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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.

## Content

• 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.
1.
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
2.
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
3.
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
4.
Trigonometry
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
5.
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
6.
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
7.
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. 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
8.
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
9.
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
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. 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.
Vectors
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
14.
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
15.
Probability
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