**Calculus**

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**991**

### practice questions

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### Calculus Course Outline

### What functions are necessary to study calculus?

Concepts Covered:

- What are the basic properties of functions?
- What are the basic classes of functions?
- How do trigonometric functions behave?
- What are inverse functions?
- What is the relationship between exponential and logarithmic functions?

### Why are limits so important in calculus?

Concepts Covered:

- How are limits used to solve problems in calculus?
- How is the limit of a function determined?
- What are the limit laws?
- At what conditions is continuity achieved?
- What is the precise definition of a limit?

### How are derivatives calculated and used in calculus?

Concepts Covered:

- How is the derivative defined?
- How is the derivative of a function calculated?
- What are the differentiation rules?
- How are derivatives used as rates of change?
- What are the derivatives of trigonometric functions?
- How is the chain rule defined, and what is its function?
- How are derivatives of inverse functions determined?
- What is implicit differentiation?
- How are derivatives of exponential and logarithmic functions found?

### What are the different applications of Derivatives?

Concepts Covered:

- How are related rates expressed in terms of derivatives?
- How are linear approximations and differentials calculated?
- What are maxima and minima?
- What is the significance of the Mean Value Theorem
- How do derivatives affect the shape of a graph?
- What do limits at infinity and asymptotes look like?
- How are applied optimization problems solved?
- When should L’Hôpital’s Rule be applied?
- What is Newton’s Method?
- How are antiderivatives of functions found?

### What is integration?

Concepts Covered:

- How are areas approximated?
- What is the definite integral?
- What is the meaning of the Fundamental Theorem of Calculus?
- How are integration formulas and the net change theorem applied?
- How is substitution used to evaluate definite and indefinite integrals?
- What are integrals involving exponential and logarithmic functions used for?
- How are integrals resulting in inverse trigonometric functions integrated?

### How is integration applied?

Concepts Covered:

- How is the area between curves calculated?
- How is volume determined by slicing?
- What are volumes of revolution as related to cylindrical shells?
- How are the arc length of a curve and surface area calculated?
- What are physical applications in calculus?
- How do centers of mass change under certain conditions and moments?
- What is the relationship between integrals, exponential functions, and logarithms?
- How is exponential growth and decay determined?
- What are Hyperbolic Functions, and how are they solved?

### What is integration?

Concepts Covered:

- How are areas approximated?
- What is the definite integral?
- What is the meaning of the Fundamental Theorem of Calculus?
- How are integration formulas and the net change theorem applied?
- How is substitution used to evaluate definite and indefinite integrals?
- What are integrals involving exponential and logarithmic functions used for?
- How are integrals resulting in inverse trigonometric functions integrated?

### How is integration applied?

Concepts Covered:

- How is the area between curves calculated?
- How is volume determined by slicing?
- What are volumes of revolution as related to cylindrical shells?
- How are the arc length of a curve and surface area calculated?
- What are physical applications in calculus?
- How do centers of mass change under certain conditions and moments?
- What is the relationship between integrals, exponential functions, and logarithms?
- How is exponential growth and decay determined?
- What are Hyperbolic Functions, and how are they solved?

### Why is understanding the different techniques of integration important?

Concepts Covered:

- When should the technique of integration by parts be used?
- What are trigonometric integrals?
- When should trigonometric substitution be used?
- What are partial fractions?
- What are some other strategies for integration?
- How is numerical integration achieved?
- How are improper integrals defined?

### How are differential equations solved and used?

Concepts Covered:

- What are the basics of differential equations?
- How are differential equations solved with direction fields and numerical methods?
- What are separable equations?
- What is the logistic equation?
- How are first-order linear equations solved?

### How are sequences and series used in calculus?

Concepts Covered:

- How are the formulas and limits of sequences found?
- What are infinite series?
- How are divergence and integral tests performed?
- Why are comparison tests significant?
- How are alternating series used and solved?
- What are ratio and root tests?

### What are power series?

Concepts Covered:

- How are power series and functions identified?
- What are the properties of power series?
- What are Taylor and Maclaurin series?
- How are Taylor Series used to solve different types of equations?

### What is the significance of parametric equations and polar coordinates?

Concepts Covered:

- How are parametric equations recognized?
- What is the calculus of parametric curves?
- What are polar coordinates?
- How are the area and arc length in polar coordinates found?
- How are conic sections evaluated?

### What is the significance of parametric equations and polar coordinates?

Concepts Covered:

- How are parametric equations recognized?
- What is the calculus of parametric curves?
- What are polar coordinates?
- How are the area and arc lenth in polar coordinates found?
- How are conic sections evaluated?

### Why are vectors in space important?

Concepts Covered:

- What are plane vectors?
- How are vectors in three dimensions located and evaluated?
- How is the dot product of given vectors calculated?
- How is the cross product of vectors determined?
- How are equations of lines and planes in space solved?
- What are quadratic surfaces?
- How are cylindrical and spherical coordinates converted to different forms?

### In what situations are vector-valued functions used?

Concepts Covered:

- What are vector-valued functions and space curves?
- What is the Calculus of Vector-Valued Functions?
- How are arc lengths and curvatures of curves in space calculated?
- How does motion behave in space?

### How is the differentiation of functions with several variables conducted?

Concepts Covered:

- How are functions with several variables recognized and calculated?
- How are the limits and continuity of functions with multiple variables determined?
- How are partial derivatives calculated?
- How are tangent planes used to make linear approximations?
- What is the Chain Rule for multivariable calculus?
- What is the relationship between directional derivatives and the gradient?
- What are the problems with maxima and minima?
- What are Lagrange Multipliers?

### How is the integration of functions with multiple variables achieved?

Concepts Covered:

- How are double integrals over a rectangular region evaluated?
- How are double integrals over general regions evaluated?
- How are double integrals in polar coordinates evaluated?
- What are triple integrals, and how are they evaluated?
- How are triple integrals in cylindrical and spherical coordinates evaluated?
- How are centers of mass and moments of inertia calculated?
- What is the change of variables in multiple integrals?

### What concepts are studied in vector calculus?

Concepts Covered:

- What are vector fields?
- How are line integrals evaluated?
- How are conservative vector fields determined?
- What is Green’s Theorem?
- How are the divergence and curl from a formula for a given vector field determined?
- How are surface integrals used to solve problems?
- What is the meaning of Stokes’ Therem?
- How is the Divergence Theorem Applied?

### How are second-order differential equations solved?

Concepts Covered:

- How are second-order linear equations recognized and solved?
- How are nonhomogeneous linear equations solved?
- What are the applications of second-order differential equations?
- What are series solutions of differential equations?

Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them.

**About the authors:**

## Senior Contributing Authors

**Gilbert Strang,** Massachusetts Institute of Technology**
Edwin “Jed” Herman,** University of Wisconsin-Stevens Point

## Contributing Authors

Alfred K. Mulzet, Florida State College at Jacksonville

Sheri J. Boyd, Rollins College

Joyati Debnath, Winona State University

Michelle Merriweather, Bronxville High School

Joseph Lakey, New Mexico State University

Elaine A. Terry, Saint Joseph’s University

David Smith, University of the Virgin Islands

Nicoleta Virginia Bila, Fayetteville State University

Valeree Falduto, Palm Beach State College

Kirsten R. Messer, Colorado State University-Pueblo

William Radulovich, Formerly at Florida State College at Jacksonville

Erica M. Rutter, Arizona State University

David Torain, Hampton University

Catherine Abbott, Keuka College

Julie Levandosky, Framingham State University

David McCune, William Jewell College