Math Academy

Upon successful completion of this course, students will have mastered the following:

Understand and analyze limits and continuity, including estimating limits from graphs, the algebra of limits, limits of functions, determining limits using algebraic manipulation, and special limits.
Understand continuity, including defining continuity at a point, classifying discontinuities, removing discontinuities, and the intermediate value theorem.
Learn the basics of differentiation, including instantaneous rates of change, calculating derivatives using the definition, the rules of differentiation, differentiating exponential, logarithmic, trigonometric, and inverse trigonometric functions, computing tangent and normal lines, the chain rule, implicit differentiation, and differentiating inverse functions.
Apply differentiation techniques to various contextual situations, such as interpreting and estimating derivatives, analyzing particle motion, and solving related rates problems.
Delve into analytical applications of differentiation, including L'Hopital's Rule, the Mean Value Theorem, and linearization.
Analyzing curves by finding and classifying critical points, deducing concavity over an interval, and finding inflection points,
Gain proficiency in using differentiation to solve optimization problems.
Understand indefinite integration as the inverse of differentiation, and apply the Fundamental Theorem of Calculus to evaluate definite integrals.
Learn techniques for approximating definite integrals, such as using Riemann sums and the trapezoidal Rule.
Master the techniques for evaluating indefinite and definite integrals, including substitution, trigonometric identities, polynomial division, and completing the square.
Solve problems involving finding the area under a curve and areas between curves.
Understand how definite integrals may be used to define accumulation functions and apply the second fundamental theorem of calculus,
Optimize functions using derivative graphs.
Integrate rates of change and interpret the results in context.
Learn to solve differential equations using direct integration, separation of variables, solve initial value problems, and model real-world situations using differential equations.
Analyze slope fields for directly integrable, autonomous, and nonautonomous differential equations, and apply qualitative techniques to understand their behavior.
Apply integration techniques to find the average value of a function, the area between two curves, volumes of solids with known cross sections, and volumes of revolution.
Gain proficiency in using graphing calculators for evaluating expressions, finding roots, intersections, extrema, derivatives, definite integrals, and exploring functions and curves.

Limits and Continuity

43 topics

1.1. Estimating Limits from Graphs

	1.1.1.	The Finite Limit of a Function
	1.1.2.	The Left and Right-Sided Limits of a Function
	1.1.3.	Finding the Existence of a Limit Using One-Sided Limits
	1.1.4.	Limits at Infinity from Graphs
	1.1.5.	Infinite Limits from Graphs

1.2. The Algebra of Limits

	1.2.1.	Limits of Power Functions, and the Constant Rule for Limits
	1.2.2.	The Sum Rule for Limits
	1.2.3.	The Product and Quotient Rules for Limits
	1.2.4.	The Power and Root Rules for Limits

1.3. Limits of Functions

	1.3.1.	Limits at Infinity of Polynomials
	1.3.2.	Limits of Reciprocal Functions
	1.3.3.	Limits of Exponential Functions
	1.3.4.	Limits of Logarithmic Functions
	1.3.5.	Limits of Radical Functions
	1.3.6.	Limits of Trigonometric Functions
	1.3.7.	Limits of Reciprocal Trigonometric Functions
	1.3.8.	Limits of Piecewise Functions

1.4. Determining Limits Using Algebraic Manipulation

	1.4.1.	Calculating Limits of Rational Functions by Factoring
	1.4.2.	Limits of Absolute Value Functions
	1.4.3.	Calculating Limits of Radical Functions Using Conjugate Multiplication
	1.4.4.	Calculating Limits Using Trigonometric Identities
	1.4.5.	Limits at Infinity and Horizontal Asymptotes of Rational Functions
	1.4.6.	Evaluating Limits at Infinity by Comparing Relative Magnitudes of Functions
	1.4.7.	Evaluating Limits at Infinity of Radical Functions
	1.4.8.	Vertical Asymptotes of Rational Functions
	1.4.9.	Connecting Infinite Limits and Vertical Asymptotes of Rational Functions

1.5. Special Limits

	1.5.1.	The Squeeze Theorem
	1.5.2.	Special Limits Involving Sine
	1.5.3.	Evaluating Special Limits Involving Sine Using a Substitution
	1.5.4.	Special Limits Involving Cosine

