Limits
Limits by direct substitution, limits of 0/0 forms by factoring and rationalizing, limits at infinity, and one-sided limits and continuity from graphs.
Limits by direct substitutionEvaluate limits of polynomial and rational functions at points where they are continuous by substituting the value.Limits by factoring (0/0 forms)Evaluate limits that give 0/0 on substitution by factoring and cancelling the common factor, or by rationalizing.Limits at infinityFind limits of rational functions as x grows without bound by comparing the degrees of the top and bottom.One-sided limits and continuityRead one-sided limits, two-sided limits and function values from a graph, and decide whether the function is continuous or has a jump or removable discontinuity.
Derivatives
The power, product, quotient and chain rules, derivatives of trigonometric, exponential and logarithmic functions, and tangent lines.
Derivatives with the power ruleDifferentiate polynomials term by term with the power rule, including negative and fractional exponents.The product and quotient rulesDifferentiate products with the product rule, (fg)′ = f′g + fg′, and quotients with the quotient rule, (f/g)′ = (f′g − fg′)/g².The chain ruleDifferentiate composite functions such as (3x + 1)⁵, (x² − 4)³ and √(2x + 5) with the chain rule.Derivatives of trigonometric, exponential and logarithmic functionsDifferentiate sine, cosine, eˣ and ln x, including multiples like e^(kx) and sin(kx), using the basic rules and the chain rule.Equations of tangent linesFind the slope of a curve at a point with the derivative, then write the equation of the tangent line there.
Applications of derivatives
Critical points and local extrema, increasing and decreasing intervals, concavity and inflection points, velocity and acceleration, optimization and related rates.
Critical points, extrema and increasing intervalsFind the critical points of a cubic, classify them as local maxima or minima, and find where the function is increasing or decreasing.Position, velocity and accelerationDifferentiate a position function to find velocity and acceleration, and find when the object is at rest.Optimization problemsWrite a function for the quantity to maximize, using a constraint to reduce it to one variable, and find its maximum with the derivative.Related ratesDifferentiate a formula with respect to time to relate the rates at which two quantities change, for growing circles, squares and spheres.Concavity and inflection pointsFind the second derivative of a cubic or quartic, its inflection points, and the intervals where the function is concave up or concave down.
Integrals
Antiderivatives, definite integrals and average value, substitution, area between curves, and Riemann sums.
Indefinite integrals (antiderivatives)Find antiderivatives of polynomials and powers with the reverse power rule, adding the constant of integration.Definite integralsEvaluate definite integrals of polynomials with the Fundamental Theorem of Calculus, and find the average value of a function on an interval.Integration by substitutionIntegrate powers of a linear or quadratic expression by substituting u for the inside and adjusting for du.Area between curvesFind where a parabola and a line meet, then integrate the top function minus the bottom one between those points.Riemann sums and the trapezoidal ruleApproximate the area under a curve with left, right and midpoint Riemann sums and with the trapezoidal rule.
Differential equations
Initial value problems and exponential growth and decay models.
Differential equations: initial value problemsSolve differential equations of the form dy/dx = f(x) by integrating, then use the initial condition to find the constant.Exponential growth and decay models (dy/dt = ky)Solve dy/dt = ky as y = y₀e^(kt), and use it to predict an amount or to find k from a doubling time.