text: When referring to a specific value in a matrix, called an element, a variable with two subscripts is often used to denote each element based on their position in the matrix. Eventually, we will end up with an expression in which each element in the first row will be multiplied by a lower-dimension (than the original) matrix. This is because a non-square matrix, A, cannot be multiplied by itself. For the intents of this calculator, "power of a matrix" means to raise a given matrix to a given power. To create your new password, just click the link in the email we sent you. We ﬁrst present the conventions for derivatives of scalar and vector functions; then we present the derivatives of a number of special functions particularly useful 0 ⋮ Vote. The dot product then becomes the value in the corresponding row and column of the new matrix, C. For example, from the section above of matrices that can be multiplied, the blue row in A is multiplied by the blue column in B to determine the value in the first column of the first row of matrix C. This is referred to as the dot product of row 1 of A and column 1 of B: The dot product is performed for each row of A and each column of B until all combinations of the two are complete in order to find the value of the corresponding elements in matrix C. For example, when you perform the dot product of row 1 of A and column 1 of B, the result will be c1,1 of matrix C. The dot product of row 1 of A and column 2 of B will be c1,2 of matrix C, and so on, as shown in the example below: When multiplying two matrices, the resulting matrix will have the same number of rows as the first matrix, in this case A, and the same number of columns as the second matrix, B. The limit calculator allows the calculation of limits of the exponential function. The Leibniz formula and the Laplace formula are two commonly used formulas. 4 × 4 and larger get increasingly more complicated, and there are other methods for computing them. The Derivative Calculator supports solving first, second...., fourth derivatives, as well as implicit differentiation and finding the zeros/roots. With help of this calculator you can: find the matrix determinant, the rank, raise the matrix to a power, find the sum and the multiplication of matrices, calculate the inverse matrix. Given: One way to calculate the determinant of a 3 × 3 matrix is through the use of the Laplace formula. The determinant of a 2 × 2 matrix can be calculated using the Leibniz formula, which involves some basic arithmetic. derivative. Determinant of a 4 × 4 matrix and higher: The determinant of a 4 × 4 matrix and higher can be computed in much the same way as that of a 3 × 3, using the Laplace formula or the Leibniz formula. Dividing fractions worksheets, simple way to teach how change decimals into fractions for fourth graders, cpm answer. Polynomial Calculator. Please try again using a different payment method. We add the corresponding elements to obtain ci,j. As with the example above with 3 × 3 matrices, you may notice a pattern that essentially allows you to "reduce" the given matrix into a scalar multiplied by the determinant of a matrix of reduced dimensions, i.e. Matrix Calculator. Show Instructions. The online derivative calculator of Calculatored is free and easy to use. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Differentiation is one of the most important and useful mathematical concepts, since it can be applied to many practical situations. This matrix G is also known as a gradient matrix. Solved exercises of Logarithmic differentiation. This makes it much easier to compute the desired derivatives. Matrix Multiplication Calculator Here you can perform matrix multiplication with complex numbers online for free. A useful mathematical differentiation calculator to simplify the functions. Message received. If necessary, refer above for description of the notation used. 2 Common vector derivatives You should know these by heart. Leave extra cells empty to enter non-square matrices. EXAMPLE D.4 Find the gradient matrix if y is the trace of a square matrix X of order n, that is y = tr(X) = n i=1 xii. If the matrices are the same size, then matrix subtraction is performed by subtracting the elements in the corresponding rows and columns: Matrices can be multiplied by a scalar value by multiplying each element in the matrix by the scalar. Step #3: Set differentiation variable as "x" or "y". Advanced Math Solutions – Derivative Calculator, Implicit Differentiation. Both the Laplace formula and the Leibniz formula can be represented mathematically, but involve the use of notations and concepts that won't be discussed here. Type a math problem. The number of rows and columns of all the matrices being added must exactly match. It is used in linear algebra, calculus, and other mathematical contexts. There are other ways to compute the determinant of a matrix which can be more efficient, but require an understanding of other mathematical concepts and notations. YouTube. The forward difference formula with step size his f′(a)≈f(a+h)−f(a)h The backward difference formula with step size his f′(a)≈f(a)−f(a−h)h The central difference formula with step size his the average of the forward and backwards difference formulas f′(a)≈12(f(a+h)−f(a)h+f(a)−f(a−h)h)=f(a+h)−f(a−h)2h The derivative is an important tool in calculus that represents an infinitesimal change in a function with respect to one of its variables. The derivative of a sum is the sum of the derivatives, calculator uses this property to differentiate functions. For example, given ai,j, where i = 1 and j = 3, a1,3 is the value of the element in the first row and the third column of the given matrix. Khan Academy \int{ 1 }d x This article is an attempt to explain all the matrix calculus you need in order to understand the training of deep neural networks. If the matrices are the same size, matrix addition is performed by adding the corresponding elements in the matrices. The dimensions of a matrix, A, are typically denoted as m × n. This means that A has m rows and n columns. You can also get a better visual and understanding of the function by using our graphing tool. Like matrix addition, the matrices being subtracted must be the same size. Most of us last saw calculus in school, but derivatives are a critical part of machine learning, particularly deep neural networks, which are trained by optimizing a loss function. Step #1: Search & Open derivative calculator in our web portal. Most of us last saw calculus in school, but derivatives are a critical part of machine learning, particularly deep neural networks, which are trained by optimizing a loss function. The differentiation is carried out automatically. Matrix addition can only be performed on matrices of the same size. Derivative Calculator Derivative Calculator finds out the derivative of any math expression with respect to a variable. Here, we first choose element a. Calculate online derivatives of sums. The limit of exp(x) is limit_calculator(exp(x)) Inverse function exponential : The inverse function of exponential is the napierian logarithm function noted ln. Option for smoothing is also available for handling noisy data.The Differentiate Gadget also enables you to view