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Mathematics LibreTexts

3.9 E Exercises

  • Page ID
    10939
  • [ "stage:draft", "article:topic" ]

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    Exercises \(\PageIndex{1}\)

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

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

    Answer

    \(F′(x)=15x^2+4x+3\)

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

    Answer

    \(F′(x)=2x+4\)

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

    Answer

    \(F′(x)=2xe^x+x^2e^x\)

    4) \(F(x)=cosx,f(x)=−sinx\)

    Answer

    \F'(x)=-sinx

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

    Answer

    \(F′(x)=e^x\)

     

    Exercises \(\PageIndex{2}\)

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

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

    Answer

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

    2) \(f(x)=e^x−3x^2+sinx\)

    Answer

    \(F(x)=e^x−x^3−cos(x)+C\)

    3) \(f(x)=e^x+3x−x^2\)

    Answer

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

    4) \(f(x)=x−1+4sin(2x)\)

    Answer

    \(F(x)=\frac{x^2}{2}−x−2cos(2x)+C\)

     

    Exercises \(\PageIndex{3}\)

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

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

    Answer

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

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

    Answer

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

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

    Answer

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

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

    Answer

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

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

    Answer

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

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

    Answer

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

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

    Answer

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

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

    Answer

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

    9) \(f(x)=sinxcosx\)

    Answer

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

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

    Answer

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

    11) \(f(x)=0\)

    Answer

    \(F(x)=C\)

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

    Answer

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

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

    Answer

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

    14) \(f(x)=4cscxcotx−secxtanx\)

    Answer

    \(F(x)=−secx−4cscx+C\)

    15) \(f(x)=8secx(secx−4tanx)\)

    Answer

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

    16) \(f(x)=\frac{1}{2}e^{−4x}+sinx\)

    Answer

    \(F(x)=−\frac{1}{8}e^{−4x}−cosx+C\)

     

    Exercises \(\PageIndex{4}\)

    For the following exercises, evaluate the integral.

    1) \(∫(−1)dx\)

    Answer

    \(-x+C\)

    2) \(∫sinxdx\)

    Answer

    \(−cosx+C\)

    3) \(∫(4x+\sqrt{x})dx\)

    Answer:

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

    4) \(∫\frac{3x^2+2}{x^2}dx\)

    Answer

    \(3x−\frac{2}{x}+C\)

    5) \(∫(secxtanx+4x)dx\)

    Answer

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

    6) \(∫(4\sqrt{x}+\sqrt[4]{x})dx\)

    Answer

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

    7) \(∫(x^{−1/3}−x^{2/3})dx\)

    Answer

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

    8) \(∫\frac{14x^3+2x+1}{x^3}dx\)

    Answer

    \(14x−\frac{2}{x}−\frac{1}{2x^2}+C\)

    9) \(∫(e^x+e^{−x})dx\)

    Answer

    \(e^x-e^{-x}+C\)

     

    Exercises \(\PageIndex{5}\)

    For the following exercises, solve the initial value problem.

    1) \(f′(x)=x^{−3},f(1)=1\)

    Answer

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

    2) \(f′(x)=\sqrt{x}+x^2,f(0)=2\)

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

    Answer

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

    4) \(f′(x)=x^3−8x^2+16x+1,f(0)=0\)

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

    Answer

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

     

    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) \(f''(x)=e^{−x}\)

    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.