Mid Chapter Review upto Product Rule
Chapter
Chapter 2
Section
Mid Chapter Review upto Product Rule
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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)

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Q1

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

\displaystyle f(x)=6 x+15

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Q2a

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

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

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Q2b

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

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

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Q2c

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

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

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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.

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Q3

Differentiate each of the following functions:

\displaystyle y=6 x^{4}

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Q4a

Differentiate each of the following functions:

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

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Q4b

Differentiate each of the following functions:

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

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Q4c

Differentiate each of the following functions:

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

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Q4d

Differentiate each of the following functions:

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

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Q4e

Differentiate each of the following functions:

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

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Q4f

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

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Q5

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

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

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Q6a

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

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

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Q6b

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

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

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Q6c

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

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

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Q6d

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

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

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Q6e

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

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

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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

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Q7a

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

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

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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

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Q7c

Determine the derivative using the product rule.

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

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Q8a

Determine the derivative using the product rule.

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

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Q8b

Determine the derivative using the product rule.

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

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Q8c

Determine the derivative using the product rule.

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

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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)

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Q9

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

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Q10

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

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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

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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} .

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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.

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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.

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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) .

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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} ?

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Q17

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.

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Q18