Section 3.9, Antiderivatives
Definition
A function $F$ is called an antiderivative of a function $f$ on an interval if $F’(x) = f(x)$ for all $x$ on the interval.
Examples:
Consider the following three three derivatives:
$\frac{d}{dx} (x^2 + 2) = 2x$ $\frac{d}{dx} (x^2 - 100) = 2x$ $\frac{d}{dx} (x^2 - 42) = 2x$
Each of the functions $x^2 + 2$, $x^2 - 100$ and $x^2 - 42$ are antiderivatives of the function $2x$.
Example
Let’s find an antiderivative of $$ f(x) = 5x^4 + 3x^2 + 1 $$
Try it first (reverse power rule) then check your answer:
Click to reveal the answer.
An answer is $F(x) = x^5 + x^3 + x + c$ where $c$ is some constant number.We call the most general antiderivative of $f$ on an interval $F(x) + c$ where $F$ is an antiderivative and $c$ is an arbitrary constant.
Examples
Find the most general antiderivative of:
- $g(x) = 2x^3 + 4x$
- $f(x) = 2 \cos x$
- $k(x) = \sec x \tan x$
- $w(x) = 3\sqrt{x} - 2 \sqrt[3]{x}$
Particular Antiderivatives:
| Function | Particular Antiderivative |
|---|---|
| $cf(x)$ | $cF(x)$ |
| $f(x) + g(x)$ | $F(x) + G(x) $ |
| $x^n \hspace{1em} (n \ne 1)$ | $\dfrac{x^{n+1}}{n+1}$ |
| $\cos x $ | $\sin x $ |
| $\sin x $ | $-\cos x $ |
| $\sec^2 x$ | $\tan x $ |
| $\sec x \tan x$ | $\sec x $ |
Example
Find all functions $g$ for which $$ g’(x) = 5 \cos x + \dfrac{2x^4 - \sqrt{x}}{x} $$
Differential Equations
If we know information about the derivative of a function (for example, we know how data is changing over time), we can find information about the function itself. An equation that involves derivatives of a function is called a differential equation and many of you will be taking an entire course on the subject. Here we’ll start with some small examples.
Example
Find $f$ if $f’(x) = x\sqrt{x}$ and $f(1) = 2$
Example
Find $f$ if $f’’(x) = 12x^2 + 6x - 4$, $f(0) = 4$ and $f(1) = 1$.
Gravity Example
For examples which involve things being dropped or thrown upward, we’ll use the gravitational constant $g = 9.8$ m/s $^2$ in metric units or $g = 32$ ft/s $^2$ for imperial measurement. We will conventionally make it negative to denote acceleration down to Earth.
Here’s an example:
A ball is thrown upward at a speed of 48 ft/s from the edge of a cliff 432 ft above the ground. Find its height above the ground $t$ seconds later. When does it reach its maximum height? When does it hit the ground?