Learn the mathematics of change that underlies science and engineering. Master limits, derivatives, and the basics of integration.

- Connect algebraic and graphical interpretations of limits, including relationships to vertical and horizontal asymptotes.
- Estimate limits numerically and compute limits using algebraic manipulation.
- Define continuity in terms of limits, determine intervals over which a function is continuous, and remove discontinuities of functions.
- Explain the behavior of a function on an interval using the Intermediate Value Theorem.

- Interpret the difference quotient geometrically and use it to compute the derivative of a function.
- Estimate derivatives numerically and apply rules to compute derivatives of a variety of algebraic functions, including higher-order derivatives.
- Understand the relationship between differentiability and continuity.
- Use implicit differentiation to solve related rates problems.
- Approximate values of functions using linearization.
- Use L’Hopital’s Rule to calculate limits of indeterminate forms.
- Relate the graph of a function to properties of its derivative.
- Solve optimization problems by using the derivative to find extrema of functions.

- Evaluate Riemann sums and interpret them geometrically and contextually.
- Compute the 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 the area between a curve and an axis.
- Interpret accumulation functions geometrically and contextually and compute their derivatives.

1.

Preliminaries
6 topics

1.1. The Hyperbolic Functions

1.1.1. | The Hyperbolic Functions | |

1.1.2. | The Reciprocal Hyperbolic Functions | |

1.1.3. | Solving Equations Containing Hyperbolic Functions | |

1.1.4. | Graphs of the Hyperbolic Functions | |

1.1.5. | Graphs of the Reciprocal Hyperbolic Functions | |

1.1.6. | The Inverse Hyperbolic Functions |

2.

Limits
33 topics

2.2. Estimating Limits from Graphs

2.2.1. | The Finite Limit of a Function | |

2.2.2. | The Left and Right-Sided Limits of a Function | |

2.2.3. | Finding the Existence of a Limit Using One-Sided Limits | |

2.2.4. | Limits at Infinity from Graphs | |

2.2.5. | Infinite Limits from Graphs |

2.3. The Algebra of Limits

2.3.1. | Limits of Power Functions, and the Constant Rule for Limits | |

2.3.2. | The Sum Rule for Limits | |

2.3.3. | The Product and Quotient Rules for Limits | |

2.3.4. | The Power and Root Rules for Limits |

2.4. Limits of Functions

2.4.1. | Limits at Infinity of Polynomials | |

2.4.2. | Limits of Reciprocal Functions | |

2.4.3. | Limits of Exponential Functions | |

2.4.4. | Limits of Logarithmic Functions | |

2.4.5. | Limits of Radical Functions | |

2.4.6. | Limits of Trigonometric Functions | |

2.4.7. | Limits of Reciprocal Trigonometric Functions | |

2.4.8. | Limits of Inverse Trigonometric Functions | |

2.4.9. | Limits of Piecewise Functions |

2.5. Determining Limits Using Algebraic Manipulation

2.5.1. | Calculating Limits of Rational Functions by Factoring | |

2.5.2. | Limits of Absolute Value Functions | |

2.5.3. | Calculating Limits of Radical Functions Using Conjugate Multiplication | |

2.5.4. | Calculating Limits Using Trigonometric Identities | |

2.5.5. | Limits at Infinity and Horizontal Asymptotes of Rational Functions | |

2.5.6. | Evaluating Limits at Infinity by Comparing Relative Magnitudes of Functions | |

2.5.7. | Evaluating Limits at Infinity of Radical Functions | |

2.5.8. | Vertical Asymptotes of Rational Functions | |

2.5.9. | Connecting Infinite Limits and Vertical Asymptotes of Rational Functions |

2.6. Special Limits

2.6.1. | The Squeeze Theorem | |

2.6.2. | Special Limits Involving Sine | |

2.6.3. | Evaluating Special Limits Involving Sine Using a Substitution | |

2.6.4. | Special Limits Involving Cosine | |

2.6.5. | Limits Involving the Exponential Function | |

2.6.6. | Further Limits Involving the Exponential Function |

3.

Continuity
13 topics

3.7. Continuity

3.7.1. | Determining Continuity from Graphs | |

3.7.2. | Defining Continuity at a Point | |

3.7.3. | Left and Right Continuity | |

3.7.4. | Further Continuity of Piecewise Functions | |

3.7.5. | Point Discontinuities | |

3.7.6. | Jump Discontinuities | |

3.7.7. | Discontinuities Due to Vertical Asymptotes | |

3.7.8. | Continuity Over an Interval | |

3.7.9. | Continuity of Functions | |

3.7.10. | The Intermediate Value Theorem |

3.8. Removing Discontinuities

3.8.1. | Removing Point Discontinuities | |

3.8.2. | Removing Jump Discontinuities | |

3.8.3. | Removing Discontinuities From Rational Functions |

4.

