3.2E: 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) $f(x)=4x+7; \quad x_1=2, \quad x_2=5$

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

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

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

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

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

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

Answer: $m_{\text{sec}}=−\frac{3}{4}$

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

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

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

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

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

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

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) $f(x)=3−4x, \quad a=2$

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

12) $f(x)=\dfrac{x}{5}+6, \quad a=−1$

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

Answer: a. $m_{\text{tan}}=3$
b. $y=3x−1$

14) $f(x)=1−x−x^2, \quad a=0$

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

Answer: 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) $f(x)=2−3x^2, \quad a=−2$

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

18) $f(x)=\dfrac{−3}{x−1}, \quad a=4$

19) $f(x)=\dfrac{2}{x+3}, \quad a=−4$

Answer: 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) $f(x)=5x+4, \quad a=−1$

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

22) $f(x)=−7x+1, \quad a=3$

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

Answer: $f'(2) = 13$

24) $f(x)=3x^2−x+2, \quad a=1$

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

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

26) $f(x)=\sqrt{x−2}, \quad a=6$

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

Answer: $f'(2) = −\frac{1}{4}$

28) $f(x)=\dfrac{1}{x−3}, \quad a=−1$

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

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

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)

Answer: $a. (i)5.100000, (ii)5.010000, (iii)5.001000, (iv)5.000100, (v)5.000010, (vi)5.000001, (vii)4.900000, (viii)4.990000, (ix)4.999000, (x)4.999900, (xi)4.999990, (x)4.999999$
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)

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

Answer: 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) $s(t)=2t^3+3$

Answer: 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. $f′(6).$

$This graph shows two connected line segments: one going from (1, 0) to (4, 6) and the other going from (4, 6) to (8, 8).$

Answer: a. $1.25$; b. $0.5$

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

$This graph shows two connected line segments: one going from (−4, 3) to (1, 3) and the other going from (1, 3) to (1.5, 4).$

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

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

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

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

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

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.

Answer: 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.

$Two functions s = g(t) and s = f(t) are graphed. The first function s = g(t) starts at (0, 0) and arcs upward through roughly (2, 1) to (4, 4). The second function s = f(t) is a straight line passing through (0, 0) and (4, 4).$

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?

Answer: 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 $m_{tan}=f′(a)=0$.

Answer

$The function starts in the third quadrant, passes through the x axis at x = −3, increases to a maximum around y = 20, decreases and passes through the x axis at x = 1, continues decreasing to a minimum around y = −13, and then increases through the x axis at x = 4, after which it continues increasing.$

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.

Answer: 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 $f′(4)$.

Answer

$The function starts in the second quadrant and gently decreases, touches the origin, and then it increases gently.$
b. $−0.028,−0.16,0.16,0.028$

\(x\)	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)