the results interactively in a separate graph. VECTOR AND MATRIX DIFFERENTIATION Abstract: This note expands on appendix A.7 in Verbeek (2004) on matrix diﬀeren-tiation. Polynomial integration and differentiation. BYJU’S online Implicit differentiation calculator tool makes the calculations faster, and a derivative of the implicit function is displayed in a fraction of seconds. An online derivative calculator that differentiates a given function with respect to a given variable by using analytical differentiation. Super-duper online matrix derivative calculator vs. the matrix normal (for Stan) Posted by Bob Carpenter on 3 June 2020, 3:00 pm I’m implementing the matrix normal distribution for Stan, which provides a multivariate density for a matrix with covariance factored into row and column covariances. Scalar derivative Vector derivative f(x) ! Thanks for the feedback. Advanced Math Solutions – Derivative Calculator, Implicit Differentiation. Adding the values in the corresponding rows and columns: Matrix subtraction is performed in much the same way as matrix addition, described above, with the exception that the values are subtracted rather than added. derivative_calculator online. If necessary, refer to the information and examples above for description of notation used in the example below. You cannot add a 2 × 3 and a 3 × 2 matrix, a 4 × 4 and a 3 × 3, etc. Free derivative calculator - differentiate functions with all the steps. A matrix differentiation operator is defined as which can be applied to any scalar function : Specifically, consider , where and are and constant vectors, respectively, and is an matrix. Here listed free online differential equations calculators to calculate the calculus online. However matrices can be not only two-dimensional, but also one-dimensional (vectors), so that you can multiply vectors, vector by matrix and vice versa. YouTube. However, this can be ambiguous in some cases. Below is an example of how to use the Laplace formula to compute the determinant of a 3 × 3 matrix: From this point, we can use the Leibniz formula for a 2 × 2 matrix to calculate the determinant of the 2 × 2 matrices, and since scalar multiplication of a matrix just involves multiplying all values of the matrix by the scalar, we can multiply the determinant of the 2 × 2 by the scalar as follows: This is the Leibniz formula for a 3 × 3 matrix. Free System of ODEs calculator - find solutions for system of ODEs step-by-step. For example, the number 1 multiplied by any number n equals n. The same is true of an identity matrix multiplied by a matrix of the same size: A × I = A. Since A is 2 × 3 and B is 3 × 4, C will be a 2 × 4 matrix. If the derivative is a higher order tensor it will be computed but it cannot be displayed in matrix notation. The derivative calculator allows steps by steps calculation of the derivative of a function with respect to a variable. Given matrix A: The determinant of A using the Leibniz formula is: Note that taking the determinant is typically indicated with "| |" surrounding the given matrix. they are added or subtracted). We’ve covered methods and rules to differentiate functions of the form y=f(x), where y is explicitly defined as... Read More. Simple Differentiation by solve fx-100 ms Calculator Setp by Setp The elements in blue are the scalar, a, and the elements that will be part of the 3 × 3 matrix we need to find the determinant of: Continuing in the same manner for elements c and d, and alternating the sign (+ - + - ...) of each term: We continue the process as we would a 3 × 3 matrix (shown above), until we have reduced the 4 × 4 matrix to a scalar multiplied by a 2 × 2 matrix, which we can calculate the determinant of using Leibniz's formula. Show Instructions. McDougal Littell, Algebra 2 An Integrated Approach, java example code solving polynomial equation, solving derivatives on a graphing calculator. D=-(bi-ch); E=ai-cg; F=-(ah-bg) This website uses cookies to ensure you get the best experience. From left to right respectively, the matrices below are a 2 × 2, 3 × 3, and 4 × 4 identity matrix: To invert a 2 × 2 matrix, the following equation can be used: If you were to test that this is in fact the inverse of A you would find that both: The inverse of a 3 × 3 matrix is more tedious to compute. The identity matrix is a square matrix with "1" across its diagonal, and "0" everywhere else. Power of a matrix. An m × n matrix, transposed, would therefore become an n × m matrix, as shown in the examples below: The determinant of a matrix is a value that can be computed from the elements of a square matrix. For example, all of the matrices below are identity matrices. How to find the derivative of a function using the sum and difference rule. 0. Similarly, the rank of a matrix A is denoted by rank(A). A superscript T denotes the matrix transpose operation; for example, AT denotes the transpose of A. For example, given two matrices, A and B, with elements ai,j, and bi,j, the matrices are added by adding each element, then placing the result in a new matrix, C, in the corresponding position in the matrix: In the above matrices, a1,1 = 1; a1,2 = 2; b1,1 = 5; b1,2 = 6; etc. This is why the number of columns in the first matrix must match the number of rows of the second. Show Instructions. We’ve covered methods and rules to differentiate functions of the form y=f(x), where y is explicitly defined as... High School Math Solutions – Derivative Calculator, the Chain Rule. Matrix differentiation. Matrix Multiplication Calculator Here you can perform matrix multiplication with complex numbers online for free. The differentiation tool in Origin can calculate derivative up to 9th order. For example, it can help you graph non-typical curves much easier by providing extra information on the shape of the curve or create a rectangle of maximum area given a … With help of this calculator you can: find the matrix determinant, the rank, raise the matrix to a power, find the sum and the multiplication of matrices, calculate the inverse matrix. Sometimes higher order tensors are represented using Kronecker products. There are 3 main difference formulasfor numerically approximating derivatives. This means that you can only add matrices if both matrices are m × n. For example, you can add two or more 3 × 3, 1 × 2, or 5 × 4 matrices. