics

The common emitter amplifier has the signal applied to the base and the output is then taken from the collector circuit. As the name implies the emitter circuit is common to both input and output.

The common emitter configuration is equally applicable to both NPN and PNP transistor variants. That said the NPN variety is more commonly used because of the more widespread us of NPN transistors.

## Common emitter transistor amplifier characteristics summary

The table below gives a summary of the major characteristics of the common emitter transistor configuration.

Common emitter transistor amplifier characteristics

Parameter Characteristics
Voltage gain Medium
Current gain Medium
Power gain High
Input / output phase relationship 180°
Input resistance Medium
Output resistance Medium

## Common emitter amplifier impedance levels

The input impedance is typically around 1kΩ, although this can vary considerably according to the circuit values and conditions. The low input impedance results from the fact that the input is applied across the base and emitter where there is a forward-biased junction,

Also the output impedance can be relatively high. Again this varies considerably upon the circuit values chosen and current levels permitted. This may be as high as 10kΩ or possibly more. However if current drain allows higher current levels to be drawn, the output impedance can be reduced considerably. The level of resistance or impedance comes from the fact that the output is taken from the collector where there is a reverse-biased junction.

## Common emitter transistor amplifier gain

Another important factor is the gain level that can be achieved. There are two forms of gain that can be determined: current gain and voltage gain.

The current gain for the common emitter amplifier circuit is denoted by the Greek symbol β. This is the ratio of collector current to base current. This may be thought of as the ratio of output current to input current. To gain an accurate figure of the gain for a signal, the current gain for small input changes in current is often used. Using this the current gain, β, and the changes in input and output current are related in the following way:

$\beta =\frac{\Delta {I}_{c}}{\Delta {I}_{b}}$

Where
β = current gain
ΔIc = change in collector current
ΔIb = change in base current

In order to look at the voltage gain of the common emitter amplifier circuit, it is necessary to look at the resistances or impedances for the input and output.

$\beta =\frac{\Delta {I}_{c}}{\Delta {I}_{b}}=\frac{\frac{\Delta {V}_{c}}{{R}_{c}}}{\frac{\Delta {V}_{b}}{{R}_{b}}}$

${A}_{v}=\frac{\Delta {V}_{c}}{\Delta {V}_{b}}$

Therefore:

Where
Av = voltage gain
Rc = collector circuit output resistance
Rb = base circuit input resistance

## Common emitter input output phase relationship

The common emitter transistor amplifier is the only configuration that gives an inversion, 180°, between the input and output signals.

The reason for this can be seen from the fact that as the input voltage rises, so the current increases through the base circuit. In turn this increases the current thought the collector circuit, i.e. it tends to turn the transistor on. This results in the voltage between the collector and emitter terminals falling.

In this way an increase in voltage between the base and emitter has resulted in a fall in voltage between the collector and emitter terminals, in other words the phase of the two signals has been inverted.

## Practical common emitter amplifier circuit

While the basic theoretical circuits shown above are able to describe the basic operation of the common emitter amplifier in concept.

However, for the circuit to be able to operate in a real system, other elements such as bias, decoupling and the like need to be added. As a result, the overall circuit for a common emitter amplifier utilises several components to ensure that it is able to operate in the fashion required.

Within the circuit there are a number of components that provide different functions to enable the overall circuit to operate in the fashion required:

 R1, R2 These resistors provide the bias for the base of the transistor. R3 This is the collector load resistor within the common emitter amplifier. R4 This resistor in the common emitter amplifier provides a measure of DC feedback to ensure that the DC conditions within the circuit are maintained. C1, C2 These capacitors provide AC coupling between stages. They need to be chosen to provide negligible reactance at the frequencies of operation. C3 This is a bypass capacitor. The effect of R4 is to reduce the gain of the circuit. Bypassing the resistor enables greater levels of AC gain to be achieved.

The circuit shown above is that if a basic AC coupled common emitter amplifier.

The common emitter circuit can be used in a variety of forms. - sometimes as a transistor logic output, a directly coupled amplifier and in many areas. It is widely used, providing a good compromise between voltage and current gain along with input and output impedance.

More Circuits & Circuit Design:
Op Amp basics     Op Amp circuits     Power supply circuits     Transistor design     Transistor Darlington     Transistor circuits     FET circuits     Circuit symbols


text:
Decentralized Publication and Consumption of Transfer Footpaths

• IDLab, Department of Electronics and Information Systems, Ghent University – imec

Abstract

Users expect route planners that combine all modes of transportation to propose good journeys to their destination. These route planners use data from several sources such as road networks and schedule-based public transit. We focus on the link between the two; specifically, the walking distances between stops. Research in this field so far has found that computing these paths dynamically is too slow, but that computing all of them results in a quadratically scaling number of edges which is prohibitively expensive in practice. The common solution is to cluster the stops into small unconnected graphs, but this restricts the amount of walking and has a significant impact on the travel times. Moreover, clustering operates on a closed-world assumption, which makes it impractical to add additional public transit services. A decentralized publishing strategy that fixes these issues should thus (i) scale gracefully with the number of stops; (ii) support unrestricted walking; (iii) make it easy to add new services and (iv) support splitting the work among several actors. We introduce a publishing strategy that is based on the Delaunay triangulation of public transit stops, where every triangle edge corresponds to a single footpath that is precomputed. This guarantees that all stops are reachable from one another, while the number of precomputed paths increases linearly with the number of stops. Each public transit service can be processed separately, and combining several operators’ can be done with a minimal amount of work. Approximating the walking distance with a path along the triangle edges overestimates the actual distance by 20% on average. Our results show that our approach is a middle-ground between completeness and practicality. It consistently overestimates the walking distances, but this seems workable since overestimating the time needed to catch a connection is arguably better than recommending an impossible journey. The estimates could still be improved by combining the great-circle distance with our approximations. Alternatively, different triangulations could be combined to create a more complete graph.

