4.1.2: Homework

Reading Questions

1. What is an antiderivative of a function \(f(x)\)?
2. If \(F(x)\) is an antiderivative of \(f(x)\), what is the most general form of an antiderivative of \(f(x)\)? What is the term for the arbitrary constant added?
3. What does the notation \(\int f(x) \, dx\) represent? Identify the integrand and the variable of integration.
4. State the Power Rule for Integrals. For which value of \(n\) does this rule not apply? What is the antiderivative in that specific case?
5. If you know the antiderivatives of \(f(x)\) and \(g(x)\), how do you find the antiderivative of \(f(x) + g(x)\)?
6. If you know the antiderivative of \(f(x)\), how do you find the antiderivative of \(k \cdot f(x)\), where \(k\) is a constant?
7. What is a differential equation?
8. What is an initial-value problem? What two components does it typically consist of?
9. How is the constant of integration \(C\) determined when solving an initial-value problem?
10. If the acceleration of an object \(a(t)\) is given, how can you find its velocity function \(v(t)\)? How can you then find its position function \(s(t)\)?

Homework

In exercises 1 - 20, find the antiderivative \(F(x)\) of each function \(f(x)\).

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

2) \(f(x)=e^x−3x^2+\sin x\)

Answer

\(F(x)=e^x−x^3−\cos x+C\)

3) \(f(x)=e^x+3x−x^2\)

4) \(f(x)=x−1+4\sin(2x)\)

Answer

\(F(x)=\dfrac{x^2}{2}−x−2\cos(2x)+C\)

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

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

Answer

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

7) \(f(x)=\dfrac{1}{\sqrt{x}}\)

8) \(f(x)=\left(\sqrt{x}\right)^3\)

Answer

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

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

10) \(f(x)=\dfrac{x^{1/3}}{x^{2/3}}\)

Answer

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

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

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

Answer

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

13) \(f(x)=\sin x\cos x\)

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

Answer

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

15) \(f(x)=0\)

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

Answer

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

17) \(f(x)=\csc x\cot x+3x\)

18) \(f(x)=4\csc x\cot x−\sec x\tan x\)

Answer

\(F(x)=−\sec x−4\csc x+C\)

19) \(f(x)=8(\sec x)\big(\sec x−4\tan x\big)\)

20) \(f(x) = \cosh(x)+\sinh(x)\)

Answer

\(F(x) = \cosh(x)+\sinh(x)+C\)

For exercises 21 - 31, evaluate the integral.

21) \(\displaystyle \int (−1)\,dx\)

22) \(\displaystyle \int \sin x\,dx\)

Answer

\(\displaystyle \int \sin x\,dx = −\cos x+C\)

23) \(\displaystyle \int \big(4x+\sqrt{x}\big)\,dx\)

24) \(\displaystyle \int \frac{3x^2+2}{x^2}\,dx\)

Answer

\(\displaystyle \int \frac{3x^2+2}{x^2}\,dx=3x−\frac{2}{x}+C\)

25) \(\displaystyle \int \big(\sec x\tan x+4x\big)\,dx\)

26) \(\displaystyle \int \big(4\sqrt{x}+\sqrt[4]{x}\big)\,dx\)

Answer

\(\displaystyle \int \big(4\sqrt{x}+\sqrt[4]{x}\big)\,dx=\frac{8}{3}x^{3/2}+\frac{4}{5}x^{5/4}+C\)

27) \(\displaystyle \int \left(x^{−1/3}−x^{2/3}\right)\,dx\)

28) \(\displaystyle \int \frac{14x^3+2x+1}{x^3}\,dx\)

Answer

\(\displaystyle \int \frac{14x^3+2x+1}{x^3}\,dx=14x−\frac{2}{x}−\frac{1}{2x^2}+C\)

29) \(\displaystyle \int \big(e^x+e^{−x}\big)\,dx\)

30) \(\displaystyle \int \frac{dx}{\sqrt{x^2+1}}\)

31) \(\displaystyle \int −\frac{dx}{x\sqrt{1−x^2}}\)

In exercises 32 - 36, solve the initial value problem.

