Functions, Graphs, and Models
Review the basics of functions including domain and range, transformation of functions, composition of functions, inverse functions, and modeling with functions.
Limits and Continuity
Find limits of functions including rational, radical, trigonometric, and composite functions. Explore the continuity of functions using limits.
Derivatives and Differentiation
Explore the need for derivatives and the formal definition of a derivative in terms of limits. Find derivatives of functions using differentiation rules. Use implicit differentiation and calculate higher order derivatives.
Applications of the Derivative
Use derivatives to solve problems involving related rates, extrema of functions, concavity of functions and optimization. Evaluate indeterminate limits using L'Hôpital's Rule. Approximate the roots of functions using Newton's Method.
Integrals and Integration
Use integrals to find antiderivatives of functions. Calculate definite integrals and explore the Fundamental Theorem of Calculus. Estimate the area under a curve using the Trapezoidal Rule and Simpson's Rule.
Applications of the Definite Integral
Use definite integrals to calculate length, area, volume, and surface area. Solve problems involving work, force, and probability using definite integrals.
Explore transcendental functions including exponential, logarithmic, trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions. Evaluate derivatives and integrals of these functions.
More Integration Techniques
Use advanced integration techniques including integration by parts, integration of powers and products of sines and cosines, and integration using trigonometric substitutions. Evaluate improper integrals and explore ordinary differential equations.
Sequences and Series
Explore sequences and series including power series, Taylor series, and Maclaurin series. Determine the convergence or divergence of infinite series using convergence tests.
Parametric Equations and Polar Coordinates
Investigate parametric equations and polar graphs. Calculate length, area, volume, and surface area of parametric equations and polar graphs.
Vectors in the Plane and Calculus
Explore vectors and vector-valued functions. Learn how to differentiate and integrate vector-valued functions. Discover how vector-valued functions can be used to model the motion of projectiles.