# 7.6E: Exercises for Section 3.1

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For exercises 1 - 10, use the equation $$m_{\text{sec}}=\dfrac{f(x)−f(a)}{x−a}$$ to find the slope of the secant line between the values $$x_1$$ and $$x_2$$ for each function $$y=f(x)$$.

1) $$f(x)=4x+7; \quad x_1=2, \quad x_2=5$$

$$m_{\text{sec}}=4$$

2) $$f(x)=8x−3;\quad x_1=−1,\quad x_2=3$$

3) $$f(x)=x^2+2x+1;\quad x_1=3,\quad x_2=3.5$$

$$m_{\text{sec}}=8.5$$

4) $$f(x)=−x^2+x+2;\quad x_1=0.5,\quad x_2=1.5$$

5) $$f(x)=\dfrac{4}{3x−1};\quad x_1=1,\quad x_2=3$$

$$m_{\text{sec}}=−\frac{3}{4}$$

6) $$f(x)=\dfrac{x−7}{2x+1};\quad x_1=−2,\quad x_2=0$$

7) $$f(x)=\sqrt{x};\quad x_1=1,\quad x_2=16$$

$$m_{\text{sec}}=0.2$$

8) $$f(x)=\sqrt{x−9};\quad x_1=10,\quad x_2=13$$

9) $$f(x)=x^{1/3}+1;\quad x_1=0,\quad x_2=8$$

$$m_{\text{sec}}=0.25$$

10) $$f(x)=6x^{2/3}+2x^{1/3};\quad x_1=1,\quad x_2=27$$

For the functions in exercises 11 - 20,

a. use the equation $$\displaystyle m_{\text{tan}}=\lim_{h→0}\frac{f(a+h)−f(a)}{h}$$ to find the slope of the tangent line $$m_{\text{tan}}=f′(a)$$, and

b. find the equation of the tangent line to $$f$$ at $$x=a$$.

11) $$f(x)=3−4x, \quad a=2$$

a. $$m_{\text{tan}}=−4$$
b. $$y=−4x+3$$

12) $$f(x)=\dfrac{x}{5}+6, \quad a=−1$$

13) $$f(x)=x^2+x, \quad a=1$$

a. $$m_{\text{tan}}=3$$
b. $$y=3x−1$$

14) $$f(x)=1−x−x^2, \quad a=0$$

15) $$f(x)=\dfrac{7}{x}, \quad a=3$$

a. $$m_{\text{tan}}=\frac{−7}{9}$$
b. $$y=\frac{−7}{9}x+\frac{14}{3}$$

16) $$f(x)=\sqrt{x+8}, \quad a=1$$

17) $$f(x)=2−3x^2, \quad a=−2$$

a. $$m_{\text{tan}}=12$$
b. $$y=12x+14$$

18) $$f(x)=\dfrac{−3}{x−1}, \quad a=4$$

19) $$f(x)=\dfrac{2}{x+3}, \quad a=−4$$

a. $$m_{\text{tan}}=−2$$
b. $$y=−2x−10$$

20) $$f(x)=\dfrac{3}{x^2}, \quad a=3$$

For the functions $$y=f(x)$$ in exercises 21 - 30, find $$f′(a)$$ using the equation $$\displaystyle f′(a)=\lim_{x→a}\frac{f(x)−f(a)}{x−a}$$.

21) $$f(x)=5x+4, \quad a=−1$$

$$f'(-1) = 5$$

22) $$f(x)=−7x+1, \quad a=3$$

23) $$f(x)=x^2+9x, \quad a=2$$

$$f'(2) = 13$$

24) $$f(x)=3x^2−x+2, \quad a=1$$

25) $$f(x)=\sqrt{x}, \quad a=4$$

$$f'(4) = \frac{1}{4}$$

26) $$f(x)=\sqrt{x−2}, \quad a=6$$

27) $$f(x)=\dfrac{1}{x}, \quad a=2$$

$$f'(2) = −\frac{1}{4}$$

28) $$f(x)=\dfrac{1}{x−3}, \quad a=−1$$

29) $$f(x)=\dfrac{1}{x^3}, \quad a=1$$

$$f'(1) = -3$$

30) $$f(x)=\dfrac{1}{\sqrt{x}}, \quad a=4$$

For the following exercises, given the function $$y=f(x)$$,

a. find the slope of the secant line $$PQ$$ for each point $$Q(x,f(x))$$ with $$x$$ value given in the table.

b. Use the answers from a. to estimate the value of the slope of the tangent line at $$P$$.

c. Use the answer from b. to find the equation of the tangent line to $$f$$ at point $$P$$.

