# Calculus 1

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

## 2 Limits and Continuity

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### 2.1 The Limit of a Function

The limit of a function $f$ is a tool for investigating the behavior of $f(x)$ as $x$ approaches some number $c$.

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

Read as “the limit of the function $f$ as $x$ approaches $c$.” 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 $x \to c$ below $\lim$ means we are taking the limit of $f$ as values of $x$ approach $c$.

One-sided limits

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.

Formal definition of a limit

$\lim\limits_{x \to c} f(x) = L$

means that to establish a specific limit, a number $\epsilon > 0$ is chosen first to establish a desired degree of proximity to $L$, then a number $\delta > 0$ is found that determines how close $x$ must be to $c$ to ensure that $f(x)$ is within $\epsilon$ units of $L$.

### 2.2 Algebraic Computation of Limits

Two special limits

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

#### Basic limit results

For any real number $a$ and any constant $c$,

Constant function rule

$\lim\limits_{x \to a} x = a$

Identity function rule

$\lim\limits_{x \to a} c = c$

#### Basic algebraic operations

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.

#### The Squeeze Theorem

Let $f(x)$, $g(x)$, and $h(x)$ be defined for all $x \neq a$ over an open interval containing $a$. If

$f(x) \leq g(x) \leq g(x)$

for all $x \neq a$ in an open interval containing $a$ and

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

where $L$ is a real number, then

$\lim\limits_{x \to a} g(x) = L$

## 3 Differentiation

### Definition of derivative

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

Study cards

### 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

## 4 Applications of Derivatives

### 4.5 L’Hôpital’s Rule

L’Hôpital’s rule is used to evaluate indeterminate forms. L’Hôpital’s rule states if the $\lim\limits_{x \to c} \frac{f(x)}{g(x)}$ yields one of the indeterminate forms of the type $\frac{0}{0}$ of $\frac{\infin}{\infin}$, then you can evaluate the limit by replacing $f$ with $f^{\prime}$ and $g$ with $g^{\prime}$. Because the rule can be applied repeatedly, you can keep applying it until you get something other than an indeterminate form.

## 5 Integration

### 5.1 Antidifferentiation

#### Antiderivative

A function $G$ is an antiderivative of $f$ on an interval $I$ if

$G^{\prime}(x) = f(x)$

for all $x$ in $I$.

If $G$ is an antiderivative of $f$, then so is $G + C$ for any constant $C$ given that the derivative of any constant is $0$.

#### Indefinite integral notation

$\int f(x)dx = G(x) + C$

The antiderivative (integral) of $f$ is $G + C$.

• $f$ is the integrand
• $dx$ is the variable of integration

#### Integration formulas

To obtain the power rule, for $n$ is any number and $n \neq -1$

$\int x^{n} dx = \frac{x^{n+1}}{n+1} + C$

### 5.5 Integration by Substitution

Integration by substitution or u-substitution is the inverse of the chain rule for diferrentiation.

Let $f$, $g$, and $u$ be differentiable functions of $x$ such that

$f(x) = g(u) \frac{du}{dx}$

Then

$\int f(x) = \int g(u) \frac{du}{dx} = \int g(u)du = G(u) + C$

Where $G$ is an antiderivative of $g$.

### 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

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