32) \(f^{\prime}(x)=x^{−3},\quad f(1)=1\)

Answer

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

33) \(f^{\prime}(x)=\sqrt{x}+x^2,\quad f(0)=2\)

34) \(f^{\prime}(x)=\cos x+\sec^2(x),\quad f(\frac{ \pi }{4})=2+\frac{\sqrt{2}}{2}\)

Answer

\(f(x)=\sin x+\tan x+1\)

35) \(f^{\prime}(x)=x^3−8x^2+16x+1,\quad f(0)=0\)

36) \(f^{\prime}(x)=\dfrac{2}{x^2}−\dfrac{x^2}{2},\quad f(1)=0\)

Answer

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

In exercises 37 - 40, find two possible functions \(f\) given the second- or third-order derivatives

37) \(f^{\prime\prime}(x)=x^2+2\)

38) \(f^{\prime\prime}(x)=e^{−x}\)

Answer

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

39) \(f^{\prime\prime}(x)=1+x\)

40) \(f^{\prime\prime\prime}(x)=\cos x\)

Answer

Answers may vary; one possible answer is \(f(x)=−\sin x\)

41) A car is being driven at a rate of \(40\) mph when the brakes are applied. The car decelerates at a constant rate of \(10\, \text{ft/sec}^2\). How long before the car stops?

Answer

\(5.867\) sec

42) In the preceding problem, calculate how far the car travels in the time it takes to stop.

43) You are merging onto the freeway, accelerating at a constant rate of \(12\, \text{ft/sec}^2\). How long does it take you to reach merging speed at \(60\) mph?

Answer

\(7.333\) sec

44) Based on the previous problem, how far does the car travel to reach merging speed?

45) 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

\(13.75\, \text{ft/sec}^2\)

46) 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.

In exercises 47 - 51, find the antiderivative of the function, assuming \(F(0)=0\).

47) [Technology Required] \(\quad f(x)=x^2+2\)

Answer

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

48) [Technology Required] \(\quad f(x)=4x−\sqrt{x}\)

49) [Technology Required] \(\quad f(x)=\sin x+2x\)

Answer

\(F(x)=x^2−\cos x+1\)

50) [Technology Required] \(\quad f(x)=e^x\)

51) [Technology Required] \(\quad f(x)=\dfrac{1}{(x+1)^2}\)

Answer

\(F(x)=−\dfrac{1}{x+1}+1\)

In exercises 52 - 55, determine whether the statement is true or false. Either prove it is true or find a counterexample if it is false.

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

Answer

True

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

54) If \(f(x)\) is the antiderivative of \(v(x)\), then \(f(x)+1\) is the antiderivative of \(v(x)+1\).

Answer

False

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

In exercises 43 - 45, use the fact that a falling body with friction equal to velocity squared obeys the equation \(\dfrac{dv}{dt}=g−v^2\).

43) Show that \(v(t)=\sqrt{g}\tanh(\sqrt{g}t)\) satisfies this equation.

Answer

Answers may vary

44) Derive the previous expression for \(v(t)\) by integrating \(\dfrac{dv}{g−v^2}=dt\).

45) [Technology Required] Estimate how far a body has fallen in \(12\)seconds by finding the area underneath the curve of \(v(t)\).

Answer

\(37.30\)

In exercises 46 - 48, use this scenario: A cable hanging under its own weight has a slope \(S=\dfrac{dy}{dx}\) that satisfies \(\dfrac{dS}{dx}=c\sqrt{1+S^2}\). The constant \(c\) is the ratio of cable density to tension.

46) Show that \(S=\sinh(cx)\) satisfies this equation.

47) Integrate \(\dfrac{dy}{dx}=\sinh(cx)\) to find the cable height \(y(x)\) if \(y(0)=1/c\).

Answer

\(y=\frac{1}{c}\cosh(cx)\)

48) Sketch the cable and determine how far down it sags at \(x=0\).