Introduction to Differentiation
22 topics

4.9. Introduction to Differentiation

4.9.1. | The Average Rate of Change of a Function | |

4.9.2. | The Average Rate of Change of a Function over a Varying Interval | |

4.9.3. | The Instantaneous Rate of Change of a Function at a Point | |

4.9.4. | Defining the Derivative Using Derivative Notation | |

4.9.5. | Connecting Differentiability and Continuity | |

4.9.6. | The Power Rule for Differentiation | |

4.9.7. | The Sum and Constant Multiple Rules for Differentiation | |

4.9.8. | Calculating the Slope of a Tangent Line Using Differentiation | |

4.9.9. | Calculating the Equation of a Tangent Line Using Differentiation | |

4.9.10. | Calculating the Equation of a Normal Line Using Differentiation |

4.10. Derivatives of Functions and the Rules of Differentiation

4.10.1. | Differentiating Exponential Functions | |

4.10.2. | Differentiating Logarithmic Functions | |

4.10.3. | Differentiating Trigonometric Functions | |

4.10.4. | Differentiating Hyperbolic Functions | |

4.10.5. | Differentiating Reciprocal Hyperbolic Functions | |

4.10.6. | Second and Higher Order Derivatives | |

4.10.7. | The Product Rule for Differentiation | |

4.10.8. | The Quotient Rule for Differentiation | |

4.10.9. | Differentiating Reciprocal Trigonometric Functions | |

4.10.10. | Calculating Derivatives From Data and Tables | |

4.10.11. | Calculating Derivatives From Graphs | |

4.10.12. | Recognizing Derivatives in Limits |

5.

Advanced Differentiation
18 topics

5.11. Differentiating Composite Functions

5.11.1. | The Chain Rule for Differentiation | |

5.11.2. | The Chain Rule With Exponential Functions | |

5.11.3. | The Chain Rule With Logarithmic Functions | |

5.11.4. | The Chain Rule With Trigonometric Functions | |

5.11.5. | Calculating Derivatives From Data Using the Chain Rule | |

5.11.6. | Calculating Derivatives From Graphs Using the Chain Rule | |

5.11.7. | Selecting Procedures for Calculating Derivatives |

5.12. Differentiating Implicit and Inverse Functions

5.12.1. | Implicit Differentiation | |

5.12.2. | Calculating Slopes of Circles, Ellipses, and Parabolas | |

5.12.3. | Calculating dy/dx Using dx/dy | |

5.12.4. | Differentiating Inverse Functions | |

5.12.5. | Differentiating an Inverse Function at a Point | |

5.12.6. | Differentiating Inverse Trigonometric Functions | |

5.12.7. | Differentiating Inverse Reciprocal Trigonometric Functions | |

5.12.8. | Differentiating Inverse Hyperbolic Functions | |

5.12.9. | Differentiating Inverse Reciprocal Hyperbolic Functions |

5.13. Differentiation Using Logarithms

5.13.1. | Logarithmic Differentiation | |

5.13.2. | Further Logarithmic Differentiation |

6.

Contextual Applications of Differentiation
16 topics

6.14. Contextual Applications of Differentiation

6.14.1. | Interpreting the Meaning of the Derivative in Context | |

6.14.2. | Rates of Change in Applied Contexts |

6.15. Estimating Derivatives

6.15.1. | Estimating Derivatives Using a Forward Difference Quotient | |

6.15.2. | Estimating Derivatives Using a Backward Difference Quotient | |

6.15.3. | Estimating Derivatives Using a Central Difference Quotient |

6.16. Related Rates of Change

6.16.1. | Introduction to Related Rates | |

6.16.2. | Related Rates With Implicit Functions | |

6.16.3. | Calculating Related Rates With Circles and Spheres | |

6.16.4. | Calculating Related Rates With Squares | |

6.16.5. | Calculating Related Rates With Rectangular Solids | |

6.16.6. | Calculating Related Rates Using the Pythagorean Theorem | |

6.16.7. | Calculating Related Rates Using Similar Triangles | |

6.16.8. | Calculating Related Rates Using Trigonometry | |

6.16.9. | Calculating Related Rates With Cones |

6.17. L'Hopital's Rule

6.17.1. | L'Hopital's Rule | |

6.17.2. | L'Hopital's Rule Applied to Tables |

7.

