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# Calculus

College

Course Outline

Single Variable Calculus I:

Course Description:

Involves a study of limits, continuity, derivatives and integrals, computations of derivatives, sum, product, and quotient formulas, chain rule, implicit differentiation, applications of derivatives to optimization problems and related rate problems, mean-value theorem, definite integrals and the fundamental theorem of calculus, application of definite integrals to computations of areas (length, surface) and volumes.

Course Outline:

1. Pre-calculus preliminaries

1. Demonstrate knowledge of basic precalculus concepts and skills

1. Limits

1. Evaluate limits

1. Continuity

1. Recognize continuity and use the properties of continuous functions

1. Differentiation

1. Find derivatives of algebraic and trigonometric functions using the definition or basic rules of differentiation

2. Find rates of change

1. Applications of differentiation

1. Solve related rate problems

2. Analyze and sketch the graphs of curves

3. Find extreme values in optimization problems

Single Variable Calculus II:

Course Description:

This second course in calculus provides further application of definite integrals, differentiation and integration of transcendental functions, techniques of integration, L'Hopital's rule and improper integrals, infinite sequences and series, Taylor and power series, polar and parametric equations.

Course Outline:

1. Solve application problems related to integration.

a. Volumes of revolution using disk and shell methods

b. Arc length

c. Surface area of revolution

d. Work

e. Centroids

f. Fluid pressure and force [Recommended/Optional]

II. Solve introductory differential equations and associated initial value problems.

a. Separation of variables

b. Initial value problems

c. Hyperbolic functions [Recommended/Optional]

III.  Apply appropriate integration techniques including integration by parts, trig substitution, and partial fractions to evaluate definite, indefinite, and improper integrals.

a. Integration by parts

b. Integrating powers of the trigonometric functions

c. Trigonometric substitutions

d. Integration of inverse trigonometric functions

e. Integration of exponential and logarithmic functions

f. Integration using partial fractions decomposition

g. Evaluating improper integrals

h. Integration tables [Recommended/Optional]

i. Numerical methods of integration [Recommended/Optional]

IV.  Demonstrate the convergence or divergence of infinite sequences and series.

1. Definition of sequence convergence

2. L'Hospital's Rule

3. Definition of an infinite series convergence

4. Convergence of geometric series

5. Sum of convergent geometric series

6. Application of the nth term test for divergence

7. Application of the integral test

8. Identification and classification of p-series as convergent or divergent

9. Use of direct and limit comparison test to determine convergence

10. Use of ratio and root tests

11. Use of alternating series test to determine convergence

12. Classification of series as absolutely or conditionally convergent

13. Hyperbolic functions [Recommended/Optional]

V.  Express functions as power series (including Taylor series) with the appropriate interval of convergence.

a. Taylor polynomials and approximations

b. Taylor series

c. Maclaurin series

d. Binomial series

e. Geometric power series

f. Manipulation of power series to express functions

g. Interval and radius of convergence of power series

h. Differentiation and integration of power series

i. Monotonic sequences [Recommended/Optional]

j. Using the integral test to control the error in an approximation [Recommended/Optional]

VI. Estimate errors in series approximations.

a. Altering series remainder theorem

b. Taylor's Remainder Theorem

c. Graph conic sections [Recommended/Optional]

d. Translation and axes rotation [Recommended/Optional]

e. Surface area of revolution in polar and parametric form [Recommended/Optional]

VII. Graph curves in polar and parametric form.

a. Plane curves and parametric equations

b. Transformation of parametric to rectangular forms and vice versa

c. Graphing polar coordinates and curves

d. Transformation of polar expressions to rectangular forms and vice versa

VIII. Analyze curves in polar and parametric form using calculus techniques.

a. Slopes and tangent lines for equations in polar coordinates

b. Area and arc length in polar coordinates

c. Slope of tangent line to a parametric curve

d. Arc length of a parametric curve

Multivariable Calculus III:

Course Description:

This third course in calculus includes vectors, lines, and simple surfaces in three-dimensional space, some linear algebra topics, vector-valued functions, partial derivatives, multiple integrals, line integrals and Green's Theorem, surface integrals and the theorems of Gauss and Stokes.

Course Outline:

1. Functions and the Geometry of Space

1. Identify, describe, and visualize equations in 3-space.

2. Use contour maps for functions of two or three variables to analyze the functions.

3. Use the algebra of vectors to study geometry in 3-space.

1. Calculus of Vector-Valued Functions

1. Use the calculus of vector-valued functions to analyze motions in 3-space

2. Find and interpret the unit tangent and unit normal vectors and curvature.

1. Calculus of Functions of Several Variables - Differentiation

1.  Find partial derivatives numerically and symbolically and use them to analyze and interpret the way a function varies.

2. Find and interpret the gradient and directional derivatives for a function at a given point.

3. Find the total differential of a function of several variables and use it to approximate incremental change in the function.

4. Analyze and solve constrained and unconstrained optimization problems.

1. Calculus of Functions of Several Variables - Integration

1. Explain the relationship between multiple and iterated integrals.

2. Evaluate multiple integrals either by using iterated integrals or approximation methods.

3. Relate rectangular coordinates in 3-space to spherical and cylindrical coordinates and use spherical and cylindrical coordinates as an aid in evaluating multiple integrals.

4. Model applied problems using multiple integrals.

1. Vector Analysis

1. Define a line integral and use it to find the total change in a function given its gradient field.

2. Calculate and interpret the flow and divergence for a vector field.

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