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# 3.9 E Exercises

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###### Exercises $$\PageIndex{1}$$

For the following exercises, show that $$F(x)$$ are anti derivatives of $$f(x)$$.

1) $$F(x)=5x^3+2x^2+3x+1,f(x)=15x^2+4x+3$$

$$F′(x)=15x^2+4x+3$$

2) $$F(x)=x^2+4x+1,f(x)=2x+4$$

$$F′(x)=2x+4$$

3) $$F(x)=x^2e^x,f(x)=e^x(x^2+2x)$$

$$F′(x)=2xe^x+x^2e^x$$

4) $$F(x)=cosx,f(x)=−sinx$$

\F'(x)=-sinx

5) $$F(x)=e^x,f(x)=e^x$$

$$F′(x)=e^x$$

###### Exercises $$\PageIndex{2}$$

For the following exercises, find the antiderivative of the function.

1) $$f(x)=\frac{1}{x^2}+x$$

$$F(x)=-\frac{1}{3x^3}+\frac{1}{2}x^2$$

2) $$f(x)=e^x−3x^2+sinx$$

$$F(x)=e^x−x^3−cos(x)+C$$

3) $$f(x)=e^x+3x−x^2$$

$$F(x)=e^x+3x-x^2$$

4) $$f(x)=x−1+4sin(2x)$$

$$F(x)=\frac{x^2}{2}−x−2cos(2x)+C$$

###### Exercises $$\PageIndex{3}$$

For the following exercises, find the antiderivative $$F(x)$$ of each function $$f(x).$$

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

$$F(x)=x^5+\frac{2}{3}x^6+C$$

2) $$f(x)=x+12x^2$$

$$F(x)=\frac{1}{2}x^2+4x^3+C$$

3) $$f(x)=\frac{1}{\sqrt{x}}$$

$$F(x)=2\sqrt{x}+C$$

4) $$f(x)=(\sqrt{x})^3$$

$$F(x)=\frac{2}{5}(\sqrt{x})^5+C$$

5) $$f(x)=x^{1/3}+(2x)^{1/3}$$

$$F(x)=\frac{3}{4}x^{4/3} + \frac{3x^{4/3}}{4 \sqrt[3]{2}}+C$$

6) $$f(x)=\frac{x^{1/3}}{x^{2/3}}$$

$$F(x)=\frac{3}{2}x^{2/3}+C$$

7) $$f(x)=2sin(x)+sin(2x)$$

$$F(x)=-2cos(x)-\frac{1}{2}cos(2x)+C$$

8) $$f(x)=sec^2(x)+1$$

$$F(x)=x+tan(x)+C$$

9) $$f(x)=sinxcosx$$

$$F(x)=\frac{1}{2}sin^2(x)+C$$

10) $$f(x)=sin^2(x)cos(x)$$

$$F(x)=\frac{1}{3}sin^3(x)+C$$

11) $$f(x)=0$$

$$F(x)=C$$

12) $$f(x)=\frac{1}{2}csc^2(x)+\frac{1}{x^2}$$

$$F(x)=−\frac{1}{2}cot(x)−\frac{1}{x}+C$$

13) $$f(x)=cscxcotx+3x$$

$$F(x)=-csc(x)+\frac{3}{2}x^2+C$$

14) $$f(x)=4cscxcotx−secxtanx$$

$$F(x)=−secx−4cscx+C$$

15) $$f(x)=8secx(secx−4tanx)$$

$$F(x)=8tan(x)-32sec(x)+C$$

16) $$f(x)=\frac{1}{2}e^{−4x}+sinx$$

$$F(x)=−\frac{1}{8}e^{−4x}−cosx+C$$

###### Exercises $$\PageIndex{4}$$

For the following exercises, evaluate the integral.

