Mid Chapter Review upto Product Rule
Chapter
Chapter 2
Section
Mid Chapter Review upto Product Rule
Solutions 36 Videos

a. Sketch the graph of \displaystyle f(x)=x^{2}-5 x

b. Calculate the slopes of the tangents to \displaystyle f(x)=x^{2}-5 x  at points with \displaystyle x -coordinates \displaystyle 0,1,2, \ldots, 5

c. Sketch the graph of the derivative function \displaystyle f^{\prime}(x)

d. Compare the graphs of \displaystyle f(x)  and \displaystyle f^{\prime}(x)

Q1

Use the definition of the derivative to find \displaystyle f^{\prime}(x)  for each function.

\displaystyle f(x)=6 x+15

Q2a

Use the definition of the derivative to find \displaystyle f^{\prime}(x)  for each function.

\displaystyle f(x)=2 x^{2}-4

Q2b

Use the definition of the derivative to find \displaystyle f^{\prime}(x)  for each function.

\displaystyle f(x)=\frac{5}{x+5}

Q2c

Use the definition of the derivative to find \displaystyle f^{\prime}(x)  for each function.

\displaystyle f(x)=\sqrt{x-2}

Q2d

a. Determine the equation of the tangent to the curve \displaystyle y=x^{2}-4 x+3  at \displaystyle x=1 .

b. Sketch the graph of the function and the tangent.

Q3

Differentiate each of the following functions:

\displaystyle y=6 x^{4}

Q4a

Differentiate each of the following functions:

\displaystyle y=10 x^{\frac{1}{2}}

Q4b

Differentiate each of the following functions:

\displaystyle g(x)=\frac{2}{x^{3}}

Q4c

Differentiate each of the following functions:

\displaystyle y=5 x+\frac{3}{x^{2}}

Q4d

Differentiate each of the following functions:

\displaystyle y=(11 t+1)^{2}

Q4e

Differentiate each of the following functions:

\displaystyle y=\frac{x-1}{x}

Q4f

Determine the equation of the tangent to the graph of \displaystyle f(x)=2 x^{4}  that has slope 1 .

Q5

Determine \displaystyle f^{\prime}(x)  for each of the following functions:

\displaystyle f(x)=4 x^{2}-7 x+8

Q6a

Determine \displaystyle f^{\prime}(x)  for each of the following functions:

\displaystyle f(x)=-2 x^{3}+4 x^{2}+5 x-6

Q6b

Determine \displaystyle f^{\prime}(x)  for each of the following functions:

\displaystyle f(x)=\frac{5}{x^{2}}-\frac{3}{x^{3}}

Q6c

Determine \displaystyle f^{\prime}(x)  for each of the following functions:

\displaystyle f(x)=\sqrt{x}+\sqrt[3]{x}

Q6d

Determine \displaystyle f^{\prime}(x)  for each of the following functions:

\displaystyle f(x)=7 x^{-2}-3 \sqrt{x}

Q6e

Determine \displaystyle f^{\prime}(x)  for each of the following functions:

\displaystyle f(x)=-4 x^{-1}+5 x-1

Q6f

Determine the equation of the tangent to the graph of each function.

\displaystyle y=-3 x^{2}+6 x+4  when \displaystyle x=1

Q7a

Determine the equation of the tangent to the graph of each function.

\displaystyle y=3-2 \sqrt{x}  when \displaystyle x=9

Q7b

Determine the equation of the tangent to the graph of each function.

\displaystyle f(x)=-2 x^{4}+4 x^{3}-2 x^{2}-8 x+9  when \displaystyle x=3

Q7c

Determine the derivative using the product rule.

\displaystyle f(x)=\left(4 x^{2}-9 x\right)\left(3 x^{2}+5\right)

Q8a

Determine the derivative using the product rule.

\displaystyle f(t)=\left(-3 t^{2}-7 t+8\right)(4 t-1)

Q8b

Determine the derivative using the product rule.

\displaystyle f(t)=\left(-3 t^{2}-7 t+8\right)(4 t-1)

Q8c

Determine the derivative using the product rule.

\displaystyle y=\left(3-2 x^{3}\right)^{3}

Q8d

Determine the equation of the tangent to \displaystyle y=\left(5 x^{2}+9 x-2\right)\left(-x^{2}+2 x+3\right)  at \displaystyle (1,48)

Q9

Determine the point(s) where the tangent to the curve \displaystyle y=2(x-1)(5-x)  is horizontal.

Q10

If \displaystyle y=5 x^{2}-8 x+4 , determine \displaystyle \frac{d y}{d x}  from first principles.

Q11

A tank holds \displaystyle 500 \mathrm{~L}  of liquid, which takes 90 min to drain from a hole in the bottom of the tank. The volume, \displaystyle V , remaining in the tank after \displaystyle t  minutes is \displaystyle V(t)=500\left(1-\frac{t}{90}\right)^{2}, \text { where } 0 \leq t \leq 90

Q12

The volume of a sphere is given by \displaystyle V(r)=\frac{4}{3} \pi r^{3}

a. Determine the average rate of change of volume with respect to radius as the radius changes from \displaystyle 10 \mathrm{~cm}  to \displaystyle 15 \mathrm{~cm} .

b. Determine the rate of change of volume when the radius is \displaystyle 8 \mathrm{~cm} .

Q13

A classmate says, "The derivative of a cubic polynomial function is a quadratic polynomial function." Is the statement always true, sometimes true, or never true? Defend your choice in words, and provide two examples to support your argument.

Q14

Show that \displaystyle \frac{d y}{d x}=(a+4 b) x^{a+4 b-1}  if \displaystyle y=\frac{x^{2 a+3 b}}{x^{a-b}}  and \displaystyle a  and \displaystyle b  are integers.

Q15

a. Determine \displaystyle f^{\prime}(3) , where \displaystyle f(x)=-6 x^{3}+4 x-5 x^{2}+10 .

b. Give two interpretations of the meaning of \displaystyle f^{\prime}(3) .

Q16

The population, \displaystyle P , of a bacteria colony at \displaystyle t  hours can be modelled by \displaystyle P(t)=100+120 t+10 t^{2}+2 t^{3}

a. What is the initial population of the bacteria colony?

b. What is the population of the colony at \displaystyle 5 \mathrm{~h} ?

c. What is the growth rate of the colony at \displaystyle 5 \mathrm{~h}  ?

The relative percent of carbon dioxide, \displaystyle C , in a carbonated soft drink at \displaystyle t  minutes can be modelled by \displaystyle C(t)=\frac{100}{t} , where \displaystyle t > 2 .  Determine \displaystyle C^{\prime}(t)  and interpret the results at 5 min, 50 min, and 100 min. Explain what is happening.