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# 3.1E: Definition of the Derivative (Exercises)

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## 3.1E: Definition of the Derivative Exercises

For the following exercises, find the slope of the secant line between the values $$x_1$$ and $$x_2$$ for each function $$y=f(x)$$.

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

$$4$$

2) $$f(x)=8x−3;x_1=−1,x_2=3$$

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

$$8.5$$

4) $$f(x)=−x^2+x+2;x_1=0.5,x_2=1.5$$

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

$$−\frac{3}{4}$$

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

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

$$0.2$$

8) $$f(x)=\sqrt{x−9};x_1=10,x_2=13$$

9) $$f(x)=x1/3+1;x_1=0,x_2=8$$

0.25

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

For the following functions,

a. find $$f′(x)$$

b. find the slope of the tangent line $$m_{tan}=f′(a)$$

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

11) $$f(x)=3−4x,a=2$$

$$a. f′(x)=-4$$ $$b. -4$$ $$c. y=3−4x$$

12) $$f(x)=\frac{x}{5}+6,a=−1$$

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

$$a. f′(x)=2x+1$$ $$b. 3$$ $$c. y=3x−1$$

14) $$f(x)=1−x−x^2, a=0$$

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

$$a. f′(x)=-\frac{7}{x^2}$$ $$b. -\frac{7}{9}$$ $$c. y=-\frac{7}{9}x+\frac{14}{3}$$

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

17) $$f(x)=2−3x^2,a=−2$$

$$a. f′(x)=-6x$$ $$b. 12$$ $$c. y=12x+14$$

18) $$f(x)=-\frac{3}{x−1},a=4$$

19) $$f(x)=\frac{2}{x+3},a=−4$$

$$a. f'(x)=-\frac{2}{(x+3)^2}$$ $$b. −2$$ $$c. y=−2x−10$$

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

For the following functions $$y=f(x)$$, (a) find $$f′(x)$$ (b) find the equation of the tangent line at $$a$$

21) $$f(x)=5x+4,a=−1$$

(a) $$f′(x)=5$$ (b) $$y+1=5(x+1) \mbox{ or } y=5x+4$$

22) $$f(x)=−7x+1,a=3$$

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

(a) $$f′(x)=2x+9$$ (b) $$y-22=13(x-2) \mbox{ or } y=13x-4$$

24) $$f(x)=3x^2−x+2,a=1$$

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

(a) $$f′(x)=\frac{1}{2\sqrt{x}}$$ (b) $$y-2=\frac{1}{4}(x-4)$$

26) $$f(x)=\sqrt{x−2},a=6$$

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

(a) $$f′(x)=-\frac{1}{x^2}$$ (b) $$y-\frac{1}{2}=-\frac{1}{4}(x-2)$$

28) $$f(x)=\frac{1}{x−3},a=−1$$

For the following functions $$y=f(x)$$, find $$f′(a)$$

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

$$−3$$

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

For the following exercises, use the definition of a derivative to find $$f′(x)$$.

J3.1.1) $$f(x)=4x^2$$

$$8x$$

there is no J1.3.2

J3.1.3) $$f(x)=\frac{1}{\sqrt{x}}$$

$$\frac{−1}{2x^{3/2}}$$

J3.1.4) $$f(x)=5x−x^2$$

J3.1.5) $$f(x)=\sqrt{2x}$$

$$\frac{1}{\sqrt{2x}}$$

there is no J1.3.6

J3.1.7) $$f(x)=\frac{9}{x}$$

$$\frac{−9}{x^2}$$

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] $$f(x)=x^2+3x+4$$, $$P(1,8)$$ (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. $$m_{tan}=5$$

c. $$y=5x+3$$

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

 $$x$$ $$Slope m_{PQ}$$ $$x$$ $$Slope m_{PQ}$$ 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)

$$a. (i)4.8771, (ii)4.9875(iii)4.9988, (iv)4.9999, (v)4.9999, (vi)4.9999$$

b. $$m_{tan}=5$$

c. $$y=5x+10$$

34) [T] $$f(x)=tan(x)$$, $$P(π,0)$$

 $$x$$ $$Slope m_{PQ}$$ 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. $$(i)0.3$$ m/s, $$(ii)0.3$$ m/s, $$(iii)0.3$$ m/s, $$(iv)0.3$$ m/s; c. $$0.3=13$$ 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) $$s(t)=\frac{16}{t^2}−\frac{4}{t}$$

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

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

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

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

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

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

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

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.

Solution: 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 $$f(x)=\frac{x}{1+x^2}$$, 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 $$m_{tan}=f′(a)=0$$.

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. $$N(x)=\frac{x}{30}$$

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 $$f(x)=\frac{x^2}{x^2+1}$$, 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 $$f′(4)$$. b. $$−0.028,−0.16,0.16,0.028$$