31) [T] $$f(x)=x^2+3x+4, \quad P(1,8)$$ (Round to $$6$$ decimal places.)

 $$x$$ Slope $$m_{PQ}$$ $$x$$ Slope $$m_{PQ}$$ 1.1 (i) 0.9 (vii) 1.01 (ii) 0.99 (viii) 1.001 (iii) 0.999 (ix) 1.0001 (iv) 0.9999 (x) 1.00001 (v) 0.99999 (xi) 1.000001 (vi) 0.999999 (xii)
$$a. (i)5.100000, (ii)5.010000, (iii)5.001000, (iv)5.000100, (v)5.000010, (vi)5.000001, (vii)4.900000, (viii)4.990000, (ix)4.999000, (x)4.999900, (xi)4.999990, (x)4.999999$$
b. $$m_{\text{tan}}=5$$
c. $$y=5x+3$$

32) [T] $$f(x)=\dfrac{x+1}{x^2−1}, \quad P(0,−1)$$

 $$x$$ Slope $$m_{PQ}$$ $$x$$ Slope $$m_{PQ}$$ 0.1 (i) −0.1 (vii) 0.01 (ii) −0.01 (viii) 0.001 (iii) −0.001 (ix) 0.0001 (iv) −0.0001 (x) 0.00001 (v) −0.00001 (xi) 0.000001 (vi) −0.000001 (xii)

33) [T] $$f(x)=10e^{0.5x}, \quad P(0,10)$$ (Round to $$4$$ decimal places.)

 $$x$$ Slope $$m_{PQ}$$ −0.1 (i) −0.01 (ii) −0.001 (iii) −0.0001 (iv) −0.00001 (v) −0.000001 (vi)
a. $$(i)4.8771, \;(ii)4.9875, \;(iii)4.9988, \;(iv)4.9999, \;(v)4.9999, \;(vi)4.9999$$
b. $$m_{\text{tan}}=5$$
c. $$y=5x+10$$

34) [T] $$f(x)=\tan(x), \quad P(π,0)$$

 $$x$$ Slope $$m_{PQ}$$ 3.1 (i) 3.14 (ii) 3.141 (iii) 3.1415 (iv) 3.14159 (v) 3.141592 (vi)

[T] For the following position functions $$y=s(t)$$, an object is moving along a straight line, where $$t$$ is in seconds and $$s$$ is in meters. Find

a. the simplified expression for the average velocity from $$t=2$$ to $$t=2+h$$;

b. the average velocity between $$t=2$$ and $$t=2+h$$, where $$(i)\;h=0.1, \;(ii)\;h=0.01, \;(iii)\;h=0.001$$, and $$(iv)\;h=0.0001$$; and

c. use the answer from a. to estimate the instantaneous velocity at $$t=2$$ second.

35) $$s(t)=\frac{1}{3}t+5$$

a. $$\frac{1}{3}$$;
b. $$(i)\;\frac{1}{3}$$ m/s, $$(ii)\;\frac{1}{3}$$ m/s, $$(iii)\;\frac{1}{3}$$ m/s, $$(iv)\;\frac{1}{3}$$ m/s;
c. $$\frac{1}{3}$$ m/s

36) $$s(t)=t^2−2t$$

37) $$s(t)=2t^3+3$$

a. $$2(h^2+6h+12)$$;
b. $$(i)\;25.22$$ m/s, $$(ii)\; 24.12$$ m/s, $$(iii)\; 24.01$$ m/s, $$(iv)\; 24$$ m/s;
c. $$24$$ m/s

38) $$s(t)=\dfrac{16}{t^2}−\dfrac{4}{t}$$

39) Use the following graph to evaluate a. $$f′(1)$$ and b. $$f′(6).$$

a. $$1.25$$; b. $$0.5$$

40) Use the following graph to evaluate a. $$f′(−3)$$ and b. $$f′(1.5)$$.

For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $$x=a$$ for each of the given functions.

41) $$f(x)=x^{1/3}, \quad x=0$$

$$\displaystyle \lim_{x→0^−}\frac{x^{1/3}−0}{x−0}=\lim_{x→0^−}\frac{1}{x^{2/3}}=∞$$

42) $$f(x)=x^{2/3}, \quad x=0$$

43) $$f(x)=\begin{cases}1, & \text{if } x<1\\x, & \text{if } x≥1\end{cases}, \quad x=1$$

$$\displaystyle \lim_{x→1^−}\frac{1−1}{x−1}=0≠1=\lim_{x→1^+}\frac{x−1}{x−1}$$

44) $$f(x)=\dfrac{|x|}{x}, \quad x=0$$

45) [T] The position in feet of a race car along a straight track after $$t$$ seconds is modeled by the function $$s(t)=8t^2−\frac{1}{16}t^3.$$

a. Find the average velocity of the vehicle over the following time intervals to four decimal places:

i. [$$4, 4.1$$]

ii. [$$4, 4.01$$]

iii. [$$4, 4.001$$]

iv. [$$4, 4.0001$$]

b. Use a. to draw a conclusion about the instantaneous velocity of the vehicle at $$t=4$$ seconds.

a. $$(i)61.7244 ft/s, \;(ii)61.0725 ft/s, \;(iii)61.0072 ft/s, \;(iv)61.0007 ft/s$$
b. At $$4$$ seconds the race car is traveling at a rate/velocity of $$61$$ ft/s.

