106 lines
6.7 KiB
Markdown
106 lines
6.7 KiB
Markdown
# Antiderivatives
|
|
An antiderivative is useful when you know the rate of change, and you want to find a point from that rate of change
|
|
|
|
> A function $F$ is said to be an *antiderivative* of $f$ if $F'(x) = f(x)$
|
|
## Notation
|
|
The collection of all antiderivatives of a function $f$ is referred to as the *indefinite integral of $f$ with respect to $x$*, and is denoted by:
|
|
$$ \int f(x) dx $$
|
|
## Examples
|
|
> Find the antiderivative of the function $y = x^2$
|
|
|
|
1. We know that to find the derivative of the above function, you'd multiply by the exponent ($2$), and subtract 1 from the exponent.
|
|
2. To perform this operation in reverse:
|
|
1. Add 1 to the exponent
|
|
2. Multiply by $\dfrac{1}{n + 1}$
|
|
3. This gives us an antiderivative of $\dfrac{1}{3}x^3$
|
|
4. To check our work, work backwards.
|
|
5. The derivative of $\dfrac{1}{3}x^3$ is $\dfrac{1}{3} (3x^2)$
|
|
6. $= \dfrac{3}{3} x^2$
|
|
|
|
|
|
## Formulas
|
|
|
|
| Differentiation Formula | Integration Formula |
|
|
| ----------------------------------------------------- | -------------------------------------------------------- |
|
|
| $\dfrac{d}{dx} x^n = nx^{x-1}$ | $\int x^n dx = \dfrac{1}{n+1}x^{n+1}+ C$ for $n \ne -1$ |
|
|
| $\dfrac{d}{dx} kx = k$ | $\int k \space dx = kx + C$ |
|
|
| $\dfrac{d}{dx} \ln \|x\| = \dfrac{1}{x}$ | <br>$\int \dfrac{1}{x}dx = \ln \|x\| + C$ |
|
|
| $\dfrac{d}{dx} e^x = e^x$ | <br>$\int e^x dx = e^x + C$ |
|
|
| $\dfrac{d}{dx} a^x = (\ln{a}) a^x$ | $\int a^xdx = \ln \|x\| + C$ |
|
|
| $\dfrac{d}{dx} \sin x = \cos x$ | $\int \cos(x) dx = \sin (x) + C$ |
|
|
| $\dfrac{d}{dx} \cos x = -\sin x$ | $\int \sin(x)dx = \sin x + C$ |
|
|
| $\dfrac{d}{dx} \tan{x} = \sec^2 x$ | $\int \sec^2(x)dx = \tan(x) + C$ |
|
|
| $\dfrac{d}{dx} \sec x = \sec x \tan x$ | $\int sec^2(x) dx = \sec(x) + C$ |
|
|
| $\dfrac{d}{dx} \sin^{-1} x = \dfrac{1}{\sqrt{1-x^2}}$ | $\int \sec(x) \tan(x) dx = \sec x + C$ |
|
|
| $\dfrac{d}{dx} \tan^{-1} x = \dfrac{1}{1+x^2}$ | $\int \dfrac{1}{\sqrt{1+x^2}}dx = \tan^{-1}x + C$ |
|
|
| $\dfrac{d}{dx} k f(x) = k f'(x)$ | $\int k*f(x)dx = k\int f(x)dx$ |
|
|
| $\dfrac{d}{dx} f(x) \pm g(x) = f'(x) \pm g'(x)$ | $\int (f(x) \pm g(x))dx = \int f(x) dx \pm \int g(x) dx$ |
|
|
# Area Under a Curve
|
|
The area under the curve $y = f(x)$ can be approximated by the equation $\sum_{i = 1}^n f(\hat{x_i})\Delta x$ where $\hat{x_i}$ is any point on the interval $[x_{i - 1}, x_i]$, and the curve is divided into $n$ equal parts of width $\Delta x$
|
|
|
|
Any sum of this form is referred to as a Reimann Sum.
|
|
|
|
To summarize:
|
|
- The area under a curve is equal to the sum of the area of $n$ rectangular subdivisions where each rectangle has a width of $\Delta x$ and a height of $f(x)$.
|
|
# Definite Integrals
|
|
Let $f$ be a continuous function on the interval $[a, b]$. Divide $[a, b]$ into $n$ equal parts of width $\Delta x = \dfrac{b - a}{n}$ . Let $x_0, x_1, x_2, \cdots, x_3$ be the endpoints of the subdivision.
|
|
|
|
The definite integral of $f(x)$ with respect to $x$ from $x = a$ to $x = b$ can be denoted:
|
|
$$ \int_{a}^b f(x) dx $$
|
|
|
|
And __can__ be defined as:
|
|
$$ \int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i = 1}^n f(x_i)\Delta x$$
|
|
|
|
$f(x_i)$ is the *height* of each sub-interval, and $\Delta x$ is the change in the *x* interval, so $f(x_i) \Delta x$ is solving for the area of each sub-interval.