1.6. Continuity

	1.6.1.	Determining Continuity from Graphs
	1.6.2.	Defining Continuity at a Point
	1.6.3.	Left and Right Continuity
	1.6.4.	Further Continuity of Piecewise Functions
	1.6.5.	Point Discontinuities
	1.6.6.	Jump Discontinuities
	1.6.7.	Discontinuities Due to Vertical Asymptotes
	1.6.8.	Continuity Over an Interval
	1.6.9.	Continuity of Functions
	1.6.10.	The Intermediate Value Theorem

1.7. Removing Discontinuities

	1.7.1.	Removing Point Discontinuities
	1.7.2.	Removing Jump Discontinuities
	1.7.3.	Removing Discontinuities From Rational Functions

Differentiation: Definition and Fundamental Properties

19 topics

2.8. Introduction to Differentiation

	2.8.1.	The Average Rate of Change of a Function over a Varying Interval
	2.8.2.	The Instantaneous Rate of Change of a Function at a Point
	2.8.3.	Defining the Derivative Using Derivative Notation
	2.8.4.	Connecting Differentiability and Continuity
	2.8.5.	The Power Rule for Differentiation
	2.8.6.	The Sum and Constant Multiple Rules for Differentiation
	2.8.7.	Calculating the Slope of a Tangent Line Using Differentiation
	2.8.8.	Calculating the Equation of a Tangent Line Using Differentiation
	2.8.9.	Calculating the Equation of a Normal Line Using Differentiation

2.9. Derivatives of Functions and the Rules of Differentiation

	2.9.1.	Differentiating Exponential Functions
	2.9.2.	Differentiating Logarithmic Functions
	2.9.3.	Differentiating Trigonometric Functions
	2.9.4.	Second and Higher Order Derivatives
	2.9.5.	The Product Rule for Differentiation
	2.9.6.	The Quotient Rule for Differentiation
	2.9.7.	Differentiating Reciprocal Trigonometric Functions
	2.9.8.	Calculating Derivatives From Data and Tables
	2.9.9.	Calculating Derivatives From Graphs
	2.9.10.	Recognizing Derivatives in Limits

Differentiation: Composite, Implicit, and Inverse Functions

13 topics

3.10. Differentiating Composite Functions

	3.10.1.	The Chain Rule for Differentiation
	3.10.2.	The Chain Rule With Exponential Functions
	3.10.3.	The Chain Rule With Logarithmic Functions
	3.10.4.	The Chain Rule With Trigonometric Functions
	3.10.5.	Calculating Derivatives From Data Using the Chain Rule
	3.10.6.	Calculating Derivatives From Graphs Using the Chain Rule
	3.10.7.	Selecting Procedures for Calculating Derivatives

3.11. Differentiating Implicit and Inverse Functions

	3.11.1.	Implicit Differentiation
	3.11.2.	Calculating dy/dx Using dx/dy
	3.11.3.	Differentiating Inverse Functions
	3.11.4.	Differentiating an Inverse Function at a Point
	3.11.5.	Differentiating Inverse Trigonometric Functions
	3.11.6.	Differentiating Inverse Reciprocal Trigonometric Functions

Contextual Applications of Differentiation

17 topics

4.12. Contextual Applications of Differentiation

	4.12.1.	Interpreting the Meaning of the Derivative in Context
	4.12.2.	Rates of Change in Applied Contexts

4.13. Estimating Derivatives

	4.13.1.	Estimating Derivatives Using a Forward Difference Quotient
	4.13.2.	Estimating Derivatives Using a Backward Difference Quotient
	4.13.3.	Estimating Derivatives Using a Central Difference Quotient

4.14. Displacement, Velocity, and Acceleration

	4.14.1.	Calculating Velocity for Straight-Line Motion Using Differentiation
	4.14.2.	Determining Characteristics of Moving Objects Using Differentiation
	4.14.3.	Calculating Acceleration for Straight-Line Motion Using Differentiation

4.15. Related Rates of Change

	4.15.1.	Introduction to Related Rates
	4.15.2.	Related Rates With Implicit Functions
	4.15.3.	Calculating Related Rates With Circles and Spheres
	4.15.4.	Calculating Related Rates With Squares
	4.15.5.	Calculating Related Rates With Rectangular Solids
	4.15.6.	Calculating Related Rates Using the Pythagorean Theorem
	4.15.7.	Calculating Related Rates Using Similar Triangles
	4.15.8.	Calculating Related Rates Using Trigonometry
	4.15.9.	Calculating Related Rates With Cones