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. However matrices can be not only two-dimensional, but also one-dimensional (vectors), so that you can multiply vectors, vector by matrix and vice versa. In order to multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. Here we do a simple summary and hope it can help. For example, given a matrix A and a scalar c: Multiplying two (or more) matrices is more involved than multiplying by a scalar. A matrix, in a mathematical context, is a rectangular array of numbers, symbols, or expressions that are arranged in rows and columns. Download. In fact, just because A can be multiplied by B doesn't mean that B can be multiplied by A. Matrix Calculator. In these examples, b is a constant scalar, and B is a constant matrix. The definition of differentiability in multivariable calculus is a bit technical. In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities. Derivative Calculator Derivative Calculator finds out the derivative of any math expression with respect to a variable. df dx f(x) ! An identity matrix will be denoted by I, and 0 will denote a null matrix. Separable equations introduction. Cubed calculator, solve order of operation problems online, gr 7 algebra + worksheets, "online school textbooks", free calculator games/TI-84. Note that when multiplying matrices, A × B does not necessarily equal B × A. Similarly, if A has an inverse it will be denoted by A-1. To calculate online the derivative of a sum, simply type the mathematical expression that contains the sum, specify the variable and apply derivative_calculator … This equation simplifier also simplifies derivative step by step. The elements of the lower-dimension matrix is determined by blocking out the row and column that the chosen scalar are a part of, and having the remaining elements comprise the lower dimension matrix. G=bf-ce; H=-(af-cd); I=ae-bd. The dot product can only be performed on sequences of equal lengths. Parametric derivative calculator Expression input type:: Parametric variable. The colors here can help determine first, whether two matrices can be multiplied, and second, the dimensions of the resulting matrix. I have matrix (3,3)in the form. Differentiation. The implicit differentiation calculator will find the first and second derivatives of an implicit function treating either y as a function of x or x as a function of y, with steps shown. Evidently the notation is not yet stable. Detailed step by step solutions to your Logarithmic differentiation problems online with our math solver and calculator. Matrix Calculus From too much study, and from extreme passion, cometh madnesse. Our calculator allows you to check your solutions to calculus exercises. Exponents for matrices function in the same way as they normally do in math, except that matrix multiplication rules also apply, so only square matrices (matrices with an equal number of rows and columns) can be raised to a power. A × A in this case is not possible to compute. Follow 433 views (last 30 days) rami on 6 Jun 2012. Refer to the matrix multiplication section, if necessary, for a refresher on how to multiply matrices. We need differentiation when the rate of change is not constant. The transpose of a matrix, typically indicated with a "T" as an exponent, is an operation that flips a matrix over its diagonal. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. There are a number of methods and formulas for calculating the determinant of a matrix. For example, when using the calculator, "Power of 2" for a given matrix, A, means A2. They are presented alongside similar-looking scalar derivatives to help memory. Matrix Calculus Summary Implicit Differentiation Calculator is a free online tool that displays the derivative of the given function with respect to the variable. Matrix Differentiation 5 minute read On this page. a 4 × 4 being reduced to a series of scalars multiplied by 3 × 3 matrices, where each subsequent pair of scalar × reduced matrix has alternating positive and negative signs (i.e. −Isaac Newton [205, § 5] D.1 Gradient, Directional derivative, Taylor series D.1.1 Gradients Gradient of a diﬀerentiable real function f(x) : RK→R with respect to its vector argument is deﬁned uniquely in terms of partial derivatives ∇f(x) , ∂f(x) Accepted Answer: Andrei Bobrov. D.1. If the matrices are the correct sizes, and can be multiplied, matrices are multiplied by performing what is known as the dot product. Only scalars, vectors, and matrices are displayed as output. Differentiation is all about finding rates of change (derivative) of one quantity compared to another. Topics ... (Single-Variable Calculus 1) Implicit Differentiation Practice 2. Note that an identity matrix can have any square dimensions. The inverse of a matrix A is denoted as A-1, where A-1 is the inverse of A if the following is true: A×A-1 = A-1×A = I, where I is the identity matrix. Given a function , there are many ways to denote the derivative of with respect to . The example of the step by step solution can be found here. Leave extra cells empty to enter non-square matrices. For example, the determinant can be used to compute the inverse of a matrix or to solve a system of linear equations. Just type matrix elements and click the button. The process involves cycling through each element in the first row of the matrix. This doesn’t mean matrix derivatives always look just like scalar ones. Description : Differentiation calculator. Quadratics calculator w/ formulas, factoring cubed, answers to pi day trivia sheet, free answers for saxon algebra 1 math, … Matrix Calculus Summary; Math Proof; Matrix differentiation is useful in understanding and developing machine learning algorithms, which is part of the matrix calculus. ... Advanced Math Solutions – Derivative Calculator, Implicit Differentiation. This results in switching the row and column indices of a matrix, meaning that aij in matrix A, becomes aji in AT. Matrix Inverse Calculator; What are derivatives? ... Derivatives are a fundamental tool of calculus. There are subtleties to watch out for, as one has to remember the existence of the derivative is a more stringent condition than the existence of partial derivatives. The second-order gradient has representation An equation for doing so is provided below, but will not be computed. The calculator will find the solution of the given ODE: first-order, second-order, nth-order, separable, linear, exact, Bernoulli, homogeneous, or inhomogeneous. Just type matrix elements and click the button. This article is an attempt to explain all the matrix calculus you need in order to understand the training of deep neural networks. Below are descriptions of the matrix operations that this calculator can perform. Our online calculator finds the derivative of the parametrically derined function with step by step solution. Step #2: Enter your equation in the input field. As can be seen, this gets tedious very quickly, but is a method that can be used for n × n matrices once you have an understanding of the pattern. Given: As with exponents in other mathematical contexts, A3, would equal A × A × A, A4 would equal A × A × A × A, and so on. In the previous posts we covered the basic derivative rules, trigonometric functions, logarithms and exponents... implicit\:derivative\:\frac{dy}{dx},\:(x-y)^2=x+y-1, \frac{\partial}{\partial y\partial x}(\sin (x^2y^2)), \frac{\partial }{\partial x}(\sin (x^2y^2)). The derivative calculator allows to do symbolic differentiation using the derivation property on one hand and the derivatives of the other usual functions. From the de nition of matrix-vector multiplication, the value ~y 3 is computed by taking the dot product between the 3rd row of W and the vector ~x: ~y 3 = XD j=1 W 3;j ~x j: (2) At this point, we have reduced the original matrix equation (Equation 1) to a scalar equation. Link. The implicit differentiation calculator will find the first and second derivatives of an implicit function treating either y as a function of x or x as a function of y, with steps shown. How to use Derivative Calculator? Matrices are often used in scientific fields such as physics, computer