Math Academy

Advanced Algebra

Leverage factoring and graphing as a strategy to solve polynomial inequalities.
Visualize two-variable linear inequalities in the coordinate plane.
Apply trigonometric identities to simplify trigonometric expressions.
Solve trigonometric equations by extending previous knowledge of algebra and leveraging properties of trigonometric functions.

Limits and Continuity

Compute limits algebraically using advanced techniques such as L'Hopital's rule and conjugate multiplication.
Understand the relationship between limits and asymptotes.
Explain the behavior of a function on an interval using the Intermediate Value Theorem.

Sequences and Series

Determine whether a sequence is convergent or divergent, and compute the limit of a convergent sequence.
Express an arithmetic or geometric series using sigma notation.
Compute the sum of an arithmetic or geometric series.

Differentiation

Relate the graph of a function to properties of its derivative and understand the relationship between differentiability and continuity.
Understand that the derivative of position is velocity, and the derivative of velocity is acceleration.
Calculate derivatives of implicit and inverse functions and use implicit differentiation to solve related rates problems.
Estimate derivatives numerically and approximate values of functions using linearization.
Solve optimization problems by using the derivative to find extrema of functions.

Integration

Approximate areas using Riemann sums and compute the definite 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 arc length and area and translate between position, velocity, and acceleration.
Interpret accumulation functions geometrically and compute their derivatives.
Solve advanced integrals using techniques like substitution, long division, completing the square, partial fractions, and integration by parts.
Evaluate improper integrals.

Differential Equations

Verify solutions of differential equations and solve elementary differential equations using direct integration.
Solve differential equations using separation of variables and fit unknown coefficients to initial conditions.
Estimate solutions to initial value problems using Euler’s method.

Conic Sections, Parametric Curves, and Polar Curves

Translate between graphs and equations of circles, parabolas, ellipses, and hyperbolas.
Extend prior knowledge of rectangular curves to parametric and polar curves.
Extend differentiation techniques to parametric and polar curves.
Use integration to calculate the area of a polar region or the arc length of a polar curve.

Complex Numbers, Vectors, and Matrices

Connect the algebraic and geometric interpretations of complex numbers through Euler’s formula.
Generalize prior intuitions about arithmetic to vectors and matrices in higher-dimensional space.
Compute dot products and cross products of vectors and interpret them geometrically.
Interpret matrices as linear transformations of points in the coordinate plane.
Compute inverses and determinants of matrices and interpret them geometrically.

Probability and Statistics

Understand independent events both conceptually and quantitatively from the perspective of conditional probability.
Define and perform computations with probability mass functions, cumulative distributions, and expected values of discrete random variables.
Apply binomial, geometric, and normal distributions in modeling contexts.

Finite Series

12 topics

1.1. Arithmetic Series

	1.1.1.	Expressing an Arithmetic Series in Sigma Notation
	1.1.2.	Finding the Sum of an Arithmetic Series
	1.1.3.	Finding the First Term of an Arithmetic Series
	1.1.4.	Calculating the Number of Terms in an Arithmetic Series

1.2. Finite Geometric Series

	1.2.1.	The Sum of a Finite Geometric Series
	1.2.2.	The Sum of the First N Terms of a Geometric Series
	1.2.3.	Writing Geometric Series in Sigma Notation
	1.2.4.	Finding the Sum of a Geometric Series Given in Sigma Notation

1.3. The Binomial Theorem

	1.3.1.	Pascal's Triangle and the Binomial Coefficients
	1.3.2.	Expanding a Binomial Using Binomial Coefficients
	1.3.3.	The Special Case of the Binomial Theorem
	1.3.4.	Approximating Values Using the Binomial Theorem

Inequalities

16 topics

2.4. Single-Variable Inequalities

	2.4.1.	Solving Elementary Quadratic Inequalities
	2.4.2.	Solving Quadratic Inequalities From Graphs
	2.4.3.	Solving Quadratic Inequalities Using the Graphical Method
	2.4.4.	Solving Quadratic Inequalities Using the Sign Table Method
	2.4.5.	Inequalities Involving Powers of Binomials
	2.4.6.	Solving Polynomial Inequalities Using a Graphical Method
	2.4.7.	Solving Inequalities Involving Exponential Functions and Polynomials
	2.4.8.	Solving Inequalities Involving Radical Functions
	2.4.9.	Solving Inequalities Involving Exponential Functions
	2.4.10.	Solving Inequalities Involving Logarithmic Functions

