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Calculus

The mathematics of change. Derivatives here are taken one rule at a time (power, product, chain…), integrals show the technique they use (substitution, parts, partial fractions), and every result is graphed against the original.

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Introductory Limits Substitution, factoring and L'Hôpital's rule. limit of sin(x)/x as x -> 0 Core Derivatives Power, product, quotient and chain rules, applied one at a time. derivative of sin(x)*x^2 Core Integrals Antiderivatives by substitution, parts and partial fractions. integral of x*e^x dx Core Definite integrals Area under a curve and the fundamental theorem of calculus. integrate x^2 dx from 0 to 1 Advanced Taylor series Approximating a function by polynomials built from its derivatives. taylor series of e^x Advanced Series and sums Partial sums, closed forms and convergence. sum of 1/n^2 for n = 1 to oo Core Maxima and minima Critical points and the second-derivative test. critical points of x^3 - 3x Core The chain rule Differentiating a function of a function: outside derivative times inside derivative. derivative of sin(x^2) Core Implicit differentiation Differentiating relations that are not solved for y. derivative of x^2 + y^2 wrt x Core Related rates and optimisation Turning a word problem into a derivative: what is changing, what is fixed, what is extreme. maximum of -x^2 + 4x Core Integration techniques: substitution, parts, partial fractions Recognising which technique an integrand wants. integral of 2x/(x^2+1) Core Applications of integration: area, volume, arc length Areas between curves, solids of revolution, average value. integrate pi*(sqrt(x))^2 dx from 0 to 4 Core Infinite series and convergence tests Geometric, p-series, ratio and comparison tests, alternating series. does n/2^n for n = 1 to oo converge

Chapters from Boelkins, Active Calculus

Every section of the book, condensed into a lesson with its own practice problems.

1. Understanding the Derivative

How do we measure velocity?The notion of limitThe derivative of a function at a pointThe derivative functionInterpreting, estimating, and using the derivativeThe second derivativeLimits, continuity, and differentiabilityThe tangent line approximation

2. Computing Derivatives

Elementary derivative rulesThe sine and cosine functionsThe product and quotient rulesDerivatives of other trigonometric functionsDerivatives of functions given implicitly

3. Using Derivatives

Using derivatives to evaluate limitsUsing derivatives to identify extreme valuesUsing derivatives to describe families of functionsGlobal optimizationApplied optimization

4. The Definite Integral

Determining distance traveled from velocityRiemann sums

5. Evaluating Integrals

Constructing accurate graphs of antiderivativesThe Second Fundamental Theorem of CalculusIntegration by substitutionOther options for finding algebraic antiderivatives

6. Using Definite Integrals

Using definite integrals to find area and lengthUsing definite integrals to find volumeDensity, mass, and center of massPhysics applications: work, force, and pressure

8. Taylor Polynomials and Taylor Series

Extending local linearizationTaylor polynomialsGeometric sumsFinding and using Taylor seriesQuantifying the accuracy of approximations

Chapters from OpenStax Calculus Volume 2

Every section of the book, condensed into a lesson with its own practice problems.

1. Integration

Approximating AreasThe Definite IntegralThe Fundamental Theorem of CalculusIntegration Formulas and the Net Change TheoremSubstitutionIntegrals Involving Exponential and Logarithmic FunctionsIntegrals Resulting in Inverse Trigonometric Functions

2. Applications of Integration

Areas between CurvesDetermining Volumes by SlicingVolumes of Revolution: Cylindrical ShellsArc Length of a Curve and Surface AreaPhysical ApplicationsMoments and Centers of MassIntegrals, Exponential Functions, and LogarithmsExponential Growth and DecayCalculus of the Hyperbolic Functions

3. Techniques of Integration

Integration by PartsTrigonometric IntegralsTrigonometric SubstitutionPartial FractionsOther Strategies for IntegrationNumerical IntegrationImproper Integrals

4. Introduction to Differential Equations

Basics of Differential EquationsDirection Fields and Numerical MethodsSeparable EquationsThe Logistic EquationFirst-Order Linear Equations

5. Sequences and Series

SequencesInfinite SeriesThe Divergence and Integral TestsComparison TestsAlternating SeriesRatio and Root Tests

6. Power Series

Power Series and FunctionsProperties of Power SeriesTaylor and Maclaurin SeriesWorking with Taylor Series

Chapters from OpenStax Calculus Volume 1

Every section of the book, condensed into a lesson with its own practice problems.

1. Functions and Graphs

Review of FunctionsBasic Classes of FunctionsTrigonometric FunctionsInverse FunctionsExponential and Logarithmic Functions

2. Limits

A Preview of CalculusThe Limit of a FunctionThe Limit LawsThe Precise Definition of a Limit

3. Derivatives

Defining the DerivativeThe Derivative as a FunctionDifferentiation RulesDerivatives as Rates of ChangeDerivatives of Trigonometric FunctionsThe Chain RuleDerivatives of Inverse FunctionsDerivatives of Exponential and Logarithmic Functions

4. Applications of Derivatives

Related RatesLinear Approximations and DifferentialsMaxima and MinimaThe Mean Value TheoremDerivatives and the Shape of a GraphLimits at Infinity and AsymptotesApplied Optimization ProblemsL’Hôpital’s RuleNewton’s MethodAntiderivatives

Chapters from OpenStax Precalculus 2e

Every section of the book, condensed into a lesson with its own practice problems.

12. Introduction to Calculus

Introduction to CalculusFinding Limits: Numerical and Graphical ApproachesFinding Limits: Properties of LimitsContinuity

Symbols used here

\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
\infty
infinity
Not a number: "grows without bound" in limits and intervals.
\sin,\ \cos,\ \tan
sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\lim_{x \to a} f(x)
limit
The value f(x) approaches as x approaches a.
f'(x),\ \frac{dy}{dx}
derivative
Instantaneous rate of change; slope of the graph.
\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
y',\ y''
first and second derivative of y
Prime notation for derivatives with respect to x (or t).
C,\ C_1,\ C_2
arbitrary constants
Constants of integration fixed by initial conditions.

Questions people ask

What is a derivative in one sentence?

The slope of the graph at a point — the rate at which the output is changing there. Speed is the derivative of position.

What is an integral in one sentence?

The accumulated total of a rate — the area under the curve. Distance travelled is the integral of speed.

Why are derivatives and integrals opposites?

That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.

When do I use substitution and when integration by parts?

Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function — a polynomial times an exponential, log or trig function.

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