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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.

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

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
19 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 Polynomial Inequalities Using the Sign Table Method
2.4.8. Solving Rational Inequalities
2.4.9. Solving Inequalities Involving Exponential Functions and Polynomials
2.4.10. Solving Radical Inequalities
2.4.11. Solving Inequalities Involving Exponential Functions
2.4.12. Solving Inequalities Involving Logarithmic Functions
2.4.13. Solving Inequalities Involving Geometric Sequences
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
Parametric & Polar Coordinates
15 topics
3.6. Parametric Equations
3.6.1. Graphing Curves Defined Parametrically
3.6.2. Cartesian Equations of Parametric Curves
3.6.3. Finding Intersections of Parametric Curves and Lines
3.6.4. Differentiating Parametric Curves
3.6.5. Calculating Tangent and Normal Lines with Parametric Equations
3.6.6. Second Derivatives of Parametric Equations
3.6.7. The Arc Length of a Parametric Curve
3.7. Polar Coordinates
3.7.1. Introduction to Polar Coordinates
3.7.2. Converting from Polar Coordinates to Cartesian Coordinates
3.7.3. Polar Equations of Circles Centered at the Origin
3.7.4. Polar Equations of Radial Lines
3.7.5. Differentiating Curves Given in Polar Form
3.7.6. Further Differentiation of Curves Given in Polar Form
3.7.7. Finding the Area of a Polar Region
3.7.8. The Arc Length of a Polar Curve
Conic Sections
27 topics
4.8. Circles
4.8.1. The Center and Radius of a Circle in the Coordinate Plane
4.8.2. Equations of Circles Centered at the Origin
4.8.3. Equations of Circles Centered at a General Point
4.8.4. Finding the Center and Radius of a Circle by Completing the Square
4.8.5. Calculating Intercepts of Circles
4.8.6. Intersections of Circles with Lines
4.8.7. Parametric Equations of Circles
4.9. Parabolas
4.9.1. Upward and Downward Opening Parabolas
4.9.2. Left and Right Opening Parabolas
4.9.3. The Vertex of a Parabola
4.9.4. Calculating the Vertex of a Parabola by Completing the Square
4.9.5. Calculating Intercepts of Parabolas
4.9.6. Parametric Equations of Parabolas
4.9.7. Parametric Equations of Parabolas Centered at (h,k)
4.10. Ellipses
4.10.1. Introduction to Ellipses
4.10.2. Equations of Ellipses Centered at the Origin
4.10.3. Equations of Ellipses Centered at a General Point
4.10.4. Finding the Center and Axes of Ellipses by Completing the Square
4.10.5. Finding Intercepts of Ellipses
4.10.6. Parametric Equations of Ellipses
4.11. Hyperbolas
4.11.1. Equations of Hyperbolas Centered at the Origin
4.11.2. Equations of Hyperbolas Centered at a General Point
4.11.3. Asymptotes of Hyperbolas Centered at the Origin
4.11.4. Asymptotes of Hyperbolas Centered at a General Point
4.11.5. Finding Intercepts and Intersections of Hyperbolas
4.11.6. Parametric Equations of Horizontal Hyperbolas
4.11.7. Parametric Equations of Vertical Hyperbolas
26 topics
5.12. The Inverse Trigonometric Functions
5.12.1. Graphing the Inverse Sine Function
5.12.2. Graphing the Inverse Cosine Function
5.12.3. Graphing the Inverse Tangent Function
5.12.4. Evaluating Expressions Containing Inverse Trigonometric Functions
5.12.5. Limits of Inverse Trigonometric Functions
5.13. Elementary Trigonometric Equations
5.13.1. Elementary Trigonometric Equations Containing Sine
5.13.2. Elementary Trigonometric Equations Containing Cosine
5.13.3. Elementary Trigonometric Equations Containing Tangent
5.13.4. Elementary Trigonometric Equations Containing Secant
5.13.5. Elementary Trigonometric Equations Containing Cosecant
5.13.6. Elementary Trigonometric Equations Containing Cotangent
5.13.7. Solving Trigonometric Equations Using the Sin-Cos-Tan Identity
5.13.8. General Solutions of Elementary Trigonometric Equations
5.13.9. General Solutions of Trigonometric Equations With Transformed Functions
5.13.10. Trigonometric Equations Containing Transformed Tangent Functions
5.14. Trigonometric Identities
5.14.1. Simplifying Expressions Using Basic Trigonometric Identities
5.14.2. Simplifying Expressions Using the Pythagorean Identity
5.14.3. Alternate Forms of the Pythagorean Identity