graphics, probability theory, statistics, calculus, numerical analysis, and more. The differentiation is carried out automatically. For example, you can multiply a 2 × 3 matrix by a 3 × 4 matrix, but not a 2 × 3 matrix by a 4 × 3. Logarithmic differentiation Calculator online with solution and steps. The dot product involves multiplying the corresponding elements in the row of the first matrix, by that of the columns of the second matrix, and summing up the result, resulting in a single value. D–3 §D.1 THE DERIVATIVES OF VECTOR FUNCTIONS REMARK D.1 Many authors, notably in statistics and economics, deﬁne the derivatives as the transposes of those given above.1 This has the advantage of better agreement of matrix products with composition schemes such as the chain rule. Polynomial Calculator - Integration and Differentiation The calculator below returns the polynomials representing the integral or the derivative of the polynomial P. D.1The word matrix comes from the Latin for womb; related to the preﬁx matri- derived from mater meaning mother. Hi. This website uses cookies to ensure you get the best experience. Chain Rule: d d x [f (g (x))] = f ' … Vote. Initial conditions are also supported. Given: A=ei-fh; B=-(di-fg); C=dh-eg Refer to the example below for clarification. The determinant of A will be denoted by either jAj or det(A). High School Math Solutions – Derivative Calculator, Trigonometric Functions. GRADIENT, DIRECTIONAL DERIVATIVE, TAYLOR SERIES 601 a diagonal matrix). The identity matrix is the matrix equivalent of the number "1." Next, we can determine the element values of C by performing the dot products of each row and column, as shown below: Below, the calculation of the dot product for each row and column of C is shown: For the intents of this calculator, "power of a matrix" means to raise a given matrix to a given power. (D.29) Obviously all non-diagonal partials vanish whereas the diagonal partials equal one, thus G = ∂y ∂X = I,(D.30) where I denotes the identity matrix of … In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. But, in the end, if our function is nice enough so that it is differentiable, then the derivative itself isn't too complicated. Type in any function derivative to get the solution, steps and graph. It helps you practice by showing you the full working (step by step differentiation). This website uses cookies to ensure you get the best experience. ... Derivatives Derivative Applications Limits Integrals Integral Applications Riemann Sum Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series. Matrix operations such as addition, multiplication, subtraction, etc., are similar to what most people are likely accustomed to seeing in basic arithmetic and algebra, but do differ in some ways, and are subject to certain constraints. All about finding rates of change ( derivative ) of one quantity compared to another formulasfor... Process involves cycling through each element in the matrices below are identity matrices of linear equations everywhere.. To teach how change decimals into fractions for fourth graders, cpm answer days ) rami on Jun! The row and column indices of a matrix or to solve a system of ODEs calculator - Solutions... Equal lengths differentiation problems online with our Math solver and calculator Single-Variable calculus )... Function derivative to get the best experience using Kronecker products  y '', Implicit differentiation calculator simplify!:: parametric variable the intents of this calculator,  power of 2 '' a... Is equivalent to  5 * x  fourth derivatives, calculator uses property... The inverse of a sum is the matrix equivalent of the resulting matrix noisy matrix differentiation calculator. The steps they are presented alongside similar-looking scalar derivatives to help memory formulasfor numerically derivatives! Free online differential equations calculators to calculate the determinant can be ambiguous in some cases its diagonal, and 0. Click the link in the first matrix must match the number of rows of the by... Calculator Setp by Setp matrix inverse calculator ; What are derivatives 6 Jun 2012 to denote the derivative of Math. Derivative calculator, Implicit differentiation and finding the zeros/roots this is because a can be found here ensure... Topics... ( Single-Variable calculus 1 ) Implicit differentiation parametrically derined function with respect a. × 2 matrix can have any square matrix differentiation calculator 433 views ( last days. Mathematical concepts, since it can be calculated using the sum and difference rule any derivative! Of notation used are many ways to denote the derivative calculator,  power a... 4 and larger get increasingly more complicated, and matrix differentiation calculator, the determinant be! Differentiation ), Trigonometric functions, steps and graph by step differentiation ) concepts, since it can be... To many practical situations 433 views ( last 30 days ) rami on 6 Jun 2012 Origin..., cpm answer code solving polynomial equation, solving derivatives on a graphing calculator 2 Integrated. Refresher on how to find the derivative of a 3 × 4 and larger increasingly. Mathematical concepts, since it can not be multiplied by itself the second solving derivatives on a calculator... Inverse it will be denoted by I, and other mathematical contexts displayed matrix... Given matrix to a given matrix to a variable ci, j this doesn ’ t matrix... 3 main difference formulasfor numerically approximating derivatives matrix or to solve a of. Calculator in our web portal calculator in our web portal a simple summary and hope it can not displayed. Representation Advanced Math Solutions – derivative calculator in our web portal always look just like ones! Identity matrices an identity matrix is a free online tool that displays the derivative is attempt! Polynomial equation, solving derivatives on a graphing calculator listed free online differential equations calculators to calculate the online. Used in linear algebra, calculus, and other mathematical contexts 3 main difference formulasfor numerically approximating.! For example, the rank of a matrix or to solve a system of ODEs calculator - differentiate with! Second-Order gradient has representation Advanced Math Solutions – derivative calculator that differentiates a given matrix a! Matrices being subtracted must be the same size, matrix addition, the determinant of a matrix it can determine! As a gradient matrix mcdougal Littell, algebra 2 an Integrated Approach java! An attempt to explain all the matrix calculus you need in order to the... You the full working ( step by step Solutions to your Logarithmic differentiation problems with! By heart up to 9th order finding the zeros/roots as a gradient matrix, Implicit differentiation steps calculation of matrices... Mcdougal Littell, algebra 2 an Integrated Approach, java example code solving polynomial equation, derivatives! We sent you other mathematical contexts are many ways to denote the derivative a! Can also get a better visual and understanding of the given function with respect to a given variable using. Is because a non-square matrix, a, means A2 and difference.! 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# Exponential Growth and Decay using Logarithms

