Calculus

Introduction

Calculus is the mathematical study of continuous change. It has two major branches: differential calculus (concerning rates of change and slopes of curves) and integral calculus (concerning accumulation of quantities and areas under curves). Together, these branches are connected by the Fundamental Theorem of Calculus.

Calculus was independently developed in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. It forms the foundation for much of modern mathematics, physics, engineering, economics, and many other fields.

Limits and Continuity

Limits

The concept of a limit is foundational to calculus. It describes the behaviour of a function as its input approaches a particular value.

Definition: We write lim(x→a) f(x) = L to mean that f(x) can be made arbitrarily close to L by taking x sufficiently close to a.

One-sided limits:

Left-hand limit: lim(x→a⁻) f(x)
Right-hand limit: lim(x→a⁺) f(x)
A two-sided limit exists only if both one-sided limits exist and are equal

Limits at infinity:

lim(x→∞) f(x) describes the behaviour as x grows without bound
Horizontal asymptotes occur when this limit exists and is finite

L’Hôpital’s Rule:
For indeterminate forms (0/0 or ∞/∞):
$lim_{x \to a} \frac{f ( x )}{g ( x )} = lim_{x \to a} \frac{f ^{'} ( x )}{g ^{'} ( x )}$
provided the limit on the right exists.

Continuity

A function f is continuous at a point a if:

f(a) is defined
lim(x→a) f(x) exists
lim(x→a) f(x) = f(a)

Types of discontinuities:

Removable: The limit exists but doesn’t equal f(a) (or f(a) is undefined)
Jump: Left and right limits exist but are not equal
Infinite: The function approaches infinity near the point
Essential: The limit does not exist in any useful sense

Intermediate Value Theorem:
If f is continuous on [a, b] and k is between f(a) and f(b), then there exists at least one c in (a, b) such that f(c) = k.

Differential Calculus

The Derivative

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

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

Geometric interpretation: The derivative f’(a) gives the slope of the tangent line to the graph of f at the point (a, f(a)).

Physical interpretation: If s(t) represents position as a function of time, then:

s’(t) = velocity
s”(t) = acceleration

Notation: The derivative can be written as f’(x), dy/dx, Df(x), or ∂f/∂x (for partial derivatives).

Differentiation Rules

Basic rules:

Constant rule: d/dx [c] = 0
Power rule: d/dx [xⁿ] = nxⁿ⁻¹
Sum rule: d/dx [f + g] = f’ + g’
Constant multiple: d/dx [cf] = cf’

Product rule:
$\frac{d}{d x} [f (x) g (x)] = f^{'} (x) g (x) + f (x) g^{'} (x)$

Quotient rule:
$\frac{d}{d x} [\frac{f ( x )}{g ( x )}] = \frac{f ^{'} ( x ) g ( x ) - f ( x ) g ^{'} ( x )}{[ g ( x ) ] ^{2}}$

Chain rule:
$\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)$

Implicit differentiation: Used when y is defined implicitly as a function of x. Differentiate both sides with respect to x, treating y as a function of x.

Logarithmic differentiation: Take the natural log of both sides, then differentiate. Useful for products, quotients, and variable exponents.

Applications of Derivatives

Finding extrema:

Critical points occur where f’(x) = 0 or f’(x) is undefined
First derivative test: f’ changes sign at local extrema
Second derivative test: f”(c) > 0 implies local minimum; f”(c) < 0 implies local maximum

Mean Value Theorem:
If f is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) such that:
$f^{'} (c) = \frac{f ( b ) - f ( a )}{b - a}$

Related rates: Problems involving rates of change of related quantities. Set up an equation relating the quantities, then differentiate with respect to time.

Optimisation problems: Find maximum or minimum values by setting the derivative equal to zero and checking endpoints and critical points.

Curve sketching: Use derivatives to determine:

Intervals of increase/decrease (first derivative)
Concavity and inflection points (second derivative)
Asymptotic behaviour (limits)

Integral Calculus

The Integral

The definite integral represents the signed area under a curve.