5.14.4. Simplifying Expressions Using the Secant-Tangent Identity
5.14.5. Alternate Forms of the Secant-Tangent Identity
5.14.6. Simplifying Trigonometric Expressions Using the Cotangent-Cosecant Identity
5.15. The Sum and Difference Formulas
5.15.1. The Sum and Difference Formulas for Sine
5.15.2. The Sum and Difference Formulas for Cosine
5.15.3. Writing Sums of Trigonometric Functions in Amplitude-Phase Form
5.15.4. The Double-Angle Formula for Sine
5.15.5. The Double-Angle Formula for Cosine
Complex Numbers
16 topics
6.16. Further Complex Numbers
6.16.1. The Complex Conjugate
6.16.2. Special Properties of the Complex Conjugate
6.16.3. The Complex Conjugate and the Roots of a Quadratic Equation
6.16.4. Dividing Complex Numbers
6.16.5. Solving Equations by Equating Real and Imaginary Parts
6.16.6. Extending Polynomial Identities to the Complex Numbers
6.17. Euler's Formula
6.17.1. The Polar Form of a Complex Number
6.17.2. De Moivre's Theorem
6.17.3. Euler's Formula
6.17.4. The Roots of Unity
6.17.5. Properties of Roots of Unity
6.18. The Fundamental Theorem of Algebra
6.18.1. The Fundamental Theorem of Algebra for Quadratic Equations
6.18.2. The Fundamental Theorem of Algebra with Cubic Equations
6.18.3. Solving Cubic Equations With Complex Roots
6.18.4. The Fundamental Theorem of Algebra with Quartic Equations
6.18.5. Solving Quartic Equations With Complex Roots
Limits & Continuity
10 topics
7.19. Limits
7.19.1. L'Hopital's Rule
7.19.2. Limits of Sequences
7.19.3. Special Limits Involving Sine
7.19.4. Limits Involving the Exponential Function
7.19.5. Vertical Asymptotes of Rational Functions
7.19.6. Limits at Infinity and Horizontal Asymptotes of Rational Functions
7.19.7. Calculating Limits of Radical Functions Using Conjugate Multiplication
7.19.8. Evaluating Limits at Infinity by Comparing Relative Magnitudes of Functions
7.20. Continuity
7.20.1. Continuity of Functions
7.20.2. The Intermediate Value Theorem
25 topics
8.21. Differentiating Implicit and Inverse Functions
8.21.1. Implicit Differentiation
8.21.2. Calculating Slopes of Circles, Ellipses, and Parabolas
8.21.3. Calculating dy/dx Using dx/dy
8.21.4. Differentiating Inverse Functions
8.21.5. Differentiating an Inverse Function at a Point
8.21.6. Differentiating Inverse Trigonometric Functions
8.21.7. Differentiating Inverse Reciprocal Trigonometric Functions
8.21.8. Integration Using Inverse Trigonometric Functions
8.22. Analytical Applications of Differentiation
8.22.1. Connecting Differentiability and Continuity
8.22.2. The Mean Value Theorem
8.22.3. Global vs. Local Extrema and Critical Points
8.22.4. The Extreme Value Theorem
8.22.5. Using Differentiation to Calculate Critical Points
8.22.6. Determining Intervals on Which a Function Is Increasing or Decreasing
8.22.7. Using the First Derivative Test to Classify Local Extrema
8.22.8. The Candidates Test
8.22.9. Intervals of Concavity
8.22.10. Relating Concavity to the Second Derivative
8.22.11. Points of Inflection
8.22.12. The Second Derivative Test
8.22.13. Approximating Functions Using Local Linearity and Linearization
8.22.14. Second-Degree Taylor Polynomials
8.23. Estimating Derivatives
8.23.1. Estimating Derivatives Using a Forward Difference Quotient
8.23.2. Estimating Derivatives Using a Backward Difference Quotient
8.23.3. Estimating Derivatives Using a Central Difference Quotient
Definite Integrals
19 topics
9.24. Approximating Areas with Riemann Sums
9.24.1. Approximating Areas With the Left Riemann Sum
9.24.2. Approximating Areas With the Right Riemann Sum
9.24.3. Left and Right Riemann Sums in Sigma Notation
9.25. Definite Integrals
9.25.1. Defining Definite Integrals Using Left and Right Riemann Sums
9.25.2. The Fundamental Theorem of Calculus
9.25.3. Applying the Fundamental Theorem of Calculus to Exponential and Trigonometric Functions
9.25.4. The Sum and Constant Multiple Rules for Definite Integrals
9.25.5. Properties of Definite Integrals Involving the Limits of Integration
9.26. The Area Under a Curve
9.26.1. The Area Bounded by a Curve and the X-Axis
9.26.2. The Area Bounded by a Curve and the Y-Axis
9.26.3. Evaluating Definite Integrals Using Symmetry
9.26.4. Finding the Area Between a Curve and the X-Axis When They Intersect