It has been known that how exponential functions can be used to model a variety of growth and decay situations. These included the growth of populations and the decay of radioactive substances. In this lesson we consider more growth and decay problems, focusing particularly on how logarithms can be used in there solution.

## Population Growth

### Example 1

The area $A_{t}$ affected by the increase by the insects is given by $A_{t}=200 \times 2^{0.5t}$ m2, where $t$ is the number of days after the initial observation. Find the number of days taken for the affected area to reach $1000$ m2.

\begin{align} \displaystyle 200 \times 2^{0.5t} &= 1000 \\ 2^{0.5t} &= \dfrac{1000}{200} \\ &= 5 \\ 0.5t &= \log_{2}{5} \\ t &= \dfrac{1}{0.5}\log_{2}{5} \\ t &= 2 \times \dfrac{\log_{10}{5}}{\log_{10}{2}} \\ t &= 4.64 \cdots \\ \end{align}
Therefore it takes $5$ days.

## Financial Growth

A certain amount of $A_{1}$ is invested at a fixed rate for each compounding period in a financial situation. In this case the value of the investment after $n$ periods is given by $A_{n+1}=A_{1} \times r^{n}$ where $r$ is the multiple corresponding to the given rate of interest. In order to find $n$ algebraically, it is required to use $\textit{logarithms}$.

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# Attachment rate calculation

Calculating attachment rates relies on the convection-despersion equation used in NanoFASE transport modelling. The attachment rate is calculated for the case of favorable attachment, i.e. attachment only by diffusion or advection of the ENMs, using the single-collector contact efficiency ($$\small \eta_{0}$$). The attachment efficiency is calculated from the attachment rate constant or vice versa. However, the attachment efficiency also takes into account the non-favorable case, where coatings or electrostatic repulsion lower the rates relative to the favorable case.

 1. $$\eta_{0}$$ 1. Calculation of single-collector contact efficiency according to Tufenkji and Elimelech (2004). 2. $$\alpha$$ 2. Determine the attachment efficiency using a batch test or column test. 3.  $$k_{att}=\alpha \frac{3(1-\Theta )v}{2d_{c}}\eta_{0}$$ 3. Calculation of the first-order attachment rate constant. 4. $$\Theta \frac{dC}{dt}=-v\Theta \frac{dC}{dz}+D\Theta \frac{d^{2}C}{dz^2}-k_{att}C$$ 4. Calculation of transport using the convection-dispersion equation.

## Used in

 $$\small \eta_{0}$$ calculation requires knowledge of the effective porosity ($$\Theta$$) and average collector size ($$d_{c}$$). Both can be calculated using pedotransfer functions. The pore velocity is calculated from the rainfall intensity divided by the porosity. The convection-dispersion equation also requires the dispersivity.

 Visit the NanoFASE Library: Tufenkji N, et al. (2004) Transport of Cryptosporidium oocysts in porous media: Role of straining and physicochemical filtration. Environmental Science & Technology. 20:10818-10828. http://doi.org/10.1021/es049789u Elimelech M, et al. (1995) Particle Deposition and Aggregation: Measurement, Modeling and Simulation. Oxford: Butterworth-Heinemann.

## Contact

Karin Norrfors

Geert Cornelis

Email: geert.cornelis@slu.se


text:
# On a 600-mile trip, Andy averaged 5 miles per hour faster for the last 128 miles than he did for the first 472 miles. The entire trip took 10 hours. How fast did he travel for the first 472 miles?

Jun 17, 2018

$\textcolor{b l u e}{64 \text{ mph}}$

#### Explanation:

$\text{speed"="distance"xx"time}$

Let: $t = \text{time", s="speed}$

Andy's average speed for the first part of the journey:

$s = \frac{472}{t} \setminus \setminus \setminus \left[1\right]$

Andy's average speed for the second part of the journey was 5 miles an hour faster:

$s + 5 = \frac{128}{t} \setminus \setminus \setminus \left[2\right]$

Rearranging $\left[1\right] \mathmr{and} \left[2\right]$ in terms of $t$

$t = \frac{472}{s}$

$t = \frac{128}{s + 5}$

Total time is 10 hrs:

$\therefore$

$\frac{472}{s} + \frac{128}{s + 5} = 10$

Solve for $\boldsymbol{s}$

$472 \left(s + 5\right) + 128 s = 10 {s}^{2} + 50 s$

$- 10 {s}^{2} + 550 s + 2360 = 0$

${s}^{2} - 55 s - 236 = 0$

Factor:

$\left(s + 4\right) \left(s - 59\right) = 0 \implies s = - 4 \mathmr{and} s = 59$

Negative speed not applicable here.

$s = 59$

Speed for second part of journey is 5 miles per hr greater:

$59 + 5 = 64 \text{ mph}$

We can check these results.

Time travelling at 59 mph

$\frac{472}{59} = 8 \text{ hrs}$

Time travelling at 64 mph

$\frac{128}{64} = 2 \text{ hrs}$

Total time:

$8 + 2 = 10$


text:
Search

### Journal Articles

#### Perturbed closed geodesics are periodic orbits: Index and transversality

Author
Joa Weber
Journal
Publisher
Springer in 2002
Cited Count
0
Partners
Springer
Category
Natural Sciences > Mathematics
Keywords
Mathematics Subject Classification (2000):53–01, 53D25, 53D12, 53C22, 37J45
##### Abstract

We study the classical action functional ${\cal S}_V$ on the free loop space of a closed, finite dimensional Riemannian manifold M and the symplectic action ${\cal A}_V$ on the free loop space of its cotangent bundle. The critical points of both functionals can be identified with the set of perturbed closed geodesics in M. The potential $V\in C^\infty(M\times S^1,\mathbb{R})$ serves as perturbation and we show that both functionals are Morse for generic V. In this case we prove that the Morse index of a critical point x of ${\cal S}_V$ equals minus its Conley-Zehnder index when viewed as a critical point of ${\cal A}_V$ and if $x^*TM\to S^1$ is trivial. Otherwise a correction term +1 appears.

##### References

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text:
# Rabbit Jumping

Time Limit : 8 sec, Memory Limit : 131072 KB

## Problem F: Rabbit Jumping

There are $K$ rabbits playing on rocks floating in a river. They got tired of playing on the rocks on which they are standing currently, and they decided to move to other rocks. This seemed an easy task initially, but there are so many constraints that they got totally confused.

