The Complete Calculus Study Guide

Everything you need to ace calculus

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Table of Contents

Chapter 1: Limits & Continuity

What is a Limit?

A limit describes the behavior of a function as the input approaches a particular value.

$$\lim_{x \to a} f(x) = L$$ This means: as x gets closer to a, f(x) gets closer to L
Key Types of Limits:
  • One-sided limits: $\lim_{x \to a^+} f(x)$ and $\lim_{x \to a^-} f(x)$
  • Infinite limits: $\lim_{x \to a} f(x) = \infty$
  • Limits at infinity: $\lim_{x \to \infty} f(x)$

Limit Laws

Sum Rule: $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$

Product Rule: $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$

Quotient Rule: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$

Power Rule: $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$

CHECK YOUR KNOWLEDGE

Question 1: If $\lim_{x \to 2} f(x) = 5$ and $\lim_{x \to 2} g(x) = 3$, then $\lim_{x \to 2} [2f(x) + g(x)] = $

Question 2: $\lim_{x \to 0} \frac{\sin x}{x} = $

Question 3: A function is continuous at $x = a$ if $\lim_{x \to a} f(x) = $

Chapter 2: Derivatives

Definition of Derivative

The derivative measures the instantaneous rate of change of a function.

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ This is the definition using the limit of difference quotients
Common Notation:
  • $f'(x)$ (Lagrange notation)
  • $\frac{df}{dx}$ (Leibniz notation)
  • $\frac{d}{dx}[f(x)]$ (operator notation)

Basic Derivative Rules

Power Rule: $$\frac{d}{dx}[x^n] = nx^{n-1}$$
Constant Rule: $$\frac{d}{dx}[c] = 0$$
Sum Rule: $$\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$$
Product Rule: $$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$
Quotient Rule: $$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$
Chain Rule: $$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$

CHECK YOUR KNOWLEDGE

Question 1: $\frac{d}{dx}[x^5] = $

Question 2: If $f(x) = 3x^2 + 2x - 1$, then $f'(x) = $

Question 3: Using the product rule, $\frac{d}{dx}[x^2 \sin x] = $

Chapter 3: Applications of Derivatives

Critical Points & Extrema

Critical Points: Points where $f'(x) = 0$ or $f'(x)$ is undefined

First Derivative Test:

  • If $f'$ changes from + to -, then local max
  • If $f'$ changes from - to +, then local min

Second Derivative Test:

  • If $f''(c) > 0$, then local min at $x = c$
  • If $f''(c) < 0$, then local max at $x = c$

CHECK YOUR KNOWLEDGE

Question 1: For $f(x) = x^3 - 3x^2 + 2$, the critical points occur when $f'(x) = $

Question 2: If $f''(c) > 0$ at a critical point, then $x = c$ is a local

Chapter 4: Integration

Antiderivatives & Indefinite Integrals

Integration is the reverse process of differentiation.

$$\int f(x) \, dx = F(x) + C$$ where $F'(x) = f(x)$ and $C$ is the constant of integration
Power Rule: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ (for $n \neq -1$)
Constant Rule: $$\int c \, dx = cx + C$$
Sum Rule: $$\int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx$$

Definite Integrals

$$\int_a^b f(x) \, dx = F(b) - F(a)$$ Fundamental Theorem of Calculus

CHECK YOUR KNOWLEDGE

Question 1: $\int x^3 \, dx = $

Question 2: $\int_0^2 x \, dx = $

Chapter 5: Applications of Integration

Area Under Curves

$$\text{Area} = \int_a^b |f(x)| \, dx$$
Volume by Disk Method: $$V = \pi \int_a^b [R(x)]^2 \, dx$$ where $R(x)$ is the radius function

CHECK YOUR KNOWLEDGE

Question 1: The area under $f(x) = x^2$ from $x = 0$ to $x = 2$ is

Chapter 6: Riemann Zeta Function

The Riemann Zeta Function

The Riemann zeta function is one of the most important functions in mathematics, connecting number theory and analysis.

$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$$ This series converges for $\text{Re}(s) > 1$
Key Properties:
  • Can be analytically continued to all complex numbers except $s = 1$
  • Has a simple pole at $s = 1$ with residue 1
  • Satisfies the functional equation relating $\zeta(s)$ and $\zeta(1-s)$
  • Connected to the distribution of prime numbers via Euler's product formula

The Critical Line

The critical line is the vertical line $\text{Re}(s) = \frac{1}{2}$ in the complex plane.

$$s = \frac{1}{2} + it \quad \text{where } t \in \mathbb{R}$$ This is where the most interesting zeros are believed to lie

Critical Strip:

The region $0 < \text{Re}(s) < 1$ where all non-trivial zeros lie

Trivial Zeros:

Located at $s = -2, -4, -6, -8, \ldots$ (negative even integers)

The Riemann Hypothesis

The Riemann Hypothesis (1859):

All non-trivial zeros of the Riemann zeta function have real part equal to $\frac{1}{2}$.

This is one of the most famous unsolved problems in mathematics, worth $1 million as a Clay Millennium Prize Problem.

$$\zeta\left(\frac{1}{2} + it\right) = 0 \quad \text{for some } t \in \mathbb{R}$$ If true, all non-trivial zeros lie exactly on the critical line
Computational Evidence:

Modern computers have verified that over $10^{13}$ zeros of $\zeta(s)$ lie on the critical line $\text{Re}(s) = \frac{1}{2}$, providing strong numerical evidence for the hypothesis.

Connection to Prime Numbers

$$\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}$$ Euler's product formula connects zeta function to prime distribution

The Riemann Hypothesis, if proven true, would give us the most precise understanding of how prime numbers are distributed among the integers.

CHECK YOUR KNOWLEDGE

Question 1: The Riemann zeta function is defined as $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$ for $\text{Re}(s) > $

Question 2: The critical line is defined by $\text{Re}(s) = $

Question 3: Computers have verified over $10^{13}$ zeros on the

Question 4: The trivial zeros of $\zeta(s)$ are located at the negative integers

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