Riemann sums: Approximate the area by dividing into n rectangles:
$\sum_{i = 1}^{n} f (x_{i}^{*}) Δ x$

Definite integral:
$\int_{a}^{b} f (x) d x = lim_{n \to \infty} \sum_{i = 1}^{n} f (x_{i}^{*}) Δ x$

Fundamental Theorem of Calculus:

If F(x) = ∫ₐˣ f(t) dt, then F’(x) = f(x)
∫ₐᵇ f(x) dx = F(b) - F(a), where F is any antiderivative of f

Integration Techniques

Substitution (u-substitution):
Let u = g(x), then du = g’(x) dx. Transform the integral into a simpler form.

Integration by parts:
$\int u d v = uv - \int v d u$

Partial fractions:
Decompose rational functions into simpler fractions that can be integrated individually.

Trigonometric substitution:

For √(a² - x²): let x = a sin θ
For √(a² + x²): let x = a tan θ
For √(x² - a²): let x = a sec θ

Improper integrals:

Type I: Infinite limits of integration
Type II: Discontinuous integrand
Evaluate as limits of proper integrals

Applications of Integration

Area between curves:
$A = \int_{a}^{b} ∣ f (x) - g (x) ∣ d x$

Volumes of revolution:

Disc method: V = π ∫ [f(x)]² dx
Washer method: V = π ∫ ([f(x)]² - [g(x)]²) dx
Shell method: V = 2π ∫ x f(x) dx

Arc length:
$L = \int_{a}^{b} 1 + [f^{'} (x)]^{2} d x$

Work: W = ∫ F(x) dx, where F(x) is the force function.

Sequences and Series

Sequences

A sequence is an ordered list of numbers {a₁, a₂, a₃, …}.

Convergence: A sequence {aₙ} converges to L if lim(n→∞) aₙ = L.

Bounded sequences: A sequence is bounded if there exists M such that |aₙ| ≤ M for all n.

Monotonic sequences: A sequence is monotonic if it is either increasing (aₙ₊₁ ≥ aₙ) or decreasing (aₙ₊₁ ≤ aₙ) for all n.

Theorem: Every bounded monotonic sequence converges.

Series

A series is the sum of a sequence: Σₙ₌₁^∞ aₙ.

Geometric series:
$\sum_{n = 0}^{\infty} a r^{n} = \frac{a}{1 - r} (converges if ∣ r ∣ < 1)$

Convergence tests:

Divergence test: If lim aₙ ≠ 0, the series diverges
Ratio test: Examine lim |aₙ₊₁/aₙ|
Root test: Examine lim |aₙ|^(1/n)
Comparison test: Compare with a known series
Integral test: Compare with an improper integral

Power series:
$\sum_{n = 0}^{\infty} c_{n} (x - a)^{n}$
Has a radius of convergence R where it converges absolutely.

Taylor series:
$f (x) = \sum_{n = 0}^{\infty} \frac{f ^{(n)} ( a )}{n !} (x - a)^{n}$

Maclaurin series: Taylor series centred at a = 0.

Multivariable Calculus

Partial Derivatives

For a function f(x, y), the partial derivative with respect to x is:
$\frac{\partial f}{\partial x} = lim_{h \to 0} \frac{f ( x + h , y ) - f ( x , y )}{h}$

Gradient vector:
$\nabla f = ⟨ \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} ⟩$

Directional derivative: The rate of change in direction u:
$D_{u} f = \nabla f \cdot u$

The gradient points in the direction of steepest ascent.

Multiple Integrals

Double integrals:
$\iint_{R} f (x, y) d A$
Compute as iterated integrals.

Triple integrals:
$∭_{E} f (x, y, z) d V$

Change of variables:
Use the Jacobian determinant when transforming coordinates:
$\iint f (x, y) d x d y = \iint f (g (u, v), h (u, v)) ∣ J ∣ d u d v$

Vector Calculus

Line integrals: Integrate a function along a curve.

Surface integrals: Integrate over a surface in three dimensions.

Fundamental theorems:

Green’s Theorem: Relates a line integral around a closed curve to a double integral over the enclosed region
Stokes’ Theorem: Generalises Green’s theorem to three dimensions
Divergence Theorem: Relates a surface integral to a triple integral

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