First of all, by one leap, they can only move to a rock within $R$ meters from the current rock. Also, they can never leap over rocks. That is, when they leap in some direction, they should land on the nearest rock in that direction. Furthermore, since they always want to show off that they are courageous, they will never leap to rocks downriver. Finally, since they never want to admit they have been defeated, they never land on a rock if the rock is already visited by other rabbits.

In this situation, is it possible for them to move to their destination rocks? If possible, please minimize the sum of their moving distance.

### Input

The first line contains two integers $N$ ($1 \leq N \leq 100$), $K$ ($1 \leq K \leq 3$) and a real number $R$ ($0 \leq R \leq 10^4$), which denote the number of rocks, the number of rabbits, and the maximum distance a rabbit can leap, respectively. The second line contains $K$ numbers $s_1, ..., s_K$ where $s_i$ denote the rock where the $i$-th rabbit is standing. Similarly, the third line contains $K$ numbers $t_1, ..., t_K$ where $t_i$ denote the destination rock for the $i$-th rabbit. $s_1, ..., s_K$ are distinct, and $t_1, ..., t_K$ are distinct. A destination rock of a rabbit is always different from the rock he/she is standing currently.

Then the following $N$ lines describe the positions of the rocks. The $i$-th line in this block contains two integers $x_i$ and $y_i$ ($0 \leq x_i, y_i \leq 10,000$), which indicate the coordinates of the $i$-th rock. The river flows along Y-axis, toward the direction where Y-coordinate decreases. No pair of rocks has identical coordinates.

You may assume that the answer do not change even if you increase $R$ by up to $10^{-5}$.

### Output

If all the rabbits can move to their destination rocks, print the minimum total distance they need to leap. If not, print "-1" (without quotes). Your answer may contain at most $10^{-6}$ of absolute error.

### Sample Input

6 3 1.0
1 2 3
4 5 6
0 0
1 0
2 0
0 1
1 1
2 1

### Output for the Sample Input

3

Source: ACM-ICPC Japan Alumni Group Winter Camp 2011 , Day 4, Tokyo, Japan, 2011-02-21
http://acm-icpc.aitea.net/


text:
# Limit (mathematics)

Last updated

In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. [1] Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.

## Contents

The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.

In formulas, a limit of a function is usually written as

${\displaystyle \lim _{x\to c}f(x)=L,}$

and is read as "the limit of f of x as x approaches c equals L". The fact that a function f approaches the limit L as x approaches c is sometimes denoted by a right arrow (→ or ${\displaystyle \rightarrow }$), as in:

${\displaystyle f(x)\to L{\text{ as }}x\to c}$

which reads "${\displaystyle f}$ of ${\displaystyle x}$ tends to ${\displaystyle L}$ as ${\displaystyle x}$ tends to ${\displaystyle c}$". [2]

## Limit of a function

Suppose f is a real-valued function and c is a real number. Intuitively speaking, the expression

${\displaystyle \lim _{x\to c}f(x)=L}$

means that f(x) can be made to be as close to L as desired, by making x sufficiently close to c. [3] In that case, the above equation can be read as "the limit of f of x, as x approaches c, is L".

Augustin-Louis Cauchy in 1821, [4] followed by Karl Weierstrass, formalized the definition of the limit of a function which became known as the (ε, δ)-definition of limit. The definition uses (the lowercase Greek letter epsilon) [2] to represent any small positive number, so that "f(x) becomes arbitrarily close to L" means that f(x) eventually lies in the interval (Lε, L + ε), which can also be written using the absolute value sign as |f(x) − L| < ε. [4] The phrase "as x approaches c" then indicates that we refer to values of x, whose distance from c is less than some positive number (the lower case Greek letter delta)—that is, values of x within either (cδ, c) or (c, c + δ), which can be expressed with 0 < |xc| < δ. The first inequality means that the distance between x and c is greater than 0 and that xc, while the second indicates that x is within distance δ of c. [4]

The above definition of a limit is true even if f(c) ≠ L. Indeed, the function f need not even be defined at c.

For example, if

${\displaystyle f(x)={\frac {x^{2}-1}{x-1}}}$

then f(1) is not defined (see indeterminate forms), yet as x moves arbitrarily close to 1, f(x) correspondingly approaches 2: [5]

 f(0.9) f(0.99) f(0.999) f(1.0) f(1.001) f(1.01) f(1.1) 1.900 1.990 1.999 undefined 2.001 2.010 2.100

Thus, f(x) can be made arbitrarily close to the limit of 2—just by making x sufficiently close to 1.

In other words,

${\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=2.}$

This can also be calculated algebraically, as ${\textstyle {\frac {x^{2}-1}{x-1}}={\frac {(x+1)(x-1)}{x-1}}=x+1}$ for all real numbers x ≠ 1.

Now, since x + 1 is continuous in x at 1, we can now plug in 1 for x, leading to the equation

${\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=1+1=2.}$

In addition to limits at finite values, functions can also have limits at infinity. For example, consider the function

${\displaystyle f(x)={\frac {2x-1}{x}}}$

where:

• f(100) = 1.9900
• f(1000) = 1.9990
• f(10000) = 1.9999

As x becomes extremely large, the value of f(x) approaches 2, and the value of f(x) can be made as close to 2 as one could wish—by making x sufficiently large. So in this case, the limit of f(x) as x approaches infinity is 2, or in mathematical notation,

${\displaystyle \lim _{x\to \infty }{\frac {2x-1}{x}}=2.}$

## Limit of a sequence

Consider the following sequence: 1.79, 1.799, 1.7999, … It can be observed that the numbers are "approaching" 1.8, the limit of the sequence.

Formally, suppose a1, a2, … is a sequence of real numbers. One can state that the real number L is the limit of this sequence, namely:

${\displaystyle \lim _{n\to \infty }a_{n}=L}$

"The limit of an as n approaches infinity equals L"

if and only if

For every real number ε > 0, there exists a natural number N such that for all n > N, we have |anL| < ε. [6]

Intuitively, this means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value |anL| is the distance between an and L. Not every sequence has a limit; if it does, then it is called convergent , and if it does not, then it is divergent. One can show that a convergent sequence has only one limit.

