Functions & graphs

Course Overview

CALCULUS is the branch of mathematics that deals with the study of continuous change. It consists of two main branches: Differential calculus and Integral calculus. This course will help you understand limits, derivatives, integrals and apply them to physics, engineering, economics and modeling real-world phenomena.

Course objectives

  • Understand limits, derivatives and integrals
  • Apply calculus techniques to solve applied problems
  • Analyze functions and model real-world phenomena

Course outline (topics/pages)

  • Differential calculus: Limits, derivatives, applications
  • Integral calculus: Indefinite/definite integrals, techniques, applications
Limits illustration

Limits

Limits describe the behavior of a function as the input approaches a point. They are foundational to derivatives and integrals and help define continuity.

Interactive: approaching a limit

This demo shows f(x)=sin(x)/x approaching 1 as x→0. Move the slider to pick x (both sides) and observe f(x).

f(x) = sin(x)/x → value: 0.9589

Continuity

f is continuous at a if limₓ→a f(x) = f(a). The limit can exist even if function isn't defined at that point (removable discontinuity).

Derivative graph

Derivatives (Interactive)

Use the controls to move a second point and see the secant slope approach the tangent (derivative). Function used: f(x)=x².

Secant → Tangent

Secant slope: 2.6 · Tangent slope: 2
Definite integral

Definite Integrals (Area)

Choose limits a and b to see area under f(x)=x² between a and b. The definite integral equals the net accumulated area (F(b)-F(a)).

Area under curve

Computed ∫ₐᵇ x² dx ≈ 1.000
Calculus notes

Meaning & Definition

Calculus, developed by Newton and Leibniz, studies rates of change (differential calculus) and accumulation (integral calculus). It discovers properties of derivatives and integrals using the idea of summing infinitesimally small changes and limits.

Short definition

A branch of mathematics focused on understanding how quantities change and how to accumulate small quantities into a whole. Central ideas: limits, derivatives, integrals.

Why it matters

Calculus underpins modern physics, engineering, data science and economic models. It provides tools to analyze motion, growth, optimization and area/volume computations.

Graphs and functions

Basic Calculus

Basic calculus combines differentiation and integration grounded on limits and continuity. Exponents and algebra are essential prerequisites. We approach problems by analyzing functions and their behavior under small changes.

Key building blocks

  • Functions & graphs
  • Limits & continuity
  • Derivatives (rate of change)
  • Integrals (area & accumulation)

Notation

dy/dx, f'(x) for derivatives; ∫ f(x) dx for integrals; limits use lim notation.

Derivative concept

Differential Calculus

Differential calculus studies how functions change when inputs change. The derivative gives the instantaneous rate of change and slope of the tangent line to the graph at a point.

Core idea

The derivative is defined as the limit of the difference quotient: f'(x)=limₕ→0 (f(x+h)-f(x))/h. It measures instantaneous change.

Rules

  • Power rule: d/dx[xⁿ]=n xⁿ⁻¹
  • Product rule, quotient rule
  • Chain rule for composite functions
Optimization

Applications of Derivatives

Derivatives power many real-world calculations: maxima/minima for optimization, marginal analysis in economics, curve sketching, and motion analysis.

Optimization

Find critical points where f'(x)=0 or undefined; use second derivative or sign analysis to classify maxima or minima.

Other uses

  • Related rates problems
  • Finding intervals of increase/decrease and concavity
  • Modeling marginal cost/revenue in economics
Area under curve

Integral Calculus

Integral calculus focuses on accumulation and area. Integration reverses differentiation and is used to compute totals from rates.

Core idea

Indefinite integrals give families of functions (plus constant). Definite integrals compute net accumulation between limits a and b.

Fundamental Theorem

If F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a).

Indefinite integral

Indefinite Integrals

Indefinite integrals return a family of antiderivatives: ∫ f(x) dx = F(x) + C, where C is an arbitrary constant.

Examples

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1). ∫ cos x dx = sin x + C. ∫ eˣ dx = eˣ + C.

When to add C

Always include constant of integration for indefinite integrals because derivative of constant is zero.

Integration techniques

Techniques of Integration

Integration requires several techniques to handle various integrands: substitution, integration by parts, partial fractions, trigonometric substitution, and numerical methods.

Common methods

  • Substitution (u-sub)
  • Integration by parts (∫ u dv = uv − ∫ v du)
  • Partial fraction decomposition
  • Trigonometric substitutions and identities
Applications of integrals

Applications of Integrals

Integrals compute areas, volumes (via disks/washers/shells), center of mass, total accumulated quantities and solve problems in physics, engineering and probability.

Examples

  • Area between curves: ∫(top − bottom) dx
  • Volume by revolution: disk/washer and shell methods
  • Work = ∫ Force · distance
Calculus summary

Summary & Reference

Calculus ties together limits, derivatives and integrals. Derivatives measure instantaneous change; integrals accumulate. Master limits, derivative rules, and integration techniques to solve applied problems.

Quick reference formulas

Derivative: f'(x)=limₕ→0 (f(x+h)−f(x))/h
Indefinite integral: ∫ f(x) dx = F(x) + C
Definite integral: ∫ₐᵇ f(x) dx = F(b) − F(a)

Study advice

  • Practice many worked examples for each technique.
  • Graph functions to build geometric intuition.
  • Use software (GeoGebra) to visualize tangents and areas.