640:135 - Calculus I

Lecture topics - Spring 2018

1 1.2, 1.3 Precalculus review: Real line, coordinate plane, distance, circles, straight lines. Trig review: Radians, definition of trig functions, graphs of sin, cos, tan, sec. (Refer to Appendix E.)
2 1.4 Precalculus review: Functions, graphs, composition of functions.
3 2.1, 2.2 Limits: Informal definition and discussion of intuitive meaning. Rules for limits, computing limits of algebraic functions. One-sided limits. Limits of trig functions. Infinite limits.
4 2.2 Topics of lecture 3, continued.
5 2.3 Definition of continuity of a function at a point. Testing continuity. Continuity of a function on an interval. Intermediate value theorem and root location theorem.
6 2.4 Exponential functions and logarithmic functions. Definition of e, properties and inverse relation of the exp and ln functions. Exponential growth. Compound interest and continuous compounding.
7 3.1 Definition of the derivative as a limit, direct calculation of derivatives using the definition. The derivative as slope of tangent line. Equation of tangent line and of normal line. Relation between the graph of f and the graph of f'. Continuity and differentiability. Notations for the derivative.
8 3.2, 3.3 Calculation of derivatives, sum, product and quotient rules. Higher-order derivatives. Differentiation of trig functions and of e^x and ln(x).
9 3.4 The derivative as rate of change. Velocity and acceleration.
10 3.5 The chain rule (for differentiating a composite function).
11   Catch up and review.
13 3.6 Implicit differentiation. Derivative of ln(|u|). Logarithmic differentiation.
14 3.7 Related rates and applications.
15 3.8 Linear approximation. Differentials. Propagation of error in measurement, relative error, percentage error. Marginal cost, marginal revenue.
16 4.1, 4.2

Absolute maximum and absolute minimum of a function defined on an interval. The extreme value theorem. Relative extrema. Critical numbers and critical points. Finding critical numbers and critical points. Finding absolute extrema among the critical numbers and the endpoints of an interval. Statements of Rolle's theorem and of the mean value theorem, and Example 1.

17 4.3 Increasing and decreasing functions. Finding intervals of increase and of decrease. The first-derivative test for a relative maximum or minimum. Concavity up or down, and the second derivative. Inflection points. The second-derivative test for a relative maximum or minimum. Application to sketching graphs.
18 4.4 Limits as x approaches plus or minus infinity, and horizontal asymptotes. Infinite limits and vertical asymptotes. Application to sketching graphs with horizontal and/or vertical asymptotes.
19 4.5 L'Hopitals's rule (for evaluating limits involving indeterminate forms).
20 4.6 Optimization applications: geometric and physical problems.
21   Catch up and review.
23 4.7 Optimization applications in business: marginal analysis, the demand function, maximizing profit or revenue, minimizing average cost.
24 5.1 Antiderivatives, indefinite integrals.
25 5.2, 5.3 Riemann sums for approximating the area under a curve.
26 5.4 The first fundamental theorem of calculus (for evaluating a definite integral using antidifferentiation). The second fundamental theorem of calculus.
27 5.5 Integration by substitution, for both indefinite and definite integrals.
28   Catch up and review.