9 Course 9 of 10
Calculus
Calculus is the mathematics of change: how fast something changes, where it peaks, how much it adds up to. It starts with limits and builds to derivatives and integrals, the two ideas behind most of modern science. Every lesson has examples worked step by step, practice problems with solutions and a quick check at the end.
Before you start: Algebra 2, coordinate geometry, trigonometry, and exponentials and logarithms.
Unit 1 Limits and continuity
- 1 Functions, a review for calculus Function notation, domain and range, even and odd functions, increasing and decreasing behavior, composition of functions and inverse functions. 13 min
- 2 Introduction to limits The intuitive idea of a limit, estimating limits from tables and graphs, one-sided limits, infinite limits, limits at infinity and when a limit does not exist. 12 min
- 3 Evaluating limits algebraically Limit laws, direct substitution, indeterminate forms, and evaluating limits by factoring, rationalizing, simplifying complex fractions and comparing degrees at infinity. 14 min
- 4 Special limits and continuity The limits of sin x over x and related trig limits, the limit that defines e, the definition of continuity, types of discontinuity, and the Intermediate and Extreme Value Theorems. 13 min
- 5 Asymptotes and limits at infinity Finding vertical, horizontal and slant asymptotes with limits, telling holes from vertical asymptotes, and end behavior of functions with radicals. 12 min
Unit 2 Derivatives
- 6 The definition of the derivative Average and instantaneous rate of change, the limit definition of the derivative, tangent lines, velocity, and where a function fails to be differentiable. 13 min
- 7 Differentiation rules The power, constant multiple and sum rules, derivatives of trig, exponential and log functions, the product and quotient rules, and the chain rule. 14 min
- 8 Increasing, decreasing and extreme values Critical numbers, the first derivative test for relative extrema, the closed interval method for absolute extrema, and optimization problems. 14 min
- 9 Concavity and inflection points Using the second derivative to find where a graph is concave up or down, locating inflection points, and the second derivative test for relative extrema. 12 min
- 10 Curve sketching A step-by-step checklist for sketching the graph of a function with calculus, worked through completely on a rational function. 15 min
Unit 3 Integrals
- 11 Antiderivatives and indefinite integrals Antiderivatives, the indefinite integral, basic integration rules, u-substitution and initial value problems. 13 min
- 12 Definite integrals and the Fundamental Theorem of Calculus Riemann sums, the definite integral as net area, both parts of the Fundamental Theorem of Calculus, area between curves and the average value of a function. 15 min