Analytical Applications of Differentiation
31 topics

7.18. Analytical Applications of Differentiation

7.18.1. | The Mean Value Theorem | |

7.18.2. | Global vs. Local Extrema and Critical Points | |

7.18.3. | The Extreme Value Theorem | |

7.18.4. | Using Differentiation to Calculate Critical Points | |

7.18.5. | Determining Intervals on Which a Function Is Increasing or Decreasing | |

7.18.6. | Using the First Derivative Test to Classify Local Extrema | |

7.18.7. | The Candidates Test | |

7.18.8. | Intervals of Concavity | |

7.18.9. | Relating Concavity to the Second Derivative | |

7.18.10. | Points of Inflection | |

7.18.11. | The Second Derivative Test |

7.19. Analysis of Curves

7.19.1. | Sketching the Derivative of a Function From the Function's Graph | |

7.19.2. | Interpreting the Graph of a Function's Derivative | |

7.19.3. | Interpreting the Graph of a Function's Derivative: Concavity and Points of Inflection | |

7.19.4. | Sketching a Function From the Graph of its Derivative | |

7.19.5. | Sketching a Function Given Some Derivative Properties |

7.20. Approximating Values of a Function

7.20.1. | Approximating Functions Using Local Linearity and Linearization | |

7.20.2. | Approximating the Roots of a Number Using Local Linearity | |

7.20.3. | Approximating Trigonometric Functions Using Local Linearity |

7.21. Optimization

7.21.1. | Solving Optimization Problems Using Derivatives | |

7.21.2. | Optimization Problems Involving Sectors of Circles | |

7.21.3. | Optimization Problems Involving Boxes and Trays | |

7.21.4. | Optimization Problems Involving Cylinders | |

7.21.5. | Optimizing Distances | |

7.21.6. | Optimizing Distances to Curves | |

7.21.7. | Optimization Problems With Inscribed Shapes | |

7.21.8. | Optimization Problems in Economics |

7.22. Displacement, Velocity, and Acceleration

7.22.1. | Calculating Velocity for Straight-Line Motion Using Differentiation | |

7.22.2. | Calculating Acceleration for Straight-Line Motion Using Differentiation | |

7.22.3. | Determining Characteristics of Moving Objects Using Differentiation | |

7.22.4. | Newton's Second Law |

8.

Integration
34 topics

8.23. Indefinite Integrals

8.23.1. | The Antiderivative | |

8.23.2. | The Constant Multiple Rule for Indefinite Integrals | |

8.23.3. | The Sum Rule for Indefinite Integrals | |

8.23.4. | Integrating the Reciprocal Function | |

8.23.5. | Integrating Exponential Functions | |

8.23.6. | Integrating Trigonometric Functions | |

8.23.7. | Integration Using Inverse Trigonometric Functions | |

8.23.8. | Integrating Hyperbolic Functions | |

8.23.9. | Integration Using Inverse Hyperbolic Functions | |

8.23.10. | Integration Using Inverse Reciprocal Hyperbolic Functions |

8.24. Approximating Areas with Riemann Sums

8.24.1. | Approximating Areas With the Left Riemann Sum | |

8.24.2. | Approximating Areas With the Right Riemann Sum | |

8.24.3. | Approximating Areas With the Midpoint Riemann Sum | |

8.24.4. | Approximating Areas With the Trapezoidal Rule | |

8.24.5. | Left and Right Riemann Sums in Sigma Notation | |

8.24.6. | Midpoint and Trapezoidal Rules in Sigma Notation | |

8.24.7. | Approximating Areas Under Graphs of Composite Functions |

8.25. Definite Integrals

8.25.1. | Defining Definite Integrals Using Left and Right Riemann Sums | |

8.25.2. | The Fundamental Theorem of Calculus | |

8.25.3. | Applying the Fundamental Theorem of Calculus to Exponential and Trigonometric Functions | |

8.25.4. | The Sum and Constant Multiple Rules for Definite Integrals | |

8.25.5. | Properties of Definite Integrals Involving the Limits of Integration |

8.26. The Area Under a Curve

8.26.1. | The Area Bounded by a Curve and the X-Axis | |

8.26.2. | Evaluating Definite Integrals Using Symmetry | |

8.26.3. | Finding the Area Between a Curve and the X-Axis When They Intersect | |

8.26.4. | The Area Bounded by a Curve and the Y-Axis | |

8.26.5. | Calculating the Definite Integral of a Function Given Its Graph | |

8.26.6. | Calculating the Definite Integral of a Function's Derivative Given its Graph | |

8.26.7. | Definite Integrals of Piecewise Functions |

8.27. Accumulation Functions

8.27.1. | The Integral as an Accumulation Function | |

8.27.2. | The Second Fundamental Theorem of Calculus | |

8.27.3. | Maximizing a Function Using the Graph of Its Derivative | |

8.27.4. | Minimizing a Function Using the Graph of its Derivative | |

8.27.5. | Further Optimizing Functions Using Graphs of Derivatives |