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# 2.2E Exercises

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###### Exercise $$\PageIndex{1}$$

For the following exercises, assume that $$f(x)$$ and $$g(x)$$ are both differentiable functions for all $$x$$. Find the derivative of each of the functions $$h(x)$$.

1) $$h(x)=4f(x)+\frac{g(x)}{7}$$

2)  $$h(x)=x^3f(x)$$

3) $$h(x)=\frac{f(x)g(x)}{2}$$

4) $$h(x)=\frac{3f(x)}{g(x)+2}$$

Answers to even numbered questions

2. $$h′(x)=3x^2f(x)+x^3f′(x)$$

4. $$h′(x)=\frac{3f′(x)(g(x)+2)−3f(x)g′(x)}{(g(x)+2)^2}$$

###### Exercise $$\PageIndex{2}$$

For the following exercises, assume that f(x) and g(x) are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives.

 $$x$$ 1 2 3 4 $$f(x)$$ 3 5 −2 0 $$g(x)$$ 2 3 −4 6 $$f′(x)$$ −1 7 8 −3 $$g′(x)$$ 4 1 2 9

1) Find $$h′(1)$$ if $$h(x)=x f(x)+4g(x)$$.

2) Find $$h′(2)$$ if $$h(x)=\frac{f(x)}{g(x)}$$.

3) Find $$h′(3)$$ if $$h(x)=2x+f(x)g(x)$$.

4) Find $$h′(4)$$ if $$h(x)=\frac{1}{x}+\frac{g(x)}{f(x)}$$.

Answers to even numbered questions

2. $$\frac{16}{9}$$

4. Undefined

###### Exercise $$\PageIndex{3}$$

For the following exercises, use the following figure to find the indicated derivatives, if they exist.

1) Let $$h(x)=f(x)+g(x)$$. Find

a. $$h′(1)$$,

b. $$h′(3)$$, and

c. $$h′(4)$$.

2) Let $$h(x)=f(x)g(x).$$ Find

a. $$h′(1),$$

b. $$h′(3)$$, and

c. $$h′(4).$$

3)  Let $$h(x)=\frac{f(x)}{g(x)}.$$ Find

a. $$h′(1),$$

b. $$h′(3)$$, and

c. $$h′(4).$$

Answers to even numbered questions

2a. $$2$$

b. does not exist

c. $$2.5$$

###### Exercise $$\PageIndex{4}$$

For the following exercises,

a. evaluate $$f′(a)$$, and

b. graph the function $$f(x)$$ and the tangent line at $$x=a.$$

1) $$f(x)=2x^3+3x−x^2,a=2$$

2) $$f(x)=\frac{1}{x}−x^2,a=1$$

3) $$f(x)=x^2−x^{12}+3x+2,a=0$$

4) $$f(x)=\frac{1}{x}−x^{2/3},a=−1$$

Answers to odd numbered questions

1a. 23

b. $$y=23x−28$$

3a. 3

b. $$y=3x+2$$

###### Exercise $$\PageIndex{5}$$

Find the derivative of the following functions and simplify your answer:

1) $$f(x)=x^7+10$$

2) $$f(x)=5x^3−x+1$$

3) $$f(x)=4x^2−7x$$

4) $$f(x)=8x^4+9x^2−1$$

5) $$f(x)=x^4+2x$$

6) $$f(x)=3x(18x^4+\frac{13}{x+1})$$

7) (f(x)=(x+2)(2x^2−3)\)

8) $$f(x)=x^2(\frac{2}{x^2}+\frac{5}{x^3})$$

9) $$f(x)=\frac{x^3+2x^2−4}{3}$$

10) $$f(x)=\frac{4x^3−2x+1}{x^2} 11) \(f(x)=\frac{x^2+4}{x^2−4}$$

12) $$f(x)=\frac{x+9}{x^2−7x+1}$$

13) $$\displaystyle{y(x)=\frac{3x^2+5}{2x^2+x-3}}$$.

14) $$\displaystyle{y(x)=\frac{3x^2+5}{2x^2+x-3}}$$

Answers to even numbered questions

2. $$f′(x)=15x^2−1$$

4. $$f′(x)=32x^3+18x$$

6.$$f′(x)=270x^4+\frac{39}{(x+1)^2}$$

8. $$f′(x)=\frac{−5}{x^2}$$

10. $$f′(x)=\frac{4x^4+2x^2−2x}{x^4}$$

12. $$f′(x)=\frac{−x^2−18x+64}{(x^2−7x+1)^2}$$

###### Exercise $$\PageIndex{6}$$

For the following exercises, find the equation of the tangent line $$T(x)$$ to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line.

