1
Let \(f(x)\) be a function with derivative \(f'(x)\text{.}\) What is the most general antiderivative of \(f'(x)\text{?}\)
Subsection 4.1.2 Exercises
Exercises — Stage 1
2
On the graph below, the black curve is \(y=f(x)\text{.}\) Which of the coloured curves is an antiderivative of \(f(x)\text{?}\)
Exercises — Stage 2
In Questions 4.1.2.3 through 4.1.2.12, you are asked to find the antiderivative of a function. Phrased like this, we mean the most general antiderivative. These will all include some added constant. The table after Example 4.1.3 might be of help.
In Questions 4.1.2.13 through 4.1.2.16, you are asked to find a specific antiderivative of a function. In this case, you should be able to solve for the entire function--no unknown constants floating about.
In Questions 4.1.2.17 through 4.1.2.19, we will explore some simple situations where antiderivatives might arise.
3
Find the antiderivative of \(f(x)=3x^2+5x^4+10x-9\text{.}\)
4
Find the antiderivative of \(f(x)=\dfrac{3}{5}x^7-18x^4+x\text{.}\)
5
Find the antiderivative of \(f(x)=4\sqrt[3]{x}-\dfrac{9}{2x^{2.7}}\text{.}\)
6
Find the antiderivative of \(f(x)=\dfrac{1}{7\sqrt{x}}\text{.}\)
7
Find the antiderivative of \(f(x)=e^{5x+11}\text{.}\)
8
Find the antiderivative of \(f(x)=3\sin(5x)+7\cos(13x)\text{.}\)
9
Find the antiderivative of \(f(x)=\sec^2(x+1)\text{.}\)
10
Find the antiderivative of \(f(x)=\dfrac{1}{x+2}\text{.}\)
11
Find the antiderivative of \(f(x)=\dfrac{7}{\sqrt{3-3x^2}}\text{.}\)
12
Find the antiderivative of \(f(x)=\dfrac{1}{1+25x^2}\)
13
Find the function \(f(x)\) with \(f'(x)=3x^2-9x+4\) and \(f(1)=10\text{.}\)
14
Find the function \(f(x)\) with \(f'(x)=\cos(2x)\) and \(f(\pi)=\pi\text{.}\)
15
Find the function \(f(x)\) with \(f'(x)=\dfrac{1}{x}\) and \(f(-1)=0\text{.}\)
16
Find the function \(f(x)\) with \(f'(x)=\dfrac{1}{\sqrt{1-x^2}}+1\) and \(f(1)=-\dfrac{\pi}{2}\text{.}\)
17
Suppose a population of bacteria at time \(t\) (measured in hours) is growing at a rate of \(100e^{2t}\) individuals per hour. Starting at time \(t=0\text{,}\) how long will it take the initial colony to increase by 300 individuals?
18
Your bank account at time \(t\) (measured in years) is growing at a rate of
\begin{equation*}
1500e^{\tfrac{t}{50}}
\end{equation*}
dollars per year. How much money is in your account at time \(t\text{?}\)
19
At time \(t\) during a particular day, \(0 \leq t \leq 24\text{,}\) your house consumes energy at a rate of
\begin{equation*}
0.5\sin\left(\frac{\pi}{24}t\right)+0.25
\end{equation*}
kW. (Your consumption was smallest in the middle of the night, and peaked at noon.) How much energy did your house consume in that day?
Exercises — Stage 3
For Questions 4.1.2.21 through 4.1.2.26, you are again asked to find the antiderivatives of certain functions. In general, finding antiderivatives can be extremely difficult--indeed, it will form the main topic of next semester's calculus course. However, you can work out the antiderivatives of the functions below using what you've learned so far about derivatives.
20 (✳)
Let \(f(x)=2\sin^{-1}\sqrt{x}\) and \(g(x)=\sin^{-1}(2x-1)\text{.}\) Find the derivative of \(f(x)-g(x)\) and simplify your answer. What does the answer imply about the relation between \(f(x)\) and \(g(x)\text{?}\)
21
Find the antiderivative of \(f(x)=2\cos(2x)\cos(3x)-3\sin(2x)\sin(3x)\text{.}\)
22
Find the antiderivative of \(f(x)=\dfrac{(x^2+1)e^x-e^x(2x)}{(x^2+1)^2}\text{.}\)
23
Find the antiderivative of \(f(x)=3x^2e^{x^3}\text{.}\)
24
Find the antiderivative of \(f(x)=5x\sin(x^2)\text{.}\)
25
Find the antiderivative of \(f(x)=e^{\log x}\text{.}\)
26
Find the antiderivative of \(f(x)=\dfrac{7}{\sqrt{3-x^2}}\text{.}\)
27
Imagine forming a solid by revolving the parabola \(y=x^2+1\) around the \(x\)-axis, as in the picture below.
Use the method of Example 4.1.7 to find the volume of such an object if the segment of the parabola that we rotate runs from \(x=-H\) to \(x=H\text{.}\)