Analytical Applications of Differentiation

29 topics

5.16. L'Hopital's Rule

	5.16.1.	L'Hopital's Rule
	5.16.2.	L'Hopital's Rule Applied to Tables

5.17. Analytical Applications of Differentiation

	5.17.1.	The Mean Value Theorem
	5.17.2.	Global vs. Local Extrema and Critical Points
	5.17.3.	The Extreme Value Theorem
	5.17.4.	Using Differentiation to Calculate Critical Points
	5.17.5.	Determining Intervals on Which a Function Is Increasing or Decreasing
	5.17.6.	Using the First Derivative Test to Classify Local Extrema
	5.17.7.	The Candidates Test
	5.17.8.	Intervals of Concavity
	5.17.9.	Relating Concavity to the Second Derivative
	5.17.10.	Points of Inflection
	5.17.11.	The Second Derivative Test

5.18. Analysis of Curves

	5.18.1.	Sketching the Derivative of a Function From the Function's Graph
	5.18.2.	Interpreting the Graph of a Function's Derivative
	5.18.3.	Interpreting the Graph of a Function's Derivative: Concavity and Points of Inflection
	5.18.4.	Sketching a Function From the Graph of its Derivative
	5.18.5.	Sketching a Function Given Some Derivative Properties

5.19. Approximating Values of a Function

	5.19.1.	Approximating Functions Using Local Linearity and Linearization
	5.19.2.	Approximating the Roots of a Number Using Local Linearity
	5.19.3.	Approximating Trigonometric Functions Using Local Linearity

5.20. Optimization

	5.20.1.	Solving Optimization Problems Using Derivatives
	5.20.2.	Optimization Problems Involving Sectors of Circles
	5.20.3.	Optimization Problems Involving Boxes and Trays
	5.20.4.	Optimization Problems Involving Cylinders
	5.20.5.	Optimizing Distances
	5.20.6.	Optimizing Distances to Curves
	5.20.7.	Optimization Problems With Inscribed Shapes
	5.20.8.	Optimization Problems in Economics

Integration

48 topics

6.21. Indefinite Integrals

	6.21.1.	The Antiderivative
	6.21.2.	The Constant Multiple Rule for Indefinite Integrals
	6.21.3.	The Sum Rule for Indefinite Integrals
	6.21.4.	Integrating the Reciprocal Function
	6.21.5.	Integrating Exponential Functions
	6.21.6.	Integrating Trigonometric Functions
	6.21.7.	Integration Using Inverse Trigonometric Functions

6.22. Approximating Areas with Riemann Sums

	6.22.1.	Approximating Areas With the Left Riemann Sum
	6.22.2.	Approximating Areas With the Right Riemann Sum
	6.22.3.	Approximating Areas With the Midpoint Riemann Sum
	6.22.4.	Approximating Areas With the Trapezoidal Rule
	6.22.5.	Left and Right Riemann Sums in Sigma Notation
	6.22.6.	Midpoint and Trapezoidal Rules in Sigma Notation
	6.22.7.	Approximating Areas Under Graphs of Composite Functions

6.23. Definite Integrals

	6.23.1.	Defining Definite Integrals Using Left and Right Riemann Sums
	6.23.2.	The Fundamental Theorem of Calculus
	6.23.3.	Applying the Fundamental Theorem of Calculus to Exponential and Trigonometric Functions
	6.23.4.	The Sum and Constant Multiple Rules for Definite Integrals
	6.23.5.	Properties of Definite Integrals Involving the Limits of Integration

6.24. The Area Under a Curve

	6.24.1.	The Area Bounded by a Curve and the X-Axis
	6.24.2.	Evaluating Definite Integrals Using Symmetry
	6.24.3.	Finding the Area Between a Curve and the X-Axis When They Intersect
	6.24.4.	The Area Bounded by a Curve and the Y-Axis
	6.24.5.	Calculating the Definite Integral of a Function Given Its Graph
	6.24.6.	Calculating the Definite Integral of a Function's Derivative Given its Graph
	6.24.7.	Definite Integrals of Piecewise Functions

6.25. Accumulation Functions

	6.25.1.	The Integral as an Accumulation Function
	6.25.2.	The Second Fundamental Theorem of Calculus
	6.25.3.	Maximizing a Function Using the Graph of Its Derivative
	6.25.4.	Minimizing a Function Using the Graph of its Derivative
	6.25.5.	Further Optimizing Functions Using Graphs of Derivatives
	6.25.6.	Integrating Rates of Change
	6.25.7.	Integrating Density Functions