1) $$y=3x^2+4x+1$$ at $$(0,1)$$

2) $$y=2\sqrt{x}+1$$ at $$(4,5)$$

3) $$y=\frac{2x}{x−1}$$ at $$(−1,1)$$

4) $$y=\frac{2}{x}−\frac{3}{x^2}$$ at $$(1,−1)$$

Answers to even numbered questions

2. $$T(x)=\frac{1}{2}x+3$$,

4. $$T(x)=4x−5$$

###### Exercise $$\PageIndex{7}$$

1) Find the equation of the tangent line to the graph of $$f(x)=2x^3+4x^2−5x−3$$ at $$x=−1.$$

2) Find the equation of the tangent line to the graph of $$f(x)=x^2+\frac{4}{x}−10$$ at $$x=8$$.

3) Find the equation of the tangent line to the graph of $$f(x)=(3x−x^2)(3−x−x^2)$$ at $$x=1$$.

4) Find the point on the graph of $$f(x)=x^3$$ such that the tangent line at that point has an x intercept of 6.

5) Find the equation of the line passing through the point $$P(3,3)$$ and tangent to the graph of $$f(x)=\frac{6}{x−1}$$.

Answers to odd numbered questions

1. $$y=−7x−3$$

3. $$y=−5x+7$$

5. $$y=−\frac{3}{2}x+\frac{15}{2}$$

###### Exercise $$\PageIndex{8}$$

Determine all points on the graph of $$f(x)=x^3+x^2−x−1$$ for which the slope of the tangent line is

a. horizontal

b. −1.

Answer
Under Construction

###### Exercise $$\PageIndex{9}$$

Find a quadratic polynomial such that $$f(1)=5,f′(1)=3$$ and $$f''(1)=−6.$$

Answer

$$y=−3x^2+9x−1$$

###### Exercise $$\PageIndex{10}$$

A car driving along a freeway with traffic has traveled $$s(t)=t^3−6t^2+9t$$ meters in $$t$$ seconds.

a. Determine the time in seconds when the velocity of the car is 0.

b. Determine the acceleration of the car when the velocity is 0.

Answer

Under Construction

###### Exercise $$\PageIndex{11}$$

A herring swimming along a straight line has traveled $$s(t)=\frac{t^2}{t^2+2}$$ feet in $$t$$ seconds.

Determine the velocity of the herring when it has traveled 3 seconds.

Answer

$$\frac{12}{121}$$ or 0.0992 ft/s

###### Exercise $$\PageIndex{12}$$

The population in millions of arctic flounder in the Atlantic Ocean is modeled by the function $$P(t)=\frac{8t+3}{0.2t^2+1}$$, where $$t$$ is measured in years.

a. Determine the initial flounder population.

b. Determine $$P′(10)$$ and briefly interpret the result.

Answer

Under Construction

###### Exercise $$\PageIndex{12}$$

The concentration of antibiotic in the bloodstream $$t$$ hours after being injected is given by the function $$C(t)=\frac{2t^2+t}{t^3+50}$$, where $$C$$ is measured in milligrams per liter of blood.

a. Find the rate of change of $$C(t).$$

b. Determine the rate of change for $$t=8,12,24$$,and $$36$$.

c. Briefly describe what seems to be occurring as the number of hours increases.

Answer

$$a. \frac{−2t^4−2t^3+200t+50}{(t^3+50)^2}$$ $$b. −0.02395$$ mg/L-hr, −0.01344 mg/L-hr, −0.003566 mg/L-hr, −0.001579 mg/L-hr c. The rate at which the concentration of drug in the bloodstream decreases is slowing to 0 as time increases.mplate active on the page.

###### Exercise $$\PageIndex{13}$$

A book publisher has a cost function given by $$C(x)=\frac{x^3+2x+3}{x^2}$$, where x is the number of copies of a book in thousands and C is the cost per book, measured in dollars. Evaluate $$C′(2)$$ and explain its meaning.

Answer

Under Construction

###### Exercise $$\PageIndex{14}$$

According to Newton’s law of universal gravitation, the force $$F$$ between two bodies of constant mass $$m_1$$ and $$m_2$$ is given by the formula $$F=\frac{Gm_1m_2}{d^2}$$, where $$G$$ is the gravitational constant and $$d$$ is the distance between the bodies.

a. Suppose that $$G,m_1,$$ and $$m_2$$ are constants. Find the rate of change of force $$F$$ with respect to distance $$d$$.

b. Find the rate of change of force $$F$$ with gravitational constant $$G=6.67×10^{−11} Nm^2/kg^2$$, on two bodies 10 meters apart, each with a mass of 1000 kilograms.

Answer

a. $$F'(d)=\frac{−2Gm_1m_2}{d_3}$$ '

b. $$−1.33×10^{−7} N/m$$

###### Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY 3/0 license. Download for free at http://cnx.org.