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4.10E: Exercises

This page is a draft and is under active development. 

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Exercises 4.10.1

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

1) F(x)=5x3+2x2+3x+1,f(x)=15x2+4x+3

Answer

F(x)=15x2+4x+3

2) F(x)=x2+4x+1,f(x)=2x+4

Answer

F(x)=2x+4

3) F(x)=x2ex,f(x)=ex(x2+2x)

Answer

F(x)=2xex+x2ex

4) F(x)=cosx,f(x)=sinx

Answer

\F'(x)=-sinx

5) F(x)=ex,f(x)=ex

Answer

F(x)=ex

Exercises 4.10.2

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

1) f(x)=1x2+x

Answer

F(x)=13x3+12x2

2) f(x)=ex3x2+sinx

Answer

F(x)=exx3cos(x)+C

3) f(x)=ex+3xx2

Answer

F(x)=ex+3xx2

4) f(x)=x1+4sin(2x)

Answer

F(x)=x22x2cos(2x)+C

Exercises 4.10.3

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

1) f(x)=5x4+4x5

Answer

F(x)=x5+23x6+C

2) f(x)=x+12x2

Answer

F(x)=12x2+4x3+C

3) f(x)=1x

Answer

F(x)=2x+C

4) f(x)=(x)3

Answer

F(x)=25(x)5+C

5) f(x)=x1/3+(2x)1/3

Answer

F(x)=34x4/3+3x4/3432+C

6) f(x)=x1/3x2/3

Answer

F(x)=32x2/3+C

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

Answer

F(x)=2cos(x)12cos(2x)+C

8) f(x)=sec2(x)+1

Answer

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

9) f(x)=sinxcosx

Answer

F(x)=12sin2(x)+C

10) f(x)=sin2(x)cos(x)

Answer

F(x)=13sin3(x)+C

11) f(x)=0

Answer

F(x)=C

12) f(x)=12csc2(x)+1x2

Answer

F(x)=12cot(x)1x+C

13) f(x)=cscxcotx+3x

Answer

F(x)=csc(x)+32x2+C

14) f(x)=4cscxcotxsecxtanx

Answer

F(x)=secx4cscx+C

15) f(x)=8secx(secx4tanx)

Answer

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

16) f(x)=12e4x+sinx

Answer

F(x)=18e4xcosx+C

Exercises 4.10.4

For the following exercises, evaluate the integral.

1) (1)dx

Answer

x+C

2) sinxdx

Answer

cosx+C

3) (4x+x)dx

Answer

2x2+23x3/2+C

4) 3x2+2x2dx

Answer

3x2x+C

5) (secxtanx+4x)dx

Answer

sec(x)+2x2+C

6) (4x+4x)dx

Answer

83x3/2+45x5/4+C

7) (x1/3x2/3)dx

Answer

32x2/335x5/3+C

8) 14x3+2x+1x3dx

Answer

14x2x12x2+C

9) (ex+ex)dx

Answer

exex+C

Exercises 4.10.5

For the following exercises, solve the initial value problem.

1) f(x)=x3,f(1)=1

Answer

f(x)=12x2+32

2) f(x)=x+x2,f(0)=2

3) f(x)=cosx+sec2(x),f(π4)=2+22

Answer

f(x)=sinx+tanx+1

4) f(x)=x38x2+16x+1,f(0)=0

5) f(x)=2x2x22,f(1)=0

Answer

f(x)=16x32x+136

Exercises 4.10.6

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

1) f

2) f''(x)=e^{−x}

Answer

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

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

4) f'''(x)=cosx

Answer

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.

Answer

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?

Answer

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.

Answer

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.

Answer

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

Answer

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

2) f(x)=4x−\sqrt{x}

3) f(x)=sinx+2x

Answer

F(x)=x^2−cosx+1

4) f(x)=e^x

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

Answer

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).

Answer

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.

Answer

False.

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

Answer

False.

Contributors

  • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.


4.10E: Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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