|
|
|
|
- If your function is always positive, then the value of a definite integral is the area under the curve.
|
|
- If the function is always negative, then the value of a definite integral is the area above the curve to zero.
|
|
- If the function has both positive and negative values, the output is equal to the area above the curve minus the area below the curve.
|
|
|
|
## Examples
|
|
> Find the exact value of the integral $\int_0^1 5x \space dx$
|
|
|
|
Relevant formulas:
|
|
$$ \sum_{i = 1}^n = \dfrac{(n)(n + 1)}{2} $$
|
|
$$ \Delta x = \dfrac{1 - 0}{n} = \dfrac{1}{n}$$$$ x_i = 0 + \Delta xi + \dfrac{1}{n} \cdot i$$
|
|
1. $\int_0^1 5x \space dx = \lim_{n \to \infty} \sum_{i=1}^n 5(x_i) \cdot \Delta x$
|
|
2. $= \lim_{n \to \infty} \sum_{i=1}^n 5(\frac{1}{n} \cdot i) \cdot \frac{1}{n}$
|
|
3. $= \lim_{n \to \infty} \sum_{i = 1}^n \dfrac{5}{n^2}\cdot i$
|
|
4. $= \lim_{n \to \infty} \dfrac{5}{n^2} \sum_{i = 1}^n i$
|
|
5. $= \lim_{x \to \infty} \dfrac{5}{n^2} \cdot \dfrac{n(n + 1)}{2}$
|
|
6. $= \lim_{n \to \infty} \dfrac{5n^2 + 5n}{2n^2}$
|
|
7. $= \dfrac{5}{2}$
|
|
|
|
# Properties of Integrals
|
|
1. $\int_a^a f(x)dx = 0$ - An integral with a domain of zero will always evaluate to zero.
|
|
2. $\int_b^a f(x)dx = -\int_a^b f(x) dx$ - The integral from $a \to b$ is equal to the integral from $-(b\to a)$
|
|
3. $\int_a^b cf(x) dx = c \int_a^b f(x) dx$ - A constant from inside of an integral can be moved outside of an integral
|
|
4. $\int_a^b f(x) \pm g(x) dx = \int_a^b f(x) dx \pm \int_a^b g(x)dx$ - Integrals can be distributed
|
|
5. $\int_a^c f(x)dx = \int_a^b f(x)dx + \int_b^c f(x)dx$ - An integral can be split into two smaller integrals covering the same domain, added together.
|
|
|
|
# Averages
|
|
To find the average value of $f(x)$ on the interval $[a, b]$ is given by the formula:
|
|
|
|
Average = $\dfrac{1}{b-a} \int_a^b f(x)dx$
|
|
|
|
# The Fundamental Theorem of Calculus
|
|
1. Let $f$ be a continuous function on the closed interval $[a, b]$ and let $F$ be any antiderivative of $f$, then:
|
|
$$\int_a^b f(x) dx = F(b) - F(a)$$
|
|
2. Let $f$ be a continuous function on $[a, b]$ and let $x$ be a point in $[a, b]$.
|
|
$$ F(x) = \int_a^x f(t)dt \Rightarrow F'(x) = f(x) $$
|
|
$$ \dfrac{d}{dx} \int_a^{g(x)} f(t) dt = f(g(x)) * g'(x)* $$
|
|
## Examples
|
|
> Finding the derivative of an integral
|
|
$$ \dfrac{d}{dx} \int_2^{7x} \cos(t^2) dt = cos((7x)^2) * 7 = 7\cos(49x^2)$$
|
|
> Finding the derivative of an integral
|
|
$$ \dfrac{d}{dx}\int_0^{\ln{x}}\tan(t) = \tan(\ln(x))*\dfrac{1}{x} $$
|
|
> $x$ and $t$ notation *(note: the bar notation is referred to as "evaluated at")*
|
|
$$ F(x) = \int_4^x 2t \space dt = t^2 \Big|_4^x = x^2 - 16$$
|
|
> $x$ in top and bottom
|
|
$$ \dfrac{d}{dx} \int_{2x}^{3x} \sin(t) dt = \dfrac{d}{dx} -\cos(t)\Big|_{2x}^{3x} = \dfrac{d}{dx} (-\cos(3x) + cos(2x) = 3\sin(3x) - 2\sin(2x) $$
|
|
|
|
# The Mean Value Theorem for Integrals
|
|
If $f(x)$ is continuous over an interval $[a, b]$ then there is at least one point $c$ in the interval $[a, b]$ such that:
|
|
$$f(c) = \dfrac{1}{b-a}\int_a^bf(x)dx $$
|
|
This formula can also be stated as $\int_a^b f(x)dx = f(c)(b-a)$
|
|
This theorem tells us that a continuous function on the closed interval will obtain its average for at least one point in the interval. |