Master the material tested on the College Board's Advanced Placement Calculus AB exam and earn up to 1 semester of university-level calculus credit. Learn the mathematics of change that underlies science and engineering.
Limits and Continuity
Connect algebraic and graphical interpretations of limits, including relationships to vertical and horizontal asymptotes.
Estimate limits numerically and compute limits using algebraic manipulation.
Define continuity in terms of limits, determine intervals over which a function is continuous, and remove discontinuities of functions.
Explain the behavior of a function on an interval using the Intermediate Value Theorem.
Interpret the difference quotient geometrically and use it to compute the derivative of a function.
Estimate derivatives numerically and apply rules to compute derivatives of a variety of algebraic functions, including higher-order derivatives.
Understand the relationship between differentiability and continuity.
Use implicit differentiation to solve related rates problems.
Approximate values of functions using linearization.
Use L’Hopital’s Rule to calculate limits of indeterminate forms.
Relate the graph of a function to properties of its derivative.
Solve optimization problems by using the derivative to find extrema of functions.
Evaluate Riemann sums and interpret them geometrically and contextually.
Compute the integral of a function as the limit of a Riemann sum.
Relate integrals and antiderivatives through the fundamental theorem of calculus.
Use integration to compute area and volume.
Interpret accumulation functions geometrically and contextually and compute their derivatives.
Solve advanced integrals using techniques like substitution, long division, and completing the square.
Construct differential equations to model real-world situations involving rates of change.
Verify solutions of differential equations, sketch slope fields, and relate solutions to slope fields.
Find solutions of initial value problems using separation of variables.
Interpret exponential growth models in context.
Limits and Continuity
1.1. Estimating Limits from Graphs
The Finite Limit of a Function
The Left and Right-Sided Limits of a Function
Finding the Existence of a Limit Using One-Sided Limits
Limits at Infinity from Graphs
Infinite Limits from Graphs
1.2. The Algebra of Limits
Limits of Power Functions, and the Constant Rule for Limits
The Sum Rule for Limits
The Product and Quotient Rules for Limits
The Power and Root Rules for Limits
1.3. Limits of Functions
Limits at Infinity of Polynomials
Limits of Reciprocal Functions
Limits of Exponential Functions
Limits of Logarithmic Functions
Limits of Radical Functions
Limits of Trigonometric Functions
Limits of Reciprocal Trigonometric Functions
Limits of Piecewise Functions
1.4. Determining Limits Using Algebraic Manipulation
Calculating Limits of Rational Functions by Factoring
Limits of Absolute Value Functions
Calculating Limits of Radical Functions Using Conjugate Multiplication
Calculating Limits Using Trigonometric Identities
Limits at Infinity and Horizontal Asymptotes of Rational Functions
Evaluating Limits at Infinity by Comparing Relative Magnitudes of Functions
Evaluating Limits at Infinity of Radical Functions
Vertical Asymptotes of Rational Functions
Connecting Infinite Limits and Vertical Asymptotes of Rational Functions
1.5. Special Limits
The Squeeze Theorem
Special Limits Involving Sine
Evaluating Special Limits Involving Sine Using a Substitution
Special Limits Involving Cosine
Determining Continuity from Graphs
Defining Continuity at a Point
Left and Right Continuity
Further Continuity of Piecewise Functions
Discontinuities Due to Vertical Asymptotes
Continuity Over an Interval
Continuity of Functions
The Intermediate Value Theorem
1.7. Removing Discontinuities
Removing Point Discontinuities
Removing Jump Discontinuities
Removing Discontinuities From Rational Functions
Differentiation: Definition and Fundamental Properties
2.8. Introduction to Differentiation
The Average Rate of Change of a Function over a Varying Interval
The Instantaneous Rate of Change of a Function at a Point
Defining the Derivative Using Derivative Notation
Connecting Differentiability and Continuity
The Power Rule for Differentiation
The Sum and Constant Multiple Rules for Differentiation
Calculating the Slope of a Tangent Line Using Differentiation
Calculating the Equation of a Tangent Line Using Differentiation
Calculating the Equation of a Normal Line Using Differentiation
2.9. Derivatives of Functions and the Rules of Differentiation