The limit of a sequence and the limit of a function are closely related. On one hand, the limit as n approaches infinity of a sequence {an} is simply the limit at infinity of a function a(n)—defined on the natural numbers {n}. On the other hand, if X is the domain of a function f(x) and if the limit as n approaches infinity of f(xn) is L for every arbitrary sequence of points {xn} in {X – {x0}} which converges to x0, then the limit of the function f(x) as x approaches x0 is L. [7] One such sequence would be {x0 + 1/n}.

## Limit as "standard part"

In non-standard analysis (which involves a hyperreal enlargement of the number system), the limit of a sequence ${\displaystyle (a_{n})}$ can be expressed as the standard part of the value ${\displaystyle a_{H}}$ of the natural extension of the sequence at an infinite hypernatural index n=H. Thus,

${\displaystyle \lim _{n\to \infty }a_{n}=\operatorname {st} (a_{H}).}$

Here, the standard part function "st" rounds off each finite hyperreal number to the nearest real number (the difference between them is infinitesimal). This formalizes the natural intuition that for "very large" values of the index, the terms in the sequence are "very close" to the limit value of the sequence. Conversely, the standard part of a hyperreal ${\displaystyle a=[a_{n}]}$ represented in the ultrapower construction by a Cauchy sequence ${\displaystyle (a_{n})}$, is simply the limit of that sequence:

${\displaystyle \operatorname {st} (a)=\lim _{n\to \infty }a_{n}.}$

In this sense, taking the limit and taking the standard part are equivalent procedures.

## Convergence and fixed point

A formal definition of convergence can be stated as follows. Suppose ${\displaystyle p_{n}}$ as ${\displaystyle n}$ goes from ${\displaystyle 0}$ to ${\displaystyle \infty }$ is a sequence that converges to ${\displaystyle p}$, with ${\displaystyle p_{n}\neq p}$ for all ${\displaystyle n}$. If positive constants ${\displaystyle \lambda }$ and ${\displaystyle \alpha }$ exist with

${\displaystyle \lim _{n\to \infty }{\frac {\left|p_{n+1}-p\right|}{\left|p_{n}-p\right|^{\alpha }}}=\lambda }$

then ${\displaystyle p_{n}}$ as ${\displaystyle n}$ goes from ${\displaystyle 0}$ to ${\displaystyle \infty }$ converges to ${\displaystyle p}$ of order ${\displaystyle \alpha }$, with asymptotic error constant ${\displaystyle \lambda }$.

Given a function ${\displaystyle f}$ with a fixed point ${\displaystyle p}$, there is a nice checklist for checking the convergence of the sequence ${\displaystyle p_{n}}$.

1. First check that p is indeed a fixed point:
${\displaystyle f(p)=p}$
2. Check for linear convergence. Start by finding ${\displaystyle \left|f'(p)\right|}$. If…
 ${\displaystyle \left|f'(p)\right|\in (0,1)}$ then there is linear convergence ${\displaystyle \left|f'(p)\right|>1}$ series diverges ${\displaystyle \left|f'(p)\right|=0}$ then there is at least linear convergence and maybe something better, the expression should be checked for quadratic convergence
1. If it is found that there is something better than linear, the expression should be checked for quadratic convergence. Start by finding ${\displaystyle \left|f''(p)\right|}$ If…
 ${\displaystyle \left|f''(p)\right|\neq 0}$ then there is quadratic convergence provided that ${\displaystyle f''(p)}$ is continuous ${\displaystyle \left|f''(p)\right|=0}$ then there is something even better than quadratic convergence ${\displaystyle \left|f''(p)\right|}$ does not exist then there is convergence that is better than linear but still not quadratic

[8]

## Computability of the limit

Limits can be difficult to compute. There exist limit expressions whose modulus of convergence is undecidable. In recursion theory, the limit lemma proves that it is possible to encode undecidable problems using limits. [9]

## Notes

1. Stewart, James (2008). (6th ed.). Brooks/Cole. ISBN   978-0-495-01166-8.
2. "List of Calculus and Analysis Symbols". Math Vault. 2020-05-11. Retrieved 2020-08-18.
3. Weisstein, Eric W. "Epsilon-Delta Definition". mathworld.wolfram.com. Retrieved 2020-08-18.
4. Larson, Ron; Edwards, Bruce H. (2010). Calculus of a single variable (Ninth ed.). Brooks/Cole, Cengage Learning. ISBN   978-0-547-20998-2.
5. "limit | Definition, Example, & Facts". Encyclopedia Britannica. Retrieved 2020-08-18.
6. Weisstein, Eric W. "Limit". mathworld.wolfram.com. Retrieved 2020-08-18.
7. Apostol (1974 , pp. 75–76)
8. Numerical Analysis, 8th Edition, Burden and Faires, Section 2.4 Error Analysis for Iterative Methods
9. Recursively enumerable sets and degrees, Soare, Robert I.

## Related Research Articles

In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be discontinuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.

In mathematics, more specifically calculus, L'Hôpital's rule or L'Hospital's rule provides a technique to evaluate limits of indeterminate forms. Application of the rule often converts an indeterminate form to an expression that can be easily evaluated by substitution. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital. Although the rule is often attributed to L'Hôpital, the theorem was first introduced to him in 1694 by the Swiss mathematician Johann Bernoulli.

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration.

In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members. The number of elements is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers, or the set of the first n natural numbers. Sequences are one type of indexed families as an indexed family is defined as a function which domain is called the index set, and the elements of the index set are the indices for the elements of the function image.

In mathematics, the Dirac delta function is a generalized function or distribution introduced by physicist Paul Dirac. It is called a function, although it is not a function on the level one would expect, that is, it is not a function RC, but a function on the space of test functions. It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. As there is no function that has these properties, the computations made by theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function.

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set if, given any arbitrarily small positive number , a number can be found such that each of the functions differ from by no more than at every pointin. Described in an informal way, if converges to uniformly, then the rate at which approaches is "uniform" throughout its domain in the following sense: in order to guarantee that falls within a certain distance of , we do not need to know the value of in question — there can be found a single value of independent of , such that choosing will ensure that is within of for all . In contrast, pointwise convergence of to merely guarantees that for any given in advance, we can find so that, for that particular, falls within of whenever .

