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Math 31A: General Course Outline |
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Catalog Description
31A. Differential and Integral Calculus. (4) Lecture, three hours; discussion, one hour. Preparation: at least three and
one-half years of high school mathematics (including some coordinate geometry
and trigonometry). Requisite: successful completion of Mathematics Diagnostic
Test or course 1 with a grade of C- or better. Differential calculus and applications;
introduction to integration. P/NP or letter grading.
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Textbook
J. Rogawski, Single Variable Calculus, (2nd Edition)
, W.H. Freeman & CO
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Reviews & Exams
The following schedule, with textbook
sections and topics, is based on 26 lectures. The remaining classroom meetings
are for leeway, reviews, and two midterm exams. These are scheduled by the individual
instructor. Often there are reviews and midterm exams about the beginning of
the 4th and 8th weeks of instruction, plus reviews for the final exam.
In certain cases (such as for coordinated classes), it may be possible to give midterm exams during additional class meetings scheduled in the evening. This has the advantage of saving class time. A decision on whether or not to do this must be made well in advance so that the extra exam sessions can be announced in the Schedule of Classes. Instructors wishing to consider this option should consult the mathematics undergraduate office for more information.
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Schedule of Lectures
Lecture |
Sections |
Topics |
1 |
2.1 |
Limits, Rates of Change, Tangent Line (A) |
2 |
2.2 - 2.3 |
Limits Numerically and Graphically, Limit Laws |
3 |
2.4 - 2.6 |
Continuity, Evaluating limits, Trigonometric limits(B) |
4 |
3.1 |
The Derivative |
5 |
3.2 - 3.3 |
The Derivative as a Function, Product and Quotient Rules |
6 |
3.4 – 3.5 |
Rates of Change, Higher Derivatives |
7 |
3.6 |
Trigonometric functions |
8 |
3.7 - 3.8 |
The Chain Rule, Implicit differentiation |
9 |
3.9 |
Related Rates |
10 |
4.1 |
Linear approximation (C) |
11 |
4.2 |
Extreme values |
12 |
4.3 |
The Mean Value Theorem, Monotonicity |
13 |
4.4 |
Shape of a graph, concavity |
14 |
4.5 |
Graph sketching |
15 |
4.6 |
Applied optimization |
16 |
4.8 |
Antiderivatives |
17 |
5.1 |
Approximating and Computing area(D) |
18 |
5.2 |
Definite integral |
19 |
5.3 |
Fundamental Theorem of Calculus I |
20 |
5.4 |
Fundamental Theorem of Calculus II |
21 |
5.5 |
Net Change |
22 |
5.6 |
Substitution Method |
23 |
6.1 |
Area between two curves |
24 |
6.2 |
Setting up integrals |
25 |
6.3 |
Volumes of Revolution (E) |
26 |
6.4 |
Method of Cylindrical Shells |
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Comments
(A) Students may be asked to memorize a small number of proofs for each exams, of which one will actually appear on the exam. Possible proofs are: the geometric proof that lim sin(x)/x = 1, proof of the Product Rule, proof that local min/max occur at critical points, part of the Fundamental Theorem (e.g., proof that the derivative of the definite integral of f(x) is f(x) itself).
(B) The text formulates the linear approximation without differentials. Differentials should be mentioned only briefly if at all.
(C) Students may be expected to evaluate limits of the right or left endpoint approximations for
linear and quadratic functions f(x). The Midpoint Rule may be omitted.
(D) Suggestion: In the “washer method”, emphasize revolution about arbitrary axes parallel to the x- and y-axes. In Lecture 26, treat cylindrical shells briefly, stressing revolution about the x- and y-axes only. This material should take two lectures.
Outline update: 10/10 |
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