Single-Variable Differential Calculus · Complete Notes

Calculus I

The full first course in calculus, functions, limits, continuity, the derivative and all its rules, curve analysis, optimization, and an introduction to integration. Every chapter has definitions, theorems, worked examples, and interactive visualizations you can play with.

7 Chapters
Complete

These notes cover an applied single-variable calculus course aimed at business, economics, and life-science students. The focus is differential calculus, what a derivative means, how to compute it, and how to use it, followed by an introduction to integration as the reverse process and as area. Worked examples lean on real applications: cost and revenue, growth and decay, compound interest, and elasticity of demand.

You can study entirely from these pages. No prior calculus is assumed, only high-school algebra. Each chapter builds on the one before it.

Unit 1, Functions

Chapter 1

Linear Functions

Slope, the equations of lines, and the linear models that underlie cost, revenue, supply, and demand. Where every calculus idea begins: rate of change.

SlopeEquations of Lines Linear FunctionsCost & Revenue Supply & DemandBreak-Even

4 sections2 animations

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Chapter 2

Nonlinear Functions

The full toolkit of functions: domain and range, quadratics, polynomials, rational functions, and the exponential and logarithmic functions that model growth, decay, and money.

Domain & RangeQuadratics PolynomialsRational Functions ExponentialsLogarithms Growth & DecayCompound Interest

6 sections3 animations

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Unit 2, The Derivative

Chapter 3

The Derivative

The heart of the course. Limits make the idea of "instantaneous" precise; continuity describes unbroken graphs; and the derivative is defined as the limit of a difference quotient, the slope of the tangent line.

LimitsOne-Sided Limits Limits at InfinityContinuity IVTRates of Change Definition of DerivativeGraphical Differentiation

5 sections4 animations

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Chapter 4

Calculating the Derivative

The mechanical rules that turn differentiation into algebra: power, product, quotient, and chain rules, plus the derivatives of exponential and logarithmic functions.

Power RuleProduct Rule Quotient RuleChain Rule Marginal AnalysisExponential Derivatives Logarithmic Derivatives

5 sections2 animations

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// Fundamental Results, Chapters 1-4

Slope of a line

$m = \dfrac{y_2-y_1}{x_2-x_1}$

The derivative

$f'(x)=\lim\limits_{h\to0}\dfrac{f(x+h)-f(x)}{h}$

Power rule

$\dfrac{d}{dx}\,x^n = n x^{n-1}$

Product rule

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

Chain rule

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

Continuous compounding

$A = Pe^{rt}$

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Unit 3, Using the Derivative

Chapter 5

Graphs & the Derivative

Reading a function from its derivatives. Where it rises and falls, where it peaks, how it curves, and how to assemble a complete graph from sign information alone.

Increasing / DecreasingCritical Numbers Relative ExtremaFirst Derivative Test ConcavityInflection Points Second Derivative TestCurve Sketching

4 sections3 animations

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Chapter 6

Applications of the Derivative

Calculus put to work: finding absolute maxima and minima, solving optimization problems, elasticity of demand, implicit differentiation, related rates, and linear approximation.

Absolute ExtremaOptimization Elasticity of DemandImplicit Differentiation Related RatesDifferentials Linear Approximation

6 sections3 animations

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Unit 4, Integration

Chapter 7

Integration

Differentiation in reverse, and the surprising fact that it computes area. Antiderivatives, substitution, the definite integral as a Riemann sum, and the Fundamental Theorem that ties it all together.

AntiderivativesIndefinite Integrals SubstitutionRiemann Sums Definite IntegralFundamental Theorem Area Under a Curve

4 sections3 animations

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// Fundamental Results, Chapters 5-7

Critical number

$f'(c)=0$ or $f'(c)$ undefined

Second derivative test

$f''(c)>0\Rightarrow$ local min

Elasticity of demand

$E = -\dfrac{p}{q}\cdot\dfrac{dq}{dp}$

Differential

$dy = f'(x)\,dx$

Definite integral

$\displaystyle\int_a^b f(x)\,dx = \lim_{n\to\infty}\sum_{i=1}^n f(x_i)\Delta x$

Fundamental Theorem

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

What you'll need

Two years of high-school algebra
No prior calculus assumed
A reference chapter on algebra is enough to fill any gaps

Approach

Applied, single-variable calculus. Examples come from business, economics, and the life sciences, cost, revenue, growth, decay, and optimization, so the math always connects to something concrete.

How to use these notes

Read in order, each chapter assumes the last. Open the worked examples, play with the interactive figures, and treat the boxed definitions and theorems as the things worth memorizing.

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