1) $$∫(−1)dx$$

$$-x+C$$

2) $$∫sinxdx$$

$$−cosx+C$$

3) $$∫(4x+\sqrt{x})dx$$

$$2x^2+\frac{2}{3}x^{3/2}+C$$

4) $$∫\frac{3x^2+2}{x^2}dx$$

$$3x−\frac{2}{x}+C$$

5) $$∫(secxtanx+4x)dx$$

$$sec(x)+2x^2+C$$

6) $$∫(4\sqrt{x}+\sqrt[4]{x})dx$$

$$\frac{8}{3}x^{3/2}+\frac{4}{5}x^{5/4}+C$$

7) $$∫(x^{−1/3}−x^{2/3})dx$$

$$\frac{3}{2}x^{2/3}-\frac{3}{5}x^{5/3}+C$$

8) $$∫\frac{14x^3+2x+1}{x^3}dx$$

$$14x−\frac{2}{x}−\frac{1}{2x^2}+C$$

9) $$∫(e^x+e^{−x})dx$$

$$e^x-e^{-x}+C$$

###### Exercises $$\PageIndex{5}$$

For the following exercises, solve the initial value problem.

1) $$f′(x)=x^{−3},f(1)=1$$

$$f(x)=−\frac{1}{2x^2}+\frac{3}{2}$$

2) $$f′(x)=\sqrt{x}+x^2,f(0)=2$$

3) $$f′(x)=cosx+sec^2(x),f(\frac{π}{4})=2+\frac{\sqrt{2}}{2}$$

$$f(x)=sinx+tanx+1$$

4) $$f′(x)=x^3−8x^2+16x+1,f(0)=0$$

5) $$f′(x)=\frac{2}{x^2}−\frac{x^2}{2},f(1)=0$$

$$f(x)=−\frac{1}{6}x^3−\frac{2}{x}+\frac{13}{6}$$

###### Exercises $$\PageIndex{6}$$

For the following exercises, find two possible functions $$f$$ given the second- or third-order derivatives

1) $$f''(x)=x^2+2$$

2) $$f''(x)=e^{−x}$$

Answers may vary; one possible answer is $$f(x)=e^{−x}$$

3) $$f''(x)=1+x$$

4) $$f'''(x)=cosx$$

Answers may vary; one possible answer is $$f(x)=−sinx$$

5) $$f'''(x)=8e^{−2x}−sinx\ ###### Exercise \(\PageIndex{7}$$

1) A car is being driven at a rate of $$40$$ mph when the brakes are applied. The car decelerates at a constant rate of $$10$$ ft/sec2. How long before the car stops?.

2) Calculate how far the car travels in the time it takes to stop.

1. $$5.867$$ sec

###### Exercise $$\PageIndex{8}$$

1) You are merging onto the freeway, accelerating at a constant rate of $$12$$ ft/sec2. How long does it take you to reach merging speed at $$60$$ mph?

2) How far does the car travel to reach merging speed?

1. $$7.333$$ sec

###### Exercise $$\PageIndex{9}$$

A car company wants to ensure its newest model can stop in $$8$$ sec when traveling at $$75$$ mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.

$$13.75 ft/sec^2$$

###### Exercise $$\PageIndex{10}$$

A car company wants to ensure its newest model can stop in less than $$450$$ ft when traveling at $$60$$ mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.

Under Construction

###### Exercises $$\PageIndex{11}$$

For the following exercises, find the antiderivative of the function, assuming $$F(0)=0.$$

1) $$f(x)=x^2+2$$

$$F(x)=\frac{1}{3}x^3+2x$$

2) $$f(x)=4x−\sqrt{x}$$

3) $$f(x)=sinx+2x$$

$$F(x)=x^2−cosx+1$$

4) $$f(x)=e^x$$

5) $$f(x)=\frac{1}{(x+1)^2}$$

$$F(x)=−\frac{1}{(x+1)}+1$$

6) $$f(x)=e^{−2x}+3x^2\ ###### Exercises \(\PageIndex{12}$$

For the following exercises, determine whether the statement is true or false. Either prove it is true or find a counterexample if it is false.

1) If $$f(x)$$ is the antiderivative of $$v(x)$$, then $$2f(x)$$ is the antiderivative of $$2v(x).$$

True.

2) If $$f(x)$$ is the antiderivative of $$v(x)$$, then $$f(2x)$$ is the antiderivative of $$v(2x).$$

3) If $$f(x)$$ is the antiderivative of $$v(x),$$ then $$f(x)+1$$ is the antiderivative of $$v(x)+1.$$

4) If $$f(x)$$ is the antiderivative of $$v(x)$$, then $$(f(x))^2$$ is the antiderivative of $$(v(x))^2.$$