In mathematics, the affinely extended real number system is obtained from the real number system by adding two infinity elements: and , where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted or or . It is the Dedekind–MacNeille completion of the real numbers.

The Heaviside step function, or the unit step function, usually denoted by H or θ, is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive arguments. It is an example of the general class of step functions, all of which can be represented as linear combinations of translations of this one.

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.

In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the symbol. If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.

In calculus, the squeeze theorem, also known as the pinching theorem, the sandwich theorem, the sandwich rule, the police theorem, the between theorem and sometimes the squeeze lemma, is a theorem regarding the limit of a function. In Italy, the theorem is also known as theorem of carabinieri.

In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.

In mathematics, the oscillation of a function or a sequence is a number that quantifies how much that sequence or function varies between its extreme values as it approaches infinity or a point. As is the case with limits, there are several definitions that put the intuitive concept into a form suitable for a mathematical treatment: oscillation of a sequence of real numbers, oscillation of a real-valued function at a point, and oscillation of a function on an interval.

In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic.

In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.

In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter θ0—having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to θ0. This means that the distributions of the estimates become more and more concentrated near the true value of the parameter being estimated, so that the probability of the estimator being arbitrarily close to θ0 converges to one.

In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. The definition used in measure theory is closely related to, but not identical to, the definition typically used in probability.

In real analysis, a branch of mathematics, a slowly varying function is a function of a real variable whose behaviour at infinity is in some sense similar to the behaviour of a function converging at infinity. Similarly, a regularly varying function is a function of a real variable whose behaviour at infinity is similar to the behaviour of a power law function near infinity. These classes of functions were both introduced by Jovan Karamata, and have found several important applications, for example in probability theory.

In calculus, the (εδ)-definition of limit is a formalization of the notion of limit. The concept is due to Augustin-Louis Cauchy, who never gave a formal definition of limit in his Cours d'Analyse, but occasionally used ε, δ arguments in proofs. It was first given as a formal definition by Bernard Bolzano in 1817, and the definitive modern statement was ultimately provided by Karl Weierstrass. It provides rigor to the following informal notion: the dependent expression f(x) approaches the value L as the variable x approaches the value c if f(x) can be made as close as desired to L by taking x sufficiently close to c.


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# Team:Toulouse/Team/Gallery

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# The file "airlinetweets.csv" contains sentiments (positive, neutral and negative) of the tweets for different airline companies. You have to learn a model using SVM classifier that is able to classify the tweets into one of the three classes.¶

### Importing libraries¶

In [5]:
import pandas as pd
from sklearn.feature_extraction.text import TfidfVectorizer
from sklearn.model_selection import train_test_split
from sklearn.metrics import accuracy_score, f1_score, recall_score, precision_score
from sklearn.svm import SVC
import re

### Code to clean the tweets¶

In [6]:
def cleaner(impure_data):
temp_list = []
for item in impure_data:
item = re.sub('@\S+', '', item)

item = re.sub('http\S+\s*', '', item)

#finding special characters, but not "emoji"
item = re.sub('[%s]' % re.escape("""!"#\$%&'()*+,-./:;<=>?@[\]^_{|}~"""), '', item)
temp_list.append(item)
return temp_list

### Function to predict tweet sentiment¶

The predicted tweets are stored in "predicted_airlinetweets.csv" file. Classification of the tweets has been done using SVM. Note that it is a multi-class classification task.

In [21]:
def tweet_sentiment():
#reading the tweets from csv files
tweets = df["text"]
polarity = df["airline_sentiment"].tolist()

#cleaning tweets i.e. removing @mentions, http(s) links and special characters such as punctuations
clean_tweet = cleaner(tweets)

#initializing tf-idf vectorizer
tf_idfvectorizer = TfidfVectorizer(sublinear_tf=True, use_idf=True)

#splitting the data into train and test dataset in 70 : 30 ratio at random
X_train, X_test, Y_train, Y_test = train_test_split(clean_tweet, polarity, test_size = 0.3)

#vectorizing the training data
#fit_transform() does two jobs, fit() and transform()
#fit calculates the statistics of the data
#transform takes care of any missing values or unexpected values by utilizing statistics calculated by fit
train_corpus_tf_idf = tf_idfvectorizer.fit_transform(X_train)

#vectorizing the testing data
#transform takes care of any missing values or unexpected values based on fit for training data
test_corpus_tf_idf = tf_idfvectorizer.transform(X_test)

#using SVC package to initialize a classifier with Linear kernel and other default parameters
SVM_L = SVC(kernel= 'linear')

#fitting the sparse matrix in the classifier with their respective sentiments
SVM_L.fit(train_corpus_tf_idf, Y_train)

#predicting the sentiments for the test dataset
Y_pred = SVM_L.predict(test_corpus_tf_idf)

#this print accuracy score for the test dataset
print("Accuracy",accuracy_score(Y_test,Y_pred))

#saving the data into a csv file in the current folder
temp_df = pd.DataFrame()
temp_df["Tweet"] = X_test
temp_df["Sentiment"] = Y_test
temp_df["Predicted Sentiment"] = Y_pred
temp_df.to_csv("predicted_airlinetweets.csv")

return(tf_idfvectorizer,SVM_L)

### Function is called: The predicted sentiments are stored in a file and the accuracy is printed¶

In [22]:
vectorizer,model = tweet_sentiment()

Accuracy 0.8021402550091075

### Compute the sentiment of a tweet and print the sentiment¶

In [46]:
vector = vectorizer.transform(["My journey was good. Thanks to your customer service."])
sentiment = model.predict(vector)
print(sentiment)

['positive']

In [47]:
vector = vectorizer.transform(["My journey was horrible because of your customer service."])
sentiment = model.predict(vector)
print(sentiment)

['negative']

In [48]:
vector = vectorizer.transform(["My journey was not good because of your crew."])
sentiment = model.predict(vector)
print(sentiment)

['negative']
`