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# 3.3E: Exercises for Section 3.3

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In exercises 1 - 12, find $$f'(x)$$ for each function.

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

2) $$f(x)=5x^3−x+1$$

$$f'(x)=15x^2−1$$

3) $$f(x)=4x^2−7x$$

4) $$f(x)=8x^4+9x^2−1$$

$$f'(x) = 32x^3+18x$$

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

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

$$f'(x) = 270x^4+\dfrac{39}{(x+1)^2}$$

7) $$f(x)=(x+2)(2x^2−3)$$

8) $$f(x)=x^2\left(\dfrac{2}{x^2}+\dfrac{5}{x^3}\right)$$

$$f'(x) = \dfrac{−5}{x^2}$$

9) $$f(x)=\dfrac{x^3+2x^2−4}{3}$$

10) $$f(x)=\dfrac{4x^3−2x+1}{x^2}$$

$$f'(x) = \dfrac{4x^4+2x^2−2x}{x^4}$$

11) $$f(x)=\dfrac{x^2+4}{x^2−4}$$

12) $$f(x)=\dfrac{x+9}{x^2−7x+1}$$

$$f'(x) = \dfrac{−x^2−18x+64}{(x^2−7x+1)^2}$$

In exercises 13 - 16, 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.

13) [T] $$y=3x^2+4x+1$$ at $$(0,1)$$

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

$$T(x)=\frac{1}{2}x+3$$ 15) [T] $$y=\dfrac{2x}{x−1}$$ at $$(−1,1)$$

16) [T] $$y=\dfrac{2}{x}−\dfrac{3}{x^2}$$ at $$(1,−1)$$

$$T(x)=4x−5$$ In exercises 17 - 20, assume that $$f(x)$$ and $$g(x)$$ are both differentiable functions for all $$x$$. Find the derivative of each of the functions $$h(x)$$.

17) $$h(x)=4f(x)+\dfrac{g(x)}{7}$$

18) $$h(x)=x^3f(x)$$

$$h'(x)=3x^2f(x)+x^3f′(x)$$

19) $$h(x)=\dfrac{f(x)g(x)}{2}$$

20) $$h(x)=\dfrac{3f(x)}{g(x)+2}$$

$$h'(x)=\dfrac{3f′(x)(g(x)+2)−3f(x)g′(x)}{(g(x)+2)^2}$$

For exercises 21 - 24, 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

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

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

$$h'(2) =\frac{16}{9}$$

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

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

$$h'(4)$$ is undefined.

In exercises 25 - 27, use the following figure to find the indicated derivatives, if they exist. 25) Let $$h(x)=f(x)+g(x)$$. Find

a) $$h′(1)$$,

b) $$h′(3)$$, and

c) $$h′(4)$$.

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

a) $$h′(1),$$

b) $$h′(3)$$, and

c) $$h′(4).$$

a. $$h'(1) = 2$$,
b. $$h'(3)$$ does not exist,
c. $$h'(4) = 2.5$$

27) Let $$h(x)=\dfrac{f(x)}{g(x)}.$$ Find

a) $$h′(1),$$

b) $$h′(3)$$, and

c) $$h′(4).$$

In exercises 28 - 31,

a) evaluate $$f′(a)$$, and

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

28) [T] $$f(x)=2x^3+3x−x^2, \quad a=2$$

a. 23
b. $$y=23x−28$$ 29) [T] $$f(x)=\dfrac{1}{x}−x^2, \quad a=1$$

30) [T] $$f(x)=x^2−x^{12}+3x+2, \quad a=0$$

a. $$3$$
b. $$y=3x+2$$ 31) [T] $$f(x)=\dfrac{1}{x}−x^{2/3}, \quad a=−1$$

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

$$y=−7x−3$$

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

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

$$y=−5x+7$$

35) Find the point on the graph of $$f(x)=x^3$$ such that the tangent line at that point has an $$x$$-intercept of $$(6,0)$$.

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

$$y=−\frac{3}{2}x+\frac{15}{2}$$

37) 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.

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

$$y=−3x^2+9x−1$$

39) 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.

40) [T] A herring swimming along a straight line has traveled $$s(t)=\dfrac{t^2}{t^2+2}$$ feet in $$t$$

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

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

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

a. Determine the initial flounder population.

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

42) [T] The concentration of antibiotic in the bloodstream $$t$$ hours after being injected is given by the function $$C(t)=\dfrac{2t^2+t}{t^3+50}$$, 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.

a. $$\dfrac{−2t^4−2t^3+200t+50}{(t^3+50)^2}$$
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.

43) A book publisher has a cost function given by $$C(x)=\dfrac{x^3+2x+3}{x^2}$$, 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.

44) [T] 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 $$F=\dfrac{Gm_1m_2}{d^2}$$, 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 $$G=6.67×10^{−11} \text{Nm}^2/\text{kg}^2$$, on two bodies 10 meters apart, each with a mass of 1000 kilograms.

a. $$F'(d)=\dfrac{−2Gm_1m_2}{d_3}$$
b. $$−1.33×10^{−7}$$ N/m