AP College Calculus AB

• Course Overview

Limits and Continuity

• Rates of Change and Limits: Understand the concept of limits, calculate limits algebraically and graphically, and utilize the limit definition of the derivative.
• Limits Involving Infinity: Evaluate limits that approach infinity or negative infinity.
• Continuity: Define continuity, determine if a function is continuous at a point, and identify types of discontinuities.
• Rates of Change and Tangent Lines: Use limits to find the slope of a tangent line and the instantaneous rate of change.

Derivatives

• Derivative of a Function: Define the derivative as the limit of the difference quotient, find the derivative using the limit definition, and interpret the derivative as the slope of the tangent line.
• Differentiability: Determine if a function is differentiable at a point and identify points of non-differentiability.
• Rules of Differentiation: Learn the power rule, product rule, quotient rule, chain rule, and other differentiation rules.
• Velocity and Other Rates of Change: Apply derivatives to find velocity, acceleration, and other rates of change.
• Derivatives of Trigonometric Functions: Find the derivatives of trigonometric functions.
• Chain Rule: Apply the chain rule to find derivatives of composite functions.
• Implicit Differentiation: Find derivatives of implicit functions.
• Derivatives of Inverse Trigonometric Functions: Calculate the derivatives of inverse trigonometric functions.
• Derivatives of Exponential and Logarithmic Functions: Find the derivatives of exponential and logarithmic functions.

Applications of Derivatives

• Extreme Values of Functions: Find maximum and minimum values of functions using the first and second derivative tests.
• Mean Value Theorem: Apply the Mean Value Theorem to analyze functions.
• Connecting f′f'f′ and f′′f''f′′ with the Graph of fff: Relate the first and second derivatives to the graph of a function.
• Modeling and Optimization: Use calculus to solve optimization problems.
• Linearization and Newton’s Method: Employ linearization to approximate functions and use Newton's method to find roots of equations.

The Definite Integral

• Estimating with Finite Sums: Approximate definite integrals using Riemann sums.
• Definite Integrals: Define the definite integral as the limit of Riemann sums, interpret it as the net signed area, and evaluate definite integrals using the definite integral theorem.
• Definite Integrals and Antiderivatives: Relate definite integrals to antiderivatives and use the Fundamental Theorem of Calculus to evaluate definite integrals.
• Fundamental Theorem of Calculus I and II: State and apply the fundamental theorem to evaluate definite integrals and find antiderivatives.
• Trapezoidal Rule: Use the trapezoidal rule to approximate definite integrals.

Differential Equations and Math Modeling

• Antiderivatives and Slope Fields: Find antiderivatives and sketch slope fields.
• Integration by Substitution: Use the substitution method to evaluate integrals.
• Integration by Parts: Apply the integration by parts method to evaluate integrals.
• Exponential Growth and Decay: Model growth and decay problems using exponential functions.
• Population Growth: Use differential equations to model population growth.
• Numerical Methods: Explore numerical methods like Euler's method to approximate solutions of differential equations.

Applications of Definite Integrals

• Integral as Net Change: Interpret the definite integral as the net change.
• Areas in the Plane: Find areas between curves using definite integrals.
• Volumes: Calculate volumes of solids using definite integrals (disk method, washer method, shell method).
• Lengths of Curves: Determine the lengths of curves using definite integrals.

L’Hôpital’s Rule

L’Hôpital’s Rule: Use L'Hôpital’s Rule to evaluate indeterminate forms.