Calculus Fundamentals

Introduction

Calculus is the mathematical study of continuous change. It has two major branches: differential calculus (concerning rates of change) and integral calculus (concerning accumulation of quantities).

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Limits

The concept of a limit is foundational to calculus. We write:

$\lim_{x \to a} f(x) = L$

This means as $x$ approaches $a$, the function $f(x)$ approaches $L$.

Important Limits

$\lim_{x \to 0} \frac{\sin x}{x} = 1$

$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$

$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$

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Derivatives

The derivative of a function $f(x)$ measures its instantaneous rate of change:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Basic Derivative Rules

Chain Rule

If $y = f(g(x))$, then:

$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$

Or using Leibniz notation:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

Product Rule

$(fg)' = f'g + fg'$

Quotient Rule

$\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$

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Integrals

Integration is the reverse process of differentiation. The indefinite integral is:

$\int f(x) \, dx = F(x) + C$

where $F'(x) = f(x)$ and $C$ is the constant of integration.

Basic Integrals

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$

$\int \frac{1}{x} \, dx = \ln|x| + C$

$\int e^x \, dx = e^x + C$

$\int \sin x \, dx = -\cos x + C$

$\int \cos x \, dx = \sin x + C$

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The Fundamental Theorem of Calculus

This theorem connects differentiation and integration:

Part 1:

If $F(x) = \int_a^x f(t) \, dt$, then $F'(x) = f(x)$.

Part 2:

$\int_a^b f(x) \, dx = F(b) - F(a)$

This is often written as:

$\int_a^b f(x) \, dx = \left[F(x)\right]_a^b$

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Definite Integrals and Area

The definite integral $\int_a^b f(x) \, dx$ represents the signed area between the curve $y = f(x)$ and the $x$-axis from $x = a$ to $x = b$.

Example

Find the area under $y = x^2$ from $x = 0$ to $x = 2$:

$\int_0^2 x^2 \, dx = \left[\frac{x^3}{3}\right]_0^2 = \frac{8}{3} - 0 = \frac{8}{3}$

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Integration by Substitution

For $\int f(g(x)) \cdot g'(x) \, dx$, let $u = g(x)$:

$\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du$

Example

$\int 2x \cos(x^2) \, dx$

Let $u = x^2$, then $du = 2x \, dx$:

$= \int \cos u \, du = \sin u + C = \sin(x^2) + C$

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Practice Questions

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• Limits define the foundation of calculus
• Derivatives measure instantaneous rate of change: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
• Integrals represent accumulation and area
• The Fundamental Theorem of Calculus connects derivatives and integrals
• Practice the chain rule, product rule, and substitution method