2.5. Two-Variable Inequalities

	2.5.1.	Graphing Strict Two-Variable Linear Inequalities
	2.5.2.	Graphing Non-Strict Two-Variable Linear Inequalities
	2.5.3.	Further Graphing of Two-Variable Linear Inequalities
	2.5.4.	Solving Systems of Linear Inequalities
	2.5.5.	Solving Two-Variable Nonlinear Inequalities
	2.5.6.	Further Solving of Two-Variable Nonlinear Inequalities

Conic Sections

16 topics

3.6. Circles

	3.6.1.	The Center and Radius of a Circle in the Coordinate Plane
	3.6.2.	Equations of Circles Centered at the Origin
	3.6.3.	Equations of Circles Centered at a General Point
	3.6.4.	Finding the Center and Radius of a Circle by Completing the Square
	3.6.5.	Calculating Intercepts of Circles
	3.6.6.	Intersections of Circles with Lines

3.7. Parabolas

	3.7.1.	Upward and Downward Opening Parabolas
	3.7.2.	Left and Right Opening Parabolas
	3.7.3.	The Vertex of a Parabola
	3.7.4.	Calculating the Vertex of a Parabola by Completing the Square

3.8. Ellipses

	3.8.1.	Introduction to Ellipses
	3.8.2.	Equations of Ellipses Centered at the Origin
	3.8.3.	Equations of Ellipses Centered at a General Point
	3.8.4.	Finding the Center and Axes of Ellipses by Completing the Square

3.9. Hyperbolas

	3.9.1.	Equations of Hyperbolas Centered at the Origin
	3.9.2.	Equations of Hyperbolas Centered at a General Point

Trigonometry

26 topics

4.10. The Inverse Trigonometric Functions

	4.10.1.	Graphing the Inverse Sine Function
	4.10.2.	Graphing the Inverse Cosine Function
	4.10.3.	Graphing the Inverse Tangent Function
	4.10.4.	Evaluating Expressions Containing Inverse Trigonometric Functions
	4.10.5.	Limits of Inverse Trigonometric Functions

4.11. Elementary Trigonometric Equations

	4.11.1.	Elementary Trigonometric Equations Containing Sine
	4.11.2.	Elementary Trigonometric Equations Containing Cosine
	4.11.3.	Elementary Trigonometric Equations Containing Tangent
	4.11.4.	Elementary Trigonometric Equations Containing Secant
	4.11.5.	Elementary Trigonometric Equations Containing Cosecant
	4.11.6.	Elementary Trigonometric Equations Containing Cotangent
	4.11.7.	Solving Trigonometric Equations Using the Sin-Cos-Tan Identity
	4.11.8.	General Solutions of Elementary Trigonometric Equations
	4.11.9.	General Solutions of Trigonometric Equations With Transformed Functions
	4.11.10.	Trigonometric Equations Containing Transformed Tangent Functions

4.12. Trigonometric Identities

	4.12.1.	Simplifying Expressions Using Basic Trigonometric Identities
	4.12.2.	Simplifying Expressions Using the Pythagorean Identity
	4.12.3.	Alternate Forms of the Pythagorean Identity
	4.12.4.	Simplifying Expressions Using the Secant-Tangent Identity
	4.12.5.	Alternate Forms of the Secant-Tangent Identity
	4.12.6.	Simplifying Trigonometric Expressions Using the Cotangent-Cosecant Identity

4.13. The Sum and Difference Formulas

	4.13.1.	The Sum and Difference Formulas for Sine
	4.13.2.	The Sum and Difference Formulas for Cosine
	4.13.3.	Writing Sums of Trigonometric Functions in Amplitude-Phase Form
	4.13.4.	The Double-Angle Formula for Sine
	4.13.5.	The Double-Angle Formula for Cosine

Complex Numbers

14 topics

5.14. Further Complex Numbers

	5.14.1.	The Complex Conjugate
	5.14.2.	Special Properties of the Complex Conjugate
	5.14.3.	The Complex Conjugate and the Roots of a Quadratic Equation
	5.14.4.	Dividing Complex Numbers
	5.14.5.	Solving Equations by Equating Real and Imaginary Parts
	5.14.6.	Extending Polynomial Identities to the Complex Numbers

5.15. Euler's Formula

	5.15.1.	The Polar Form of a Complex Number
	5.15.2.	De Moivre's Theorem
	5.15.3.	Euler's Formula
	5.15.4.	Roots of Unity
	5.15.5.	Properties of Roots of Unity

5.16. The Fundamental Theorem of Algebra

	5.16.1.	The Fundamental Theorem of Algebra for Quadratic Equations
	5.16.2.	The Fundamental Theorem of Algebra with Cubic Equations
	5.16.3.	Solving Cubic Equations With Complex Roots