6.26. Integration Using Substitution

	6.26.1.	Integrating Algebraic Functions Using Substitution
	6.26.2.	Integrating Linear Rational Functions Using Substitution
	6.26.3.	Integration Using Substitution
	6.26.4.	Calculating Definite Integrals Using Substitution
	6.26.5.	Further Integration of Algebraic Functions Using Substitution
	6.26.6.	Integrating Exponential Functions Using Linear Substitution
	6.26.7.	Integrating Exponential Functions Using Substitution
	6.26.8.	Integrating Trigonometric Functions Using Substitution
	6.26.9.	Integrating Logarithmic Functions Using Substitution
	6.26.10.	Integration by Substitution With Inverse Trigonometric Functions

6.27. Integration Using Trigonometric Identities

	6.27.1.	Integration Using Basic Trigonometric Identities
	6.27.2.	Integration Using the Pythagorean Identities
	6.27.3.	Integration Using the Double-Angle Formulas

6.28. Special Techniques for Integration

	6.28.1.	Integrating Functions Using Polynomial Division
	6.28.2.	Integrating Functions by Completing the Square

Differential Equations

18 topics

7.29. Introduction to Differential Equations

	7.29.1.	Introduction to Differential Equations
	7.29.2.	Verifying Solutions of Differential Equations
	7.29.3.	Solving Differential Equations Using Direct Integration
	7.29.4.	Solving First-Order ODEs Using Separation of Variables
	7.29.5.	Solving Initial Value Problems Using Separation of Variables
	7.29.6.	Modeling With Differential Equations
	7.29.7.	Further Modeling With Differential Equations
	7.29.8.	Exponential Growth and Decay Models With Differential Equations
	7.29.9.	Exponential Growth and Decay Models With Differential Equations: Calculating Unknown Times and Initial Values
	7.29.10.	Exponential Growth and Decay Models With Differential Equations: Half-Life Problems

7.30. Qualitative Techniques for Differential Equations

	7.30.1.	Qualitative Analysis of Differential Equations
	7.30.2.	Equilibrium Solutions of Differential Equations

7.31. Slope Fields

	7.31.1.	Slope Fields for Directly Integrable Differential Equations
	7.31.2.	Slope Fields for Autonomous Differential Equations
	7.31.3.	Slope Fields for Nonautonomous Differential Equations
	7.31.4.	Analyzing Slope Fields for Directly Integrable Differential Equations
	7.31.5.	Analyzing Slope Fields for Autonomous Differential Equations
	7.31.6.	Analyzing Slope Fields for Nonautonomous Differential Equations

Applications of Integration

18 topics

8.32. Applications of Integration

	8.32.1.	The Average Value of a Function
	8.32.2.	The Area Between Curves Expressed as Functions of X
	8.32.3.	The Area Between Curves Expressed as Functions of Y
	8.32.4.	Finding Areas Between Curves that Intersect at More Than Two Points

8.33. Connecting Position, Velocity and Acceleration Using Integrals

	8.33.1.	Calculating Velocity Using Integration
	8.33.2.	Determining Characteristics of Moving Objects Using Integration
	8.33.3.	Calculating the Position Function of a Particle Using Integration
	8.33.4.	Calculating the Displacement of a Particle Using Integration
	8.33.5.	Calculating the Total Distance Traveled by a Particle
	8.33.6.	Average Position, Velocity, and Acceleration

8.34. Volumes of Solids With Known Cross Sections

	8.34.1.	Volumes of Solids with Square Cross Sections
	8.34.2.	Volumes of Solids with Rectangular Cross Sections
	8.34.3.	Volumes of Solids with Triangular Cross Sections
	8.34.4.	Volumes of Solids with Circular Cross Sections

8.35. Volumes of Revolution

	8.35.1.	Volumes of Revolution Using the Disc Method: Rotation About the Coordinate Axes
	8.35.2.	Volumes of Revolution Using the Disc Method: Rotation About Other Axes
	8.35.3.	Volumes of Revolution Using the Washer Method: Rotation About the Coordinate Axes
	8.35.4.	Volumes of Revolution Using the Washer Method: Rotation About Other Axes

Applications of Technology

7 topics

9.36. Using Graphing Calculators

	9.36.1.	Evaluating Expressions Using a Graphing Calculator
	9.36.2.	Finding Roots of Functions Using a Graphing Calculator
	9.36.3.	Finding Intersections of Functions Using a Graphing Calculator
	9.36.4.	Finding Extrema of Functions Using a Graphing Calculator
	9.36.5.	Finding Derivatives Using a Graphing Calculator
	9.36.6.	Finding Definite Integrals Using a Graphing Calculator
	9.36.7.	Exploring Functions Using Technology

AP Calculus AB

Overview

Outcomes

Content