9.26.5. Calculating the Definite Integral of a Function Given Its Graph
9.26.6. Definite Integrals of Piecewise Functions
9.27. Accumulation Functions
9.27.1. The Integral as an Accumulation Function
9.27.2. The Second Fundamental Theorem of Calculus
9.28. Applications of Integration
9.28.1. The Average Value of a Function
9.28.2. The Area Between Curves Expressed as Functions of X
9.28.3. The Arc Length of a Planar Curve
Integration Techniques
29 topics
10.29. Integration Using Substitution
10.29.1. Integrating Algebraic Functions Using Substitution
10.29.2. Integrating Linear Rational Functions Using Substitution
10.29.3. Integration Using Substitution
10.29.4. Calculating Definite Integrals Using Substitution
10.29.5. Further Integration of Algebraic Functions Using Substitution
10.29.6. Integrating Exponential Functions Using Linear Substitution
10.29.7. Integrating Exponential Functions Using Substitution
10.29.8. Integrating Trigonometric Functions Using Substitution
10.29.9. Integrating Logarithmic Functions Using Substitution
10.29.10. Integration by Substitution With Inverse Trigonometric Functions
10.30. Integration Using Trigonometric Identities
10.30.1. Integration Using Basic Trigonometric Identities
10.30.2. Integration Using the Pythagorean Identities
10.30.3. Integration Using the Double-Angle Formulas
10.31. Special Techniques for Integration
10.31.1. Integrating Functions Using Polynomial Division
10.31.2. Integrating Functions by Completing the Square
10.32. Integration by Parts
10.32.1. Introduction to Integration by Parts
10.32.2. Using Integration by Parts to Calculate Integrals With Logarithms
10.32.3. Applying the Integration By Parts Formula Twice
10.32.4. Integration by Parts in Cyclic Cases
10.33. Integration Using Partial Fractions
10.33.1. Expressing Rational Functions as Sums of Partial Fractions
10.33.2. Expressing Rational Functions with Repeated Factors as Sums of Partial Fractions
10.33.3. Expressing Rational Functions with Irreducible Quadratic Factors as Sums of Partial Fractions
10.33.4. Integrating Rational Functions Using Partial Fractions
10.33.5. Integrating Rational Functions with Repeated Factors
10.33.6. Integrating Rational Functions with Irreducible Quadratic Factors
10.34. Improper Integrals
10.34.1. Improper Integrals
10.34.2. Improper Integrals Involving Exponential Functions
10.34.3. Improper Integrals Involving Arctangent
10.34.4. Improper Integrals Over the Real Line
Contextual Applications of Calculus
21 topics
11.35. Displacement, Velocity, and Acceleration
11.35.1. Calculating Velocity for Straight-Line Motion Using Differentiation
11.35.2. Calculating Acceleration for Straight-Line Motion Using Differentiation
11.35.3. Determining Characteristics of Moving Objects Using Differentiation
11.35.4. Calculating Velocity Using Integration
11.35.5. Determining Characteristics of Moving Objects Using Integration
11.35.6. Calculating the Position Function of a Particle Using Integration
11.35.7. Calculating the Displacement of a Particle Using Integration
11.36. The Planar Motion of a Particle
11.36.1. Velocity and Acceleration for Plane Motion
11.36.2. Calculating Displacement for Plane Motion
11.36.3. Calculating Velocity for Plane Motion Using Differentiation
11.36.4. Calculating Acceleration for Plane Motion Using Differentiation
11.36.5. Finding Velocity Vectors in Two Dimensions Using Integration
11.36.6. Finding Displacement Vectors in Two Dimensions Using Integration
11.37. Related Rates and Optimization
11.37.1. Rates of Change in Applied Contexts
11.37.2. Introduction to Related Rates
11.37.3. Related Rates With Implicit Functions
11.37.4. Calculating Related Rates With Circles and Spheres
11.37.5. Calculating Related Rates Using the Pythagorean Theorem
11.37.6. Solving Optimization Problems Using Derivatives
11.37.7. Optimizing Distances
11.37.8. Optimizing Distances to Curves
Differential Equations
9 topics
12.38. Introduction to Differential Equations
12.38.1. Introduction to Differential Equations
12.38.2. Verifying Solutions of Differential Equations
12.38.3. Solving Differential Equations Using Direct Integration
12.38.4. Solving First-Order ODEs Using Separation of Variables