Limits & Continuity

9 topics

6.17. Limits

	6.17.1.	L'Hopital's Rule
	6.17.2.	Limits of Sequences
	6.17.3.	Special Limits Involving Sine
	6.17.4.	Vertical Asymptotes of Rational Functions
	6.17.5.	Limits at Infinity and Horizontal Asymptotes of Rational Functions
	6.17.6.	Calculating Limits of Radical Functions Using Conjugate Multiplication
	6.17.7.	Evaluating Limits at Infinity by Comparing Relative Magnitudes of Functions

6.18. Continuity

	6.18.1.	Continuity of Functions
	6.18.2.	The Intermediate Value Theorem

Differentiation

23 topics

7.19. Differentiating Implicit and Inverse Functions

	7.19.1.	Implicit Differentiation
	7.19.2.	Calculating Slopes of Circles, Ellipses, and Parabolas
	7.19.3.	Calculating dy/dx Using dx/dy
	7.19.4.	Differentiating Inverse Functions
	7.19.5.	Differentiating Inverse Trigonometric Functions
	7.19.6.	Differentiating Inverse Reciprocal Trigonometric Functions
	7.19.7.	Integration Using Inverse Trigonometric Functions

7.20. Analytical Applications of Differentiation

	7.20.1.	Connecting Differentiability and Continuity
	7.20.2.	The Mean Value Theorem
	7.20.3.	Global vs. Local Extrema and Critical Points
	7.20.4.	The Extreme Value Theorem
	7.20.5.	Using Differentiation to Calculate Critical Points
	7.20.6.	Determining Intervals on Which a Function Is Increasing or Decreasing
	7.20.7.	Using the First Derivative Test to Classify Local Extrema
	7.20.8.	The Candidates Test
	7.20.9.	Intervals of Concavity
	7.20.10.	Relating Concavity to the Second Derivative
	7.20.11.	Points of Inflection
	7.20.12.	Using the Second Derivative Test to Determine Extrema
	7.20.13.	Approximating Functions Using Local Linearity and Linearization

7.21. Estimating Derivatives

	7.21.1.	Estimating Derivatives Using a Forward Difference Quotient
	7.21.2.	Estimating Derivatives Using a Backward Difference Quotient
	7.21.3.	Estimating Derivatives Using a Central Difference Quotient

Definite Integrals

18 topics

8.22. Approximating Areas with Riemann Sums

	8.22.1.	Approximating Areas With the Left Riemann Sum
	8.22.2.	Approximating Areas With the Right Riemann Sum
	8.22.3.	Left and Right Riemann Sums in Sigma Notation

8.23. Definite Integrals

	8.23.1.	Defining Definite Integrals Using Left and Right Riemann Sums
	8.23.2.	The Fundamental Theorem of Calculus
	8.23.3.	Applying the Fundamental Theorem of Calculus to Exponential and Trigonometric Functions
	8.23.4.	The Sum and Constant Multiple Rules for Definite Integrals
	8.23.5.	Properties of Definite Integrals Involving the Limits of Integration

8.24. The Area Under a Curve

	8.24.1.	The Area Bounded by a Curve and the X-Axis
	8.24.2.	Evaluating Definite Integrals Using Symmetry
	8.24.3.	Finding the Area Between a Curve and the X-Axis When They Intersect
	8.24.4.	Calculating the Definite Integral of a Function Given Its Graph
	8.24.5.	Definite Integrals of Piecewise Functions

8.25. Accumulation Functions

	8.25.1.	The Integral as an Accumulation Function
	8.25.2.	The Second Fundamental Theorem of Calculus

8.26. Applications of Integration

	8.26.1.	The Average Value of a Function
	8.26.2.	The Area Between Curves Expressed as Functions of X
	8.26.3.	The Arc Length of a Smooth Planar Curve

Integration Techniques

29 topics

9.27. Integration Using Substitution

	9.27.1.	Integrating Algebraic Functions Using Substitution
	9.27.2.	Integrating Linear Rational Functions Using Substitution
	9.27.3.	Integration Using Substitution
	9.27.4.	Calculating Definite Integrals Using Substitution
	9.27.5.	Further Integration of Algebraic Functions Using Substitution
	9.27.6.	Integrating Exponential Functions Using Linear Substitution
	9.27.7.	Integrating Exponential Functions Using Substitution
	9.27.8.	Integrating Trigonometric Functions Using Substitution
	9.27.9.	Integrating Logarithmic Functions Using Substitution
	9.27.10.	Integration by Substitution With Inverse Trigonometric Functions

