Calculus

Limits, derivatives, integrals, series and optimisation.

Isomo

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

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.

Birenzeho

Understanding the Derivative

How do we measure velocity?The notion of limit The derivative of a function at a point The derivative function Interpreting, estimating, and using the derivative The second derivative Limits, continuity, and differentiability The tangent line approximation

Functions and Graphs

Review of Functions Basic Classes of Functions Trigonometric Functions Inverse Functions Exponential and Logarithmic Functions

Integration

Approximating Areas The Definite Integral The Fundamental Theorem of Calculus Integration Formulas and the Net Change Theorem Substitution Integrals Involving Exponential and Logarithmic Functions Integrals Resulting in Inverse Trigonometric Functions

Limits

A Preview of Calculus The Limit of a Function The Limit Laws The Precise Definition of a Limit

Computing Derivatives

Elementary derivative rules The sine and cosine functions The product and quotient rules Derivatives of other trigonometric functions Derivatives of functions given implicitly

Applications of Integration

Areas between Curves Determining Volumes by Slicing Volumes of Revolution: Cylindrical Shells Arc Length of a Curve and Surface Area Physical Applications Moments and Centers of Mass Integrals, Exponential Functions, and Logarithms Exponential Growth and Decay Calculus of the Hyperbolic Functions

Derivatives

Defining the Derivative The Derivative as a Function Derivatives as Rates of Change Derivatives of Trigonometric Functions The Chain Rule Derivatives of Inverse Functions Derivatives of Exponential and Logarithmic Functions

Using Derivatives

Using derivatives to evaluate limits Using derivatives to identify extreme values Using derivatives to describe families of functions Global optimization Applied optimization

Techniques of Integration

Trigonometric Integrals Partial Fractions Other Strategies for Integration Numerical Integration Improper Integrals

The Definite Integral

Determining distance traveled from velocity Riemann sums

Applications of Derivatives

Linear Approximations and Differentials Maxima and Minima The Mean Value Theorem Derivatives and the Shape of a Graph Limits at Infinity and Asymptotes Applied Optimization Problems Newton’s Method Antiderivatives

Evaluating Integrals

Constructing accurate graphs of antiderivatives The Second Fundamental Theorem of Calculus Other options for finding algebraic antiderivatives

Using Definite Integrals

Using definite integrals to find area and length Using definite integrals to find volume Density, mass, and center of mass Physics applications: work, force, and pressure

Sequences and Series

Sequences Infinite Series The Divergence and Integral Tests Comparison Tests Alternating Series Ratio and Root Tests

Power Series

Power Series and Functions Properties of Power Series Taylor and Maclaurin Series Working with Taylor Series

Taylor Polynomials and Taylor Series

Extending local linearization Taylor polynomials Geometric sums Finding and using Taylor series Quantifying the accuracy of approximations

Introduction to Calculus

Introduction to Calculus Finding Limits: Numerical and Graphical Approaches Finding Limits: Properties of Limits Continuity

Foundations

Calculus Limit (mathematics)Continuous function Derivative Differentiation rules Product rule Quotient rule Related rates Linear approximation

Integrals

Integral Antiderivative Fundamental theorem of calculus Integration by substitution Integration by parts Partial fraction decomposition Trigonometric substitution Improper integral Arc length Solid of revolution Riemann sum

Series and applications

Maclaurin series Power series Convergence tests Optimization (mathematics)Mean value theorem L'Hôpital's rule Curve sketching Exponential growth Logistic function

Bya Kuva: Gufungura Umwandiko: Boelkins, Active Calculus, OpenStax Calculus Volume 1, OpenStax Calculus Volume 2, OpenStax Precalculus 2e, Wikipedia. Each lesson page names its source and licence.

Byakoreshejwe

Icyo ari cyo cyose Ibimenyetso ya: i Byuzuye Insobanuro:, A Ishusho, na buri Ibaruwa: in.

Abaturage

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.

A Bya?
Cyangwa Gushushanya ku i Ipaji:: Intera, A Ishusho, i Gihishwe & Kugeza & Kubaza. Gusiba →

Ikindi

＋ Arithmetic 𝑥 Algebra △ Geometry ∿ Trigonometry ƒ Precalculus σ Statistics & Probability ⊢ Discrete Math & Logic ⊞ Linear Algebra ℕ Number Theory ẏ Differential Equations ∇ Multivariable Calculus ε Real Analysis ⋔ Combinatorics & Graph Theory ∈ Set Theory & Logic 𝔾 Abstract Algebra 𝑃 Probability Theory ≈ Numerical Methods ℂ Complex Analysis ◎ Topology μ Measure Theory ‖·‖ Functional Analysis κ Differential Geometry π₁ Algebraic Topology → Category Theory ∂ Partial Differential Equations ≋ Fluid Dynamics ∞ Frontiers