Introduction

Fueled by the need for more sustainable transportation, users can pick between more forms of transport than ever before – and they expect that route planners can combine these to find the ideal journey. Those route planners integrate various kinds of data such as road networks and schedule-based public transit. In this article, we focus on the link between the two, specifically on how to walk between stops. The walking time from one stop to another is commonly referred to as a footpath [1, 2].

Computing these paths dynamically is often too slow in practice [3, 4], yet computing them beforehand quickly becomes prohibitively expensive in practice [5, 1, 2]. A small country such as Belgium already contains 100,000 public transit stops, yielding 10 billion pairs of stops. These scaling issues are often avoided by only evaluating algorithms on small unconnected footpath graphs. However, this restricts the amount of walking in a journey and has a significant impact on travel times [4]. On top of that, determining which stops belong to the same graph ultimately relies on a closed-world assumption; which makes it impractical to add additional public transit services.

Our goal is to find a method to compute and publish these footpaths in a scalable and decentralized manner. This implies that we need not publish all the paths, but ideally the ones that are published should allow for unrestricted walking. In practice this means that every stop in the footpath graph must be reachable from every other stop if there is a path between those stops in the real world. Various actors should be able to add information to this footpath graph, and adding a new service should not require changes in existing information. These actors would need a common frame of reference to combine their efforts, which can be provided by Semantic Web technologies.

Method

We use a Delaunay triangulation of the public transit stop locations to approximate the full footpath graph, based on the insights from the related work section. The triangle edges represent which stops will be directly connected through a footpath. Our representation of a footpath differs slightly from the conventional one; we use walking distances instead of walking times to avoid making assumptions about walking speeds. A road network route planner is used to compute the walking distances between the stops.

Delaunay triangulations can be generalized to any metric space, but the walking distance is not a mathematical metric because it could, in theory, lack the required symmetry property. Even on foot, you cannot always retrace your steps – either due to legal restrictions or due to physical limitations. And, equally importantly, this would bring us back to the case where an impractical number of path lengths have to be precomputed. This is why we the triangulation uses the Euclidean distance between the stops’ WGS84 coordinates instead. Other metrics such as the great-circle distance can be used, but none of them have any relation to the walking distance so we opted for the most simple metric to implement. This also means that the theoretical guarantees of the Delaunay triangulation only hold for the Euclidean distance because that’s the metric was used for its construction.

The paths between stops of a single operator can be calculated independently from the rest. Combining the graphs of two operators is done by comparing their individual triangulations to the triangulation of their combined stops – and computing the missing footpaths. This process is illustrated in Fig. 1 where the graph of a city’s public transit network is merged with the one from the region around it. The combined graph is the union of each operators’ graph with the additional edges that connect the two graphs. Operators typically have their own service area, which means that the vast majority of paths will be between stops of the same operator, thus making it possible for anyone to combine several operators’ graphs with minimal work. The results can be shared and reused, which stands in stark contrast to the conventional approaches. The paths can be published similar to the Routable Tiles dataset so that sharing them does not come at a significant extra cost.

Results

We aim to verify two claims about our method: (i) the graph we construct is almost as good as the complete graph for route planning and (ii) footpath graphs can be merged with minimal effort. To verify the first claim we have used a graph of paths between bus stops to approximate the distances between railway stations. Edges between train stations and bus stops have also been added to make the railway stations reachable. Fig 2 shows that the approximation overestimates the actual distance by about 20% on average. This seems reasonable for route planners – it is better to underestimate the time needed to catch a connection than to recommend an impossible route.

Table 1: The number of paths that need to be computed to connect two operators.
Case Paths of first operator Paths of second operator Additional required paths
Flanders + Brussels 107,171 7,969 2,508
Flanders + Wallonia 107,171 94,730 4,020

To verify that combining the graphs from two operators does not require excessive work, we consider both the usual and the worst case scenario, respectively being mostly disjoint service areas and entirely overlapping service areas. We have taken the public transit operators from the Belgian regions of Flanders and Wallonia for the usual case and we revisit the case from Fig. 1 for the worst case, where the Brussels public transit network is added to the one from Flanders. The results in Table 1 show that in the usual scenario merging two operators is relatively straight-forward as most work has already been done. The worst case scenario shows that it is possible that merging two operators can still require a significant amount of work, although the vast majority has already been done. Our implementation contains some additional experiments and is available on https://github.com/hdelva/planner.js/tree/footpaths as a frozen branch of an experimental serverless route planner.

Conclusion and future work

This paper introduced a decentralized publishing method for transit footpaths based on Delaunay triangulations. We have shown that different actors can add information to the footpath graph, splitting the work between them. The amount of paths scales linearly with the amount of public transit stops, and every stop is reachable from every other. This last property could prove useful in its own right, as it allows for unrestricted walking unlike conventional methods. However, this comes at the cost of precision; the walking distance between stops is consistently slightly overestimated which could cause route planners to discard the most optimal journeys. Future work could focus on comparing the results from a route planner with a complete footpath graph to the one as we describe in this paper. If the difference in quality is negligible our methods could prove useful even for centralized services.

The approximations may still be improved with different kinds of point set triangulations and several triangulations could even be combined to create a more complete graph. Alternatively, combining the haversine distance and the currently approximated distance seems promising as well – although this could lead to underestimations. Another open question is how scalable this approach is when more modes of transportation are required. It seems undesirable to create separate graphs for bicycles, electric bikes, wheel chairs, etc. but they might be able to share information.