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Mathematical Foundations III

Learn advanced calculus techniques for computing limits, derivatives, and integrals, and apply calculus to solve problems in the context of related rates, optimization, particle motion, and differential equations. Dive deeper into complex numbers, vectors, matrices, parametric and polar curves, probability, and statistics.



Advanced Algebra

Limits and Continuity

Sequences and Series



Differential Equations

Conic Sections, Parametric Curves, and Polar Curves

Complex Numbers, Vectors, and Matrices

Probability and Statistics

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
14 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
2.5. 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
25 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
4.11. 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
13 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. The 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
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. Using the Candidates Test to Determine Global Extrema
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 Functions with Discontinuities
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
27 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 Over the Real Line
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
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
Parametric & Polar Coordinates
15 topics
12.37. Parametric Equations
12.37.1. Graphing Curves Defined Parametrically
12.37.2. Finding the Cartesian Equation of Curves Defined Parametrically
12.37.3. Finding Intersections of Parametric Curves and Lines
12.37.4. Defining and Differentiating Parametric Equations
12.37.5. Calculating Tangent and Normal Lines with Parametric Equations
12.37.6. Second Derivatives of Parametric Equations
12.37.7. Finding the Arc Lengths of Curves Given by Parametric Equations
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 topics
13.39. Vectors in 3D Cartesian Coordinates
13.39.1. Three-Dimensional Vectors Expressed 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
Linear Algebra
34 topics
14.42. Introduction to Matrices
14.42.1. Introduction to Matrices
14.42.2. Index Notation for Matrices
14.42.3. Adding and Subtracting Matrices
14.42.4. Properties of Matrix Addition
14.42.5. Scalar Multiplication of Matrices
14.42.6. Zero, Square, Diagonal and Identity Matrices
14.42.7. The Transpose of a Matrix
14.43. Matrix Multiplication
14.43.1. Multiplying a Matrix by a Column Vector
14.43.2. Multiplying Square Matrices
14.43.3. Conformability for Matrix Multiplication
14.43.4. Multiplying Matrices
14.43.5. Powers of Matrices
14.43.6. Multiplying a Matrix by the Identity Matrix
14.43.7. Properties of Matrix Multiplication
14.43.8. Representing 2x2 Systems of Equations Using a Matrix Product
14.43.9. Representing 3x3 Systems of Equations Using a Matrix Product
14.44. Determinants
14.44.1. The Determinant of a 2x2 Matrix
14.44.2. The Geometric Interpretation of the 2x2 Determinant
14.44.3. The Minors of a 3x3 Matrix
14.44.4. The Determinant of a 3x3 Matrix
14.45. The Inverse of a Matrix
14.45.1. Introduction to the Inverse of a Matrix
14.45.2. Calculating the Inverse of a 2x2 Matrix
14.45.3. Solving 2x2 Systems of Equations Using Inverse Matrices
14.46. Linear Transformations
14.46.1. Introduction to Linear Transformations
14.46.2. The Standard Matrix of a Linear Transformation
14.46.3. Linear Transformations of Points and Lines in the Plane
14.46.4. Linear Transformations of Objects in the Plane
14.46.5. Dilations and Reflections as Linear Transformations
14.46.6. Shear and Stretch as Linear Transformations
14.46.7. Right-Angle Rotations as Linear Transformations
14.46.8. Rotations as Linear Transformations
14.46.9. Combining Linear Transformations Using 2x2 Matrices
14.46.10. Using Inverse Matrices to Reverse Linear Transformations
14.46.11. Determining the Area Scale Factor of a Linear Transformation
16 topics
15.47. Conditional Probability
15.47.1. Conditional Probabilities From Venn Diagrams
15.47.2. The Multiplication Law for Conditional Probability
15.47.3. The Law of Total Probability
15.47.4. Independent Events
15.48. Discrete Random Variables
15.48.1. Probability Mass Functions of Discrete Random Variables
15.48.2. Cumulative Distribution Functions for Discrete Random Variables
15.48.3. Expected Values of Discrete Random Variables
15.48.4. The Binomial Distribution
15.48.5. Modeling With the Binomial Distribution
15.48.6. The Geometric Distribution
15.48.7. Modeling With the Geometric Distribution
15.49. The Normal Distribution
15.49.1. The Standard Normal Distribution
15.49.2. The Normal Distribution
15.49.3. Mean and Variance of the Normal Distribution
15.49.4. Percentage Points of the Standard Normal Distribution
15.49.5. Modeling With the Normal Distribution