Math 01.130 - Calculus I
Math 01.130 Calculus I 4 s.h.
Prerequisites: Math #01122 or 60 on the CLM exam
This course begins with a discussion of functions, the limit concept
and continuity. The concept of a derivative is introduced and the
student learns to differentiate algebraic functions, exponential, functions,
logarithmic and trigonometric functions. Differentiation is applied to
analysis of functions, extreme problems and to problems in related rates.
The integral as the unit of a sum is linked to the antiderivative by the
Fundamental Theorem of Calculus and used to find areas. A graphing calculator
is required for this course, and so is the use of a computer software,
such as Mathematica. Students are expected to have completed an equivalent
of (Math 01.122) Precalculus.
Students will demonstrate the ability to: (i) compute limits; (ii) differentiate
and integrate polynomial, rational, algebraic, exponential, logarithmic
and trigonometric functions; (iii) use differentiation to solve extreme
and related rate problems, and (iv) use integration to find areas and volumes.
1: Prerequisites for Calculus
A brief review of functions and their graphs.
2: Limits and Continuity
It is recommended that the emphasis here be on definitions and on intuitive understanding of concepts, though formal delta and epsilon proofs should be demonstrated.
The derivative definition and the rules for finding derivatives of polynomial, rational, exponential, logarithmic, trigonometric and algebraic functions.
4: Logs and exponential functions
Inverse functions, logs and exponential functions, derivatives
of logs and exponential functions.
Implicit differentiation, related rates.
5: Analysis of functions and their graphs
Increasing and decreasing functions, concavity, Relative extreme, First and 2nd derivative tests, Rolle's Theorem and the Mean Value Theorem.
6: Applications of Derivatives
Applications of the derivative to maxima and minima, the relationships between distance, velocity and acceleration.
The definite integral is formally defined and the definition is used to evaluate the integral. The Fundamental Theorem of Calculus is proved and techniques for evaluation of both definite and indefinite integrals by means of the Fundamental Theorem is discussed. The process of integration by means of a change of variable is then introduced.
8: Applications of Definite Integrals
Areas between two curves by the use of the definite integral.
REMARKS: In each chapter we will be studying a little about
the history of the development of Calculus through a brief study of the
biographies of the great mathematicians who developed it. In addition,
we will begin to learn to use Mathematica as a tool.
Rogawski, Jon, Calculus: Early Transcendentals Combo (Mathematica) & CalPortal, 2008, Freeman
Stewart, James, CALCULUS: CONCEPTS AND CONTEXTS, 3rd edition, Brooks, Cole, 2005.
Anton, H., CALCULUS, 6th edition, John Wiley and Sons, Inc., New York, 1998.
Larson, Hostetler & Edwards: Calculus, 6th edition, D.C. Heath Company, Lexington, 1998.
(Note: There are many suitable texts available that cover the
same material at the same level. Among these are those by Finney/Thomas,
Stein, Hunt and Leithold).