# Calculus I

The $\frac{0}{0}$ form is the basis of differentiation; the $\infty \times 0$ form is the basis of integration.

## Limits and continuity

The limit of a function is the value that $f(x)$ gets closer to as $x$ approaches some number.

$\lim\limits_{x \to a} f(x)$

"The limit of the function $f$ as $x$ approaches $a$." The symbol $\lim$ means we are taking the limit of something. The expression to the right is the expression we are taking the limit of. The expression $x \to a$ below $\lim$ means we are taking the limit of $f$ as values of $x$ approach $a$.

### Approach

As values increase towards $a$, this is called approaching from the left (notated $x \to a^{-}$).

As values decrease towards $a$, this is called approaching from the right (notated $x \to a^{+}$).

When a limit doesn't approach the same value from both sides, we say that the limit doesn't exist.

### Limit theorems

#### Arithmetic

The limit of a sum is the sum of the individual limits.

The limit of a difference is the difference of the individual limits.

The limit of a product is the product of the individual limits.

The limit of a quotient is the quotient of the individual limits, provided each individual limit exists and the limit of the denominator is nonzero.

### Two special limits

• $\lim\limits_{\theta \to 0} \frac{\sin(\theta)}{\theta} = 1$
• $\lim\limits_{\theta \to 0} \frac{\cos(\theta)}{\theta} = 0$

## Derivatives

### Definition of derivative

$f^{\prime}(x) = \lim\limits_{h \to 0} \frac{f(x + h) - f(x)}{h}$

### Derivatives of trig functions

• $\frac{d}{dx} \sin(x) = \cos(x)$
• $\frac{d}{dx} \cos(x) = -\sin(x)$
• $\frac{d}{dx} \tan(x) = \sec^2(x)$
• $\frac{d}{dx} \cot(x) = -\csc^2(x)$
• $\frac{d}{dx} \sec(x) = \sec(x)\tan(x)$
• $\frac{d}{dx} \csc(x) = -\csc(x)\cot(x)$

### Derivatives of inverse trig functions

• $\frac{d}{dx} \sin^{-1}(u) = \frac{du/dx}{\sqrt{1 - u^2}}$
• $\frac{d}{dx} \cos^{-1}(u) = \frac{-du/dx}{\sqrt{1 - u^2}}$
• $\frac{d}{dx} \tan^{-1}(u) = \frac{du/dx}{1 + u^2}$
• $\frac{d}{dx} \cot^{-1}(u) = \frac{-du/dx}{1 + u^2}$
• $\frac{d}{dx} \sec^{-1}(u) = \frac{du/dx}{\lvert u \rvert \sqrt{u^2 - 1}}$
• $\frac{d}{dx} \csc^{-1}(u) = \frac{-du/dx}{\lvert u \rvert \sqrt{u^2 - 1}}$

### Derivatives of exponential functions

General rule for the derivative of exponential function: write it down x derivative of the exponent x natural log of the base

• $\frac{d}{dx} 8e^{2x} = 8e^{2x} \cdot 2 \cdot \ln(e)$

### Derivatives of logarithmic functions

General rule for the derivative of natural log functions: 1 over the object x derivative of the object

• $\frac{d}{dx} \ln(u) = \frac{1}{u} \cdot \frac{du}{dx}$
• $\frac{d}{dx} \log_{a} x$ use the base-changing formula first

## Integrals

### Trigonometric power reduction identities

• $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$
• $\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$

### Integration of logarithmic trigonometric functions

• $\int \tan(\theta) dx = \int \frac{\sin(\theta)}{\cos(\theta)} dx$
• $\int \sec(\theta) dx \cdot \frac{\sec(\theta) + \tan(\theta)}{\sec(\theta) + \tan(\theta)}$
• $\int \csc(\theta) dx \cdot \frac{\csc(\theta) + \cot(\theta)}{\csc(\theta) + \cot(\theta)}$

### Integration resulting in inverse trionometric functions

#### Essential formulas

• $\int \frac{du}{\sqrt{a^2 + u^2}} = \sin^{-1}(\frac{u}{a}) + c$
• $\int \frac{du}{a^2 + u^2} = \frac{1}{a} \tan^{-1}(\frac{u}{a}) + c$
• $\int \frac{du}{u \sqrt{u^2 - a^2}} = \frac{1}{a} \sec^{-1}(\frac{\lvert u \rvert}{a}) + c$