In Active Calculus, we endeavor to actively engage students in learning the subject through an activity-driven approach in which the vast majority of the examples are completed by students.
This approach is consistent with the following goals:
- To have students engage in an active, inquiry-driven approach, where learners strive to construct solutions and approaches to ideas on their own, with appropriate support through questions posed, hints, and guidance from the instructor and text.
- To build in students intuition for why the main ideas in calculus are natural and true. Often, we do this through consideration of the instantaneous position and velocity of a moving object, a scenario that is common and familiar.
- To challenge students to acquire deep, personal understanding of calculus through reading the text and completing preview activities on their own, through working on activities in small groups in class, and through doing substantial exercises outside of class time.
- To strengthen students’ written and oral communicating skills by having them write about and explain aloud the key ideas of calculus.
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- 1 Understanding the Derivative
- 1.1 How do we measure velocity?
- 1.2 The notion of limit
- 1.3 The derivative of a function at a point
- 1.4 The derivative function
- 1.5 Interpreting, estimating, and using the derivative
- 1.6 The second derivative
- 1.7 Limits, Continuity, and Differentiability
- 1.8 The Tangent Line Approximation
- 2 Computing Derivatives
- 2.1 Elementary derivative rules
- 2.2 The sine and cosine functions
- 2.3 The product and quotient rules
- 2.4 Derivatives of other trigonometric functions
- 2.5 The chain rule
- 2.6 Derivatives of Inverse Functions
- 2.7 Derivatives of Functions Given Implicitly
- 2.8 Using Derivatives to Evaluate Limits
- 3 Using Derivatives
- 3.1 Using derivatives to identify extreme values of a function
- 3.2 Using derivatives to describe families of functions
- 3.3 Global Optimization
- 3.4 Applied Optimization
- 3.5 Related Rates
- 4 The Deﬁnite Integral
- 4.1 Determining distance traveled from velocity
- 4.2 Riemann Sums
- 4.3 The Deﬁnite Integral
- 4.4 The Fundamental Theorem of Calculus
- 5 Finding Antiderivatives and Evaluating Integrals
- 5.1 Constructing Accurate Graphs of Antiderivatives
- 5.2 The Second Fundamental Theorem of Calculus
- 5.3 Integration by Substitution
- 5.4 Integration by Parts
- 5.5 Other Options for Finding Algebraic Antiderivatives
- 5.6 Numerical Integration
- 6 Using Deﬁnite Integrals
- 6.1 Using Deﬁnite Integrals to Find Area and Length
- 6.2 Using Deﬁnite Integrals to Find Volume
- 6.3 Density, Mass, and Center of Mass
- 6.4 Physics Applications: Work, Force, and Pressure
- 6.5 Improper Integrals
- 7 Differential Equations
- 7.1 An Introduction to Differential Equations
- 7.2 Qualitative behavior of solutions to differential equations
- 7.3 Euler’s method
- 7.4 Separable differential equations
- 7.5 Modeling with differential equations
- 7.6 Population Growth and the Logistic Equation
- 8 Sequences and Series
- 8.1 Sequences
- 8.2 Geometric Series
- 8.3 Series of Real Numbers
- 8.4 Alternating Series
- 8.5 Taylor Polynomials and Taylor Series
- 8.6 Power Series