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# Calculus I

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

## Content

### Limits and Continuity

• 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.

### Differentiation

• 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.

### Integration

• 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 Hyperbolic Functions 1.1.5. Graphs of the Reciprocal Hyperbolic Functions 1.1.6. The Inverse Hyperbolic Functions
2.
Limits
32 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 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. Special Limits Involving the Exponential Function 2.6.6. Further Special 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. 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.
17 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 and Graphs Using the Chain Rule 5.11.6. 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. More Logarithmic Differentiation
6.
Contextual Applications of Differentiation
15 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. Calculating Related Rates With Circles and Spheres 6.16.3. Calculating Related Rates With Squares 6.16.4. Calculating Related Rates With Rectangular Solids 6.16.5. Calculating Related Rates Using the Pythagorean Theorem 6.16.6. Calculating Related Rates Using Similar Triangles 6.16.7. Calculating Related Rates Using Trigonometry 6.16.8. 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
30 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. Using the Candidates Test to Determine Global Extrema 7.18.8. Intervals of Concavity 7.18.9. Relating Concavity to the Second Derivative 7.18.10. Points of Inflection 7.18.11. Using the Second Derivative Test to Determine Extrema
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. Optimization Problems Involving Rectangles 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. Finding Minimum Distances 7.21.6. Optimization Problems With Inscribed Shapes 7.21.7. 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 Applied to a Particle Moving With Nonuniform Acceleration
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 of Composite Functions Using Riemann Sums 8.24.5. Approximating Areas With the Trapezoidal Rule 8.24.6. Left and Right Riemann Sums in Sigma Notation 8.24.7. Midpoint and Trapezoidal Rules in Sigma Notation
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 Functions with Discontinuities
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