46) [T] The distance in feet that a ball rolls down an incline is modeled by the function $$s(t)=14t^2$$,

where t is seconds after the ball begins rolling.

a. Find the average velocity of the ball over the following time intervals:

i. [5, 5.1]

ii. [5, 5.01]

iii. [5, 5.001]

iv. [5, 5.0001]

b. Use the answers from a. to draw a conclusion about the instantaneous velocity of the ball at $$t=5$$ seconds.

47) Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the following graph, are given by $$s=f(t)$$ and $$s=g(t)$$, where s is measured in feet and t is measured in seconds.

a. Which vehicle has traveled farther at $$t=2$$ seconds?

b. What is the approximate velocity of each vehicle at $$t=3$$ seconds?

c. Which vehicle is traveling faster at $$t=4$$ seconds?

d. What is true about the positions of the vehicles at $$t=4$$ seconds?

a. The vehicle represented by $$f(t)$$, because it has traveled $$2$$ feet, whereas $$g(t)$$ has traveled $$1$$ foot.
b. The velocity of $$f(t)$$ is constant at $$1$$ ft/s, while the velocity of $$g(t)$$ is approximately $$2$$ ft/s.
c. The vehicle represented by $$g(t)$$, with a velocity of approximately $$4$$ ft/s.
d. Both have traveled $$4$$ feet in $$4$$ seconds.

48) [T] The total cost $$C(x)$$, in hundreds of dollars, to produce $$x$$ jars of mayonnaise is given by $$C(x)=0.000003x^3+4x+300$$.

a. Calculate the average cost per jar over the following intervals:

i. [100, 100.1]

ii. [100, 100.01]

iii. [100, 100.001]

iv. [100, 100.0001]

b. Use the answers from a. to estimate the average cost to produce $$100$$ jars of mayonnaise.

49) [T] For the function $$f(x)=x^3−2x^2−11x+12$$, do the following.

a. Use a graphing calculator to graph $$f$$ in an appropriate viewing window.

b. Use the ZOOM feature on the calculator to approximate the two values of $$x=a$$ for which $$m_{tan}=f′(a)=0$$.

a.

b. $$a≈−1.361,\;2.694$$

50) [T] For the function $$f(x)=\dfrac{x}{1+x^2}$$, do the following.

a. Use a graphing calculator to graph $$f$$ in an appropriate viewing window.

b. Use the ZOOM feature on the calculator to approximate the values of $$x=a$$ for which $$m_{\text{tan}}=f′(a)=0$$.

51) Suppose that $$N(x)$$ computes the number of gallons of gas used by a vehicle traveling $$x$$ miles. Suppose the vehicle gets $$30$$ mpg.

a. Find a mathematical expression for $$N(x)$$.

b. What is $$N(100)$$? Explain the physical meaning.

c. What is $$N′(100)$$? Explain the physical meaning.

a. $$N(x)=\dfrac{x}{30}$$
b. ∼$$3.3$$ gallons. When the vehicle travels $$100$$ miles, it has used $$3.3$$ gallons of gas.
c. $$\frac{1}{30}$$. The rate of gas consumption in gallons per mile that the vehicle is achieving after having traveled $$100$$ miles.

52) [T] For the function $$f(x)=x^4−5x^2+4$$, do the following.

a. Use a graphing calculator to graph $$f$$ in an appropriate viewing window.

b. Use the $$nDeriv$$ function, which numerically finds the derivative, on a graphing calculator to estimate $$f′(−2),\;f′(−0.5),\;f′(1.7)$$, and $$f′(2.718)$$.

53) [T] For the function $$f(x)=\dfrac{x^2}{x^2+1}$$, do the following.

a. Use a graphing calculator to graph $$f$$ in an appropriate viewing window.

b. Use the $$nDeriv$$ function on a graphing calculator to find $$f′(−4),\;f′(−2),\;f′(2)$$, and $$f′(4)$$.

a.

b. $$−0.028,−0.16,0.16,0.028$$

7.6E: Exercises for Section 3.1 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.