9.28. Integration Using Trigonometric Identities

	9.28.1.	Integration Using Basic Trigonometric Identities
	9.28.2.	Integration Using the Pythagorean Identities
	9.28.3.	Integration Using the Double Angle Formulas

9.29. Special Techniques for Integration

	9.29.1.	Integrating Functions Using Polynomial Division
	9.29.2.	Integrating Functions by Completing the Square

9.30. Integration by Parts

	9.30.1.	Introduction to Integration by Parts
	9.30.2.	Using Integration by Parts to Calculate Integrals With Logarithms
	9.30.3.	Applying the Integration By Parts Formula Twice
	9.30.4.	Integration by Parts in Cyclic Cases

9.31. Integration Using Partial Fractions

	9.31.1.	Expressing Rational Functions as Sums of Partial Fractions
	9.31.2.	Expressing Rational Functions with Repeated Factors as Sums of Partial Fractions
	9.31.3.	Expressing Rational Functions with Irreducible Quadratic Factors as Sums of Partial Fractions
	9.31.4.	Integrating Rational Functions Using Partial Fractions
	9.31.5.	Integrating Rational Functions with Repeated Factors
	9.31.6.	Integrating Rational Functions with Irreducible Quadratic Factors

9.32. Improper Integrals

	9.32.1.	Improper Integrals
	9.32.2.	Improper Integrals Involving Exponential Functions
	9.32.3.	Improper Integrals Involving Arctangent
	9.32.4.	Improper Integrals Over the Real Line

10.

Contextual Applications of Calculus

14 topics

10.33. Displacement, Velocity, and Acceleration

	10.33.1.	Distance-Time Graphs
	10.33.2.	Calculating Acceleration From a Speed-Time Graph
	10.33.3.	Calculating Distance From a Speed-Time Graph
	10.33.4.	Calculating Velocity for Straight-Line Motion Using Differentiation
	10.33.5.	Calculating Acceleration for Straight-Line Motion Using Differentiation
	10.33.6.	Calculating Velocity Using Integration
	10.33.7.	Calculating the Position Function of a Particle Using Integration
	10.33.8.	Calculating the Displacement of a Particle Using Integration

10.34. Related Rates and Optimization

	10.34.1.	Rates of Change in Applied Contexts
	10.34.2.	Introduction to Related Rates
	10.34.3.	Calculating Related Rates With Circles and Spheres
	10.34.4.	Calculating Related Rates Using the Pythagorean Theorem
	10.34.5.	Optimization Problems Involving Rectangles
	10.34.6.	Finding Minimum Distances

11.

Differential Equations

8 topics

11.35. Introduction to Differential Equations

	11.35.1.	Introduction to Differential Equations
	11.35.2.	Verifying Solutions of Differential Equations
	11.35.3.	Solving Differential Equations Using Direct Integration
	11.35.4.	Solving First-Order ODEs Using Separation of Variables
	11.35.5.	Solving Initial Value Problems Using Separation of Variables
	11.35.6.	Qualitative Analysis of Differential Equations

11.36. Numerical Solutions of Differential Equations

	11.36.1.	Euler's Method: Calculating One Step
	11.36.2.	Euler's Method: Calculating Multiple Steps

12.

Parametric & Polar Coordinates

15 topics

12.37. Parametric Equations

	12.37.1.	Graphing Curves Defined Parametrically
	12.37.2.	Cartesian Equations of Parametric Curves
	12.37.3.	Finding Intersections of Parametric Curves and Lines
	12.37.4.	Differentiating Parametric Curves
	12.37.5.	Calculating Tangent and Normal Lines with Parametric Equations
	12.37.6.	Second Derivatives of Parametric Equations
	12.37.7.	Arc Lengths of Parametric Curves

12.38. Polar Coordinates

	12.38.1.	Introduction to Polar Coordinates
	12.38.2.	Converting from Polar Coordinates to Cartesian Coordinates
	12.38.3.	Polar Equations of Circles Centered at the Origin
	12.38.4.	Polar Equations of Radial Lines
	12.38.5.	Differentiating Curves Given in Polar Form
	12.38.6.	Further Differentiation of Curves Given in Polar Form
	12.38.7.	Finding the Area of a Polar Region
	12.38.8.	The Arc Length of a Polar Curve

13.

