Problem Set: Defining the Derivative

For the following exercises (1-10), use the definition of a derivative to find the slope of the secant line between the values $x_1$ and $x_2$ for each function $y=f(x)$ .

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

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2. $f(x)=8x-3; \,\,\, x_1=-1, \,\,\, x_2=3$

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

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4. $f(x)=\text{−}{x}^{2}+x+2;{x}_{1}=0.5,{x}_{2}=1.5$

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

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$-\frac{3}{4}$

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

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

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

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

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

For the following functions (11-20),

Use $m_{\tan}=\underset{h\to 0}{\lim}\dfrac{f(a+h)-f(a)}{h}$ to find the slope of the tangent line $m_{\tan}=f^{\prime}(a)$ , and
find the equation of the tangent line to $f$ at $x=a$ .

11. $f(x)=3-4x, \,\,\, a=2$

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a. -4 b. $y=3-4x$

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

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

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a. 3 b. $y=3x-1$

14. $f(x)=1-x-x^2, \,\,\, a=0$

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

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a. $\frac{-7}{9}$ b. $y=\frac{-7}{9}x+\frac{14}{3}$

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

17. $f(x)=2-3x^2, \,\,\, a=-2$

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a. 12 b. $y=12x+14$

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

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

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a. -2 b. $y=-2x-10$

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

For the following functions $y=f(x)$ (21-30), find $f^{\prime}(a)$ using $f^{\prime}(a)=\underset{x\to a}{\lim}\dfrac{f(x)-f(a)}{x-a}$ .

21. $f(x)=5x+4, \,\,\, a=-1$

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22. $f(x)=-7x+1, \,\,\, a=3$

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

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24. $f(x)=3x^2-x+2, \,\,\, a=1$

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

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$\frac{1}{4}$

26. $f(x)=\sqrt{x-2}, \,\,\, a=6$

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

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$-\frac{1}{4}$

28. $f(x)=\dfrac{1}{x-3}, \,\,\, a=-1$

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

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

For the following exercises (31-34), given the function $y=f(x)$ ,

find the slope of the secant line $PQ$ for each point $Q(x,f(x))$ with $x$ value given in the table.
Use the answers from a. to estimate the value of the slope of the tangent line at $P$ .
Use the answer from b. to find the equation of the tangent line to $f$ at point $P$ .

31. [T] $f(x)=x^2+3x+4, \,\,\, P(1,8)$ (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)

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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) $4.999990$ , (xii) $4.999999$

b. $m_{\tan}=5$
c. $y=5x+3$

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

$x$	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] $f(x)=10e^{0.5x}, \,\,\, P(0,10)$ (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)

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a. (i) $4.8771$ , (ii) $4.9875$ , (iii) $4.9988$ , (iv) $4.9999$ , (v) $4.9999$ , (vi) $4.9999$
b. $m_{\tan}=5$
c. $y=5x+10$

34. [T] $f(x)= \tan (x), \,\,\, P(\pi,0)$

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

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

the simplified expression for the average velocity from $t=2$ to $t=2+h$ ;
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
use the answer from a. to estimate the instantaneous velocity at $t=2$ seconds.

35. [T] $s(t)=\frac{1}{3}t+5$

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a. $\frac{1}{3}$ ;
b. (i) $0.\bar{3}$ m/s, (ii) $0.\bar{3}$ m/s, (iii) $0.\bar{3}$ m/s, (iv) $0.\bar{3}$ m/s;
c. $0.\bar{3}=\frac{1}{3}$ m/s

36. [T] $s(t)=t^2-2t$

37. [T] $s(t)=2t^3+3$

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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. [T] $s(t)=\dfrac{16}{t^2}-\dfrac{4}{t}$

39. Use the following graph to evaluate a. $f^{\prime}(1)$ and b. $f^{\prime}(6)$ .

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).

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a. $1.25$ ; b. 0.5

40. Use the following graph to evaluate a. $f^{\prime}(-3)$ and b. $f^{\prime}(1.5)$ .

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. $f(x)=x^{\frac{1}{3}}, \, x=0$

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$\underset{x\to 0^-}{\lim}\frac{x^{1/3}-0}{x-0}=\underset{x\to 0^-}{\lim}\frac{1}{x^{2/3}}=\infty$

42. $f(x)=x^{\frac{2}{3}}, \, x=0$

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

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$\underset{x\to 1^-}{\lim}\frac{1-1}{x-1}=0\ne 1=\underset{x\to 1^+}{\lim}\frac{x-1}{x-1}$

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

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

Find the average velocity of the vehicle over the following time intervals to four decimal places:
1. [4, 4.1]
2. [4, 4.01]
3. [4, 4.001]
4. [4, 4.0001]
Use a. to draw a conclusion about the instantaneous velocity of the vehicle at $t=4$ seconds.

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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.

Find the average velocity of the ball over the following time intervals:
1. [5, 5.1]
2. [5, 5.01]
3. [5, 5.001]
4. [5, 5.0001]
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).

Which vehicle has traveled farther at $t=2$ seconds?
What is the approximate velocity of each vehicle at $t=3$ seconds?
Which vehicle is traveling faster at $t=4$ seconds?
What is true about the positions of the vehicles at $t=4$ seconds?

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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 $C(x)=0.000003x^3+4x+300$ .

Calculate the average cost per jar over the following intervals:
1. [100, 100.1]
2. [100, 100.01]
3. [100, 100.001]
4. [100, 100.0001]
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.

Use a graphing calculator to graph $f$ in an appropriate viewing window.
Use the ZOOM feature on the calculator to approximate the two values of $x=a$ for which $m_{\tan}=f^{\prime}(a)=0$ .

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b. $a\approx -1.361, \, 2.694$

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

Use a graphing calculator to graph $f$ in an appropriate viewing window.
Use the ZOOM feature on the calculator to approximate the values of $x=a$ for which $m_{\tan}=f^{\prime}(a)=0$ .

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.

Find a mathematical expression for $N(x)$ .
What is $N(100)$ ? Explain the physical meaning.
What is $N^{\prime}(100)$ ? Explain the physical meaning.

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a. $N(x)=\frac{x}{30}$
b. $\sim 3.3$ 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.

Use a graphing calculator to graph $f$ in an appropriate viewing window.
Use the $\text{nDeriv}$ function, which numerically finds the derivative, on a graphing calculator to estimate $f^{\prime}(-2), \, f^{\prime}(-0.5), \, f^{\prime}(1.7)$ , and $f^{\prime}(2.718)$ .

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

Use a graphing calculator to graph $f$ in an appropriate viewing window.
Use the $\text{nDeriv}$ function on a graphing calculator to find $f^{\prime}(-4), \, f^{\prime}(-2), \, f^{\prime}(2)$ , and $f^{\prime}(4)$ .

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b. $-0.028, \, -0.16, \, 0.16, \, 0.028$

Derivatives: Supplemental Content

Problem Set: Defining the Derivative

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