MATH 5A – Calculus I
Spring 2008
Student Learning Outcomes:
Upon successful completion of the course, the student will be able to:
1. Demonstrate the rigorous definition of the limit, and how it applies to differential and integral
calculus.
2. Calculate the derivative using the definition and the properties of differentiation and apply
derivatives to geometric and dynamic problems.
3. Integrate, use the integral in application and relate the integral to the derivative.
Student Performance Objectives:
1. State and use definitions for a function, the limit of a function, continuity, the derivative,
indefinite and definite integral.
2. State the mean value theorem for derivatives, Rolle's theorem, the mean value theorem for
integrals and the fundamental theorem of calculus.
3. Evaluate limits of algebraic and trigonometric functions.
4. Use basic theorems of limits to justify steps in evaluating limits.
5. Differentiate algebraic and trigonometric functions.
6. State, and use in proofs, the result that differentiability implies continuity.
7. Apply differentiation in solving problems.
8. Use the definition of an integral to find integrals of first and second degree polynomials .
9. Explain the relationship between definite integrals and the area under a curve, using sigma
notation and limits.
10. Perform integrations of algebraic and trigonometric functions using the techniques of
algebraic substitution and change of variable.
11. State and apply the first and second fundamental theorems of calculus.
12. Use the fundamental theorems of calculus to calculate average value of a function, and to
differentiate integrals with variable limits.
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13. Apply the definite integral to solve problems.
Course Content Outline:
Functions
Operations on Functions and Relations
Elements of Analytic Geometry
Limits and Continuity
Derivatives of Algebraic and Trigonometric Functions
The Chain Rule
Implicit Differentiation
Linear Approximation
The Differential of a Real Function
Applications of the Derivative
Tangent and normal lines
Mean Value Theorem
First and Second Derivative Tests
a. Graphing functions
b. Maximum and minimum
Velocity and Acceleration
Related Rates
The Definite Integral
The Mean Value Theorem for Definite Integrals
The Fundamental Theorem of Calculus
Numerical Integration by Rectangles
Properties of the Integral
Integration by Substitution
Application of the Definite Integral
Area
Volumes of Solids (Disk, Washer, Shells, Slices)
Arc Length
Areas of Surface of Revolution
Work
Moments and Centroids
Introduction to Differential Equations
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