Vectors

16 topics

13.39. Vectors in 3D Cartesian Coordinates

	13.39.1.	Three-Dimensional Vectors in Component Form
	13.39.2.	Addition and Scalar Multiplication of Cartesian Vectors in 3D
	13.39.3.	Calculating the Magnitude of Cartesian Vectors in 3D

13.40. The Dot Product

	13.40.1.	Calculating the Dot Product Using Angle and Magnitude
	13.40.2.	Calculating the Dot Product Using Components
	13.40.3.	The Angle Between Two Vectors
	13.40.4.	Calculating a Scalar Projection
	13.40.5.	Calculating a Vector Projection

13.41. The Cross Product

	13.41.1.	Calculating the Cross Product of Two Vectors Using the Definition
	13.41.2.	Calculating the Cross Product Using Determinants
	13.41.3.	Finding Areas Using the Cross Product
	13.41.4.	The Scalar Triple Product
	13.41.5.	Volumes of Parallelepipeds

13.42. Vector-Valued Functions

	13.42.1.	Defining Vector-Valued Functions
	13.42.2.	Differentiating Vector-Valued Functions
	13.42.3.	Integrating Vector-Valued Functions

14.

Linear Algebra

35 topics

14.43. Introduction to Matrices

	14.43.1.	Introduction to Matrices
	14.43.2.	Index Notation for Matrices
	14.43.3.	Adding and Subtracting Matrices
	14.43.4.	Properties of Matrix Addition
	14.43.5.	Scalar Multiplication of Matrices
	14.43.6.	Zero, Square, Diagonal and Identity Matrices
	14.43.7.	The Transpose of a Matrix

14.44. Matrix Multiplication

	14.44.1.	Multiplying a Matrix by a Column Vector
	14.44.2.	Multiplying Square Matrices
	14.44.3.	Conformability for Matrix Multiplication
	14.44.4.	Multiplying Matrices
	14.44.5.	Powers of Matrices
	14.44.6.	Multiplying a Matrix by the Identity Matrix
	14.44.7.	Properties of Matrix Multiplication
	14.44.8.	Representing 2x2 Systems of Equations Using a Matrix Product
	14.44.9.	Representing 3x3 Systems of Equations Using a Matrix Product

14.45. Determinants

	14.45.1.	The Determinant of a 2x2 Matrix
	14.45.2.	The Geometric Interpretation of the 2x2 Determinant
	14.45.3.	The Minors of a 3x3 Matrix
	14.45.4.	The Determinant of a 3x3 Matrix

14.46. The Inverse of a Matrix

	14.46.1.	Introduction to the Inverse of a Matrix
	14.46.2.	Inverses of 2x2 Matrices
	14.46.3.	Solving 2x2 Systems of Equations Using Inverse Matrices

14.47. Linear Transformations

	14.47.1.	Introduction to Linear Transformations
	14.47.2.	The Standard Matrix of a Linear Transformation
	14.47.3.	Linear Transformations of Points and Lines in the Plane
	14.47.4.	Linear Transformations of Objects in the Plane
	14.47.5.	Dilations and Reflections as Linear Transformations
	14.47.6.	Shear and Stretch as Linear Transformations
	14.47.7.	Right-Angle Rotations as Linear Transformations
	14.47.8.	Rotations as Linear Transformations
	14.47.9.	Combining Linear Transformations Using 2x2 Matrices
	14.47.10.	Inverting Linear Transformations
	14.47.11.	Area Scale Factors of Linear Transformations
	14.47.12.	Singular Linear Transformations in the Plane

15.

Probability

16 topics

15.48. Conditional Probability

	15.48.1.	Conditional Probabilities From Venn Diagrams
	15.48.2.	The Multiplication Law for Conditional Probability
	15.48.3.	The Law of Total Probability
	15.48.4.	Independent Events

15.49. Discrete Random Variables

	15.49.1.	Probability Mass Functions of Discrete Random Variables
	15.49.2.	Cumulative Distribution Functions for Discrete Random Variables
	15.49.3.	Expected Values of Discrete Random Variables
	15.49.4.	The Binomial Distribution
	15.49.5.	Modeling With the Binomial Distribution
	15.49.6.	The Geometric Distribution
	15.49.7.	Modeling With the Geometric Distribution

15.50. The Normal Distribution

	15.50.1.	The Standard Normal Distribution
	15.50.2.	The Normal Distribution
	15.50.3.	Mean and Variance of the Normal Distribution
	15.50.4.	Percentage Points of the Standard Normal Distribution
	15.50.5.	Modeling With the Normal Distribution

Mathematical Foundations III

Outcomes

Content

Advanced Algebra

Limits and Continuity

Sequences and Series

Differentiation

Integration

Differential Equations

Conic Sections, Parametric Curves, and Polar Curves

Complex Numbers, Vectors, and Matrices

Probability and Statistics