Our Calculus I course provides in-depth coverage of differential calculus including limits, continuity, differentiation techniques, and applications of differentiation. This comprehensive course, together with the following course Calculus II, will prepare students for further studies in advanced mathematics, engineering, statistics, machine learning, and other fields requiring a solid foundation in differential calculus.
Learn the mathematics of change that underlies science and engineering. Master limits, derivatives, and the basics of integration.
| 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 |
| 2.2.1. | 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 |
| 2.3.1. | 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 |
| 2.4.1. | 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 |
| 2.5.1. | 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 |
| 2.6.1. | 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 |
| 3.7.1. | 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 |
| 3.8.1. | Removing Point Discontinuities | |
| 3.8.2. | Removing Jump Discontinuities | |
| 3.8.3. | Removing Discontinuities From Rational Functions |
| 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 |
| 4.10.1. | 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 |
| 5.11.1. | 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 |
| 5.12.1. | 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 |
| 5.13.1. | Logarithmic Differentiation | |
| 5.13.2. | Further Logarithmic Differentiation |
| 6.14.1. | Interpreting the Meaning of the Derivative in Context | |
| 6.14.2. | Rates of Change in Applied Contexts |
| 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 |
| 6.16.1. | 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 |
| 6.17.1. | L'Hopital's Rule | |
| 6.17.2. | L'Hopital's Rule Applied to Tables |
| 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 |
| 7.19.1. | 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 |
| 7.20.1. | 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 |
| 7.21.1. | 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 |
| 7.22.1. | 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 |
| 8.23.1. | 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 |
| 8.24.1. | 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 |
| 8.25.1. | 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 |
| 8.26.1. | 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 |
| 8.27.1. | 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 |