12.38.5. Solving Initial Value Problems Using Separation of Variables
12.38.6. Qualitative Analysis of Differential Equations
12.38.7. Modeling With Differential Equations
12.39. Numerical Solutions of Differential Equations
12.39.1. Euler's Method: Calculating One Step
12.39.2. Euler's Method: Calculating Multiple Steps
17 topics
13.40. Vectors in 3D Cartesian Coordinates
13.40.1. Three-Dimensional Vectors in Component Form
13.40.2. Addition and Scalar Multiplication of Cartesian Vectors in 3D
13.40.3. Calculating the Magnitude of Cartesian Vectors in 3D
13.41. The Dot Product
13.41.1. Calculating the Dot Product Using Angle and Magnitude
13.41.2. Calculating the Dot Product Using Components
13.41.3. The Angle Between Two Vectors
13.41.4. Calculating a Scalar Projection
13.41.5. Calculating a Vector Projection
13.42. The Cross Product
13.42.1. The Cross Product of Two Vectors
13.42.2. Properties of the Cross Product
13.42.3. Calculating the Cross Product Using Determinants
13.42.4. Finding Areas Using the Cross Product
13.42.5. The Scalar Triple Product
13.42.6. Volumes of Parallelepipeds
13.43. Vector-Valued Functions
13.43.1. Defining Vector-Valued Functions
13.43.2. Differentiating Vector-Valued Functions
13.43.3. Integrating Vector-Valued Functions
Linear Algebra
37 topics
14.44. Introduction to Matrices
14.44.1. Introduction to Matrices
14.44.2. Index Notation for Matrices
14.44.3. Adding and Subtracting Matrices
14.44.4. Properties of Matrix Addition
14.44.5. Scalar Multiplication of Matrices
14.44.6. Zero, Square, Diagonal and Identity Matrices
14.44.7. The Transpose of a Matrix
14.45. Matrix Multiplication
14.45.1. Multiplying a Matrix by a Column Vector
14.45.2. Multiplying Square Matrices
14.45.3. Conformability for Matrix Multiplication
14.45.4. Multiplying Matrices
14.45.5. Powers of Matrices
14.45.6. Multiplying a Matrix by the Identity Matrix
14.45.7. Properties of Matrix Multiplication
14.45.8. Representing 2x2 Systems of Equations Using a Matrix Product
14.45.9. Representing 3x3 Systems of Equations Using a Matrix Product
14.46. Determinants
14.46.1. The Determinant of a 2x2 Matrix
14.46.2. The Geometric Interpretation of the 2x2 Determinant
14.46.3. The Minors of a 3x3 Matrix
14.46.4. The Determinant of a 3x3 Matrix
14.47. The Inverse of a Matrix
14.47.1. Introduction to the Inverse of a Matrix
14.47.2. Inverses of 2x2 Matrices
14.47.3. Calculating the Inverse of a 3x3 Matrix Using the Cofactor Method
14.47.4. Solving 2x2 Systems of Equations Using Inverse Matrices
14.47.5. Solving Systems of Equations Using Inverse Matrices
14.48. Linear Transformations
14.48.1. Introduction to Linear Transformations
14.48.2. The Standard Matrix of a Linear Transformation
14.48.3. Linear Transformations of Points and Lines in the Plane
14.48.4. Linear Transformations of Objects in the Plane
14.48.5. Dilations and Reflections as Linear Transformations
14.48.6. Shear and Stretch as Linear Transformations
14.48.7. Right-Angle Rotations as Linear Transformations
14.48.8. Rotations as Linear Transformations
14.48.9. Combining Linear Transformations Using 2x2 Matrices
14.48.10. Inverting Linear Transformations
14.48.11. Area Scale Factors of Linear Transformations
14.48.12. Singular Linear Transformations in the Plane
17 topics
15.49. Conditional Probability
15.49.1. Conditional Probabilities From Venn Diagrams
15.49.2. The Multiplication Law for Conditional Probability
15.49.3. The Law of Total Probability
15.49.4. Independent Events
15.50. Discrete Random Variables
15.50.1. Probability Mass Functions of Discrete Random Variables
15.50.2. Cumulative Distribution Functions for Discrete Random Variables
15.50.3. Expected Values of Discrete Random Variables
15.50.4. The Binomial Distribution
15.50.5. Modeling With the Binomial Distribution
15.50.6. The Geometric Distribution
15.50.7. Modeling With the Geometric Distribution
15.51. The Normal Distribution
15.51.1. The Standard Normal Distribution
15.51.2. Symmetry Properties of the Standard Normal Distribution
15.51.3. The Normal Distribution
15.51.4. Mean and Variance of the Normal Distribution
15.51.5. Percentage Points of the Standard Normal Distribution
15.51.6. Modeling With the Normal Distribution