Course Description

Math 172 is the second of a three semester beginning calculus sequence, which is taken, for the most part, by math, chemistry, and physics majors. The department expects that students passing Math 172 will be able to set up an appropriate definite integral to solve the applied problems (areas, volumes, arclength, work, and force) discussed in the course. Students must understand the relationship between definite integrals and Riemann sums, and be able to clearly state (write) this relationship. Regarding infinite series: students are expected to know what an infinite series is, how to use the convergence tests, be able to clearly state them, and explain (prove) why they work. Students are expected to know the alternating series test, including the error estimate for this test and the error estimate from the integral test for positive term series. Students will also learn to solve basic first order differential equations using the method of separation of variables and integrating factors.

Textbook

The textbook is James Stewart Calculus: Early Transcendentals; 9th Edition

Key Points

Students should be taught and expected to express mathematical ideas in grammatically correct, complete sentences.
Students should be able to explain the reasoning behind their computations and justify each step conceptually.
Students should develop proficiency in working with inequalities, both algebraically and graphically.
Students should demonstrate proficiency with key definitions and be able to reproduce and apply them accurately on exams.
Students should build intuition throughout the semester leading them to being able to make conjectures and predictions (at an appropriate level).
In formulas for integrals, an emphasis should be placed on computations involving parameters (for example, students should be able to derive reduction rules for integrals such as ∫cos^N(x)dx).
As a foundational calculus course, it should also develop students’ proficiency in standard computations.
Students should begin to distinguish between formal proofs, informal arguments, and heuristic reasoning.
As many math majors will be taking Math 300 the following semester, points of contact with that course should be emphasized (for example, the logic in various theorems should be clearly explicated.

Weekly Schedule

Math 172 Weekly Schedule
Week	Sections	Topics	Notes
1	Chapter 5	Integrals	This should be an intense review. Students should see Riemann sums again and should able to compute the integrals of basic functions like x and x² using Riemann sums. The substitution rule should be reviewed and proof given. Note that this material is ostensibly covered at the end of Math 171 but it might not have solidified in the minds of students as well as other topics.
2	6.1,6.2,6.3	Area Between Curves and Volume	Students should be able to both set up the Riemann sum for these problems as well as the integral to which the sum converges.
3	6.4, 6.5	Work and Average Value of a Function	Tedious and “tricky” computations should be avoided in the section on work. The basic principles should be emphasized (including setting up a Riemann sum for the problem). Examples in pumping (with simple geometries) and rope lifting should be emphasized. Section 5.5 is on the Mean Value Theorem for integrals. This should be proven in class via the Intermediate Value Theorem. Optional: Instructors may spend more time on different work problems if time permits.
4	7.1,7.2,7.3	Integration by Parts; Trigonometric Integrals; Trigonometric Substitution	Students should be able to derive the IBP formula from the product rule. In Section 7.2, emphasis should be placed on integrals involving sin and cos . Students should be able to derive “reduction of order” formulas for powers of sin and cos functions. In 7.3, focus should be placed on sin and tan substitutions and having students derive general formulas for integrals with parameters (e.g. expressions involving a² + x²)Optional: If time permits, more time can be spent on things like sec substitution and so on.
5	7.3, 7.4	Trigonometric Substitution; Partial Fractions; Test 1 (Thru Section 7.2)	Finish 7.3 if needed; the test is on Wednesday
6	7.7, 7.8	Numerical Integration; Improper Integrals	The midpoint rule and trapezoid rule should be emphasized. The error estimates should be given and students are expected to be able to use them. Improper integrals at ± ∞ should be emphasized rather than local improper integrals. Students should learn and be able to apply DCT and should see LCT as a way to prepare them for what they will see with series (note: Stewart does not have LCT for integrals). Students should know the p test for improper integrals on an infinite domain. Students should minimally know the p test for local singularities. Optional: Instructors may spend more time on local singularities and may also add things like Simpson’s rule if time permits.
7	11.1, 11.2	Sequences; Series	These chapters are foundational. Time should be spent on sequences and ε–N proofs should be given and students should be expected to give ε–N proofs for rational functions. Students should know what a monotonic sequence is and what a bounded sequence is and the Monotonic Sequence Theorem; students should be able to “prove” that a sequence is increasing and bounded and apply Monotonic Sequence Theorem to find a limit. Students should be able to find limit of a recursive sequence including being able to determine which solution to the corresponding algebraic equation is the correct one. Students need to be comfortable using Σ notation and dealing with index set.
8	11.3	Integral Test; Test 2 (7.3 thru 11.2)	Students should be able to apply the integral test as well as demonstrate they know why the integral test is true; this includes understanding the purpose of each hypothesis in the Integral Test. Students should be able to use the remainder estimate from the Integral Test. There is less material this week and the extra time can be used for review and/or catch up.
9	11.4; 11.5	Comparison Tests; Alternating Series Test	Students should understand the purpose of each hypothesis in each test (e.g. it is reasonable to ask them to come up with a series that satisfies some of the hypotheses but does not converge). Students should be able to prove the AST. Students should be able to use AST to estimate remainder and do things like compute the number of terms needed to reach a given precision.
10	11.6, 11.8	Absolute Convergence and Ratio Test; Power Series	Root Test is not covered; Similar to the previous tests, students need to be able to use the test but also understand the hypotheses of the test and their purpose. Optional: Root test
11	11.9, 11.10	Representation of Functions as Power Series; Taylor and MacLaurin Series	$\frac{1}{1-x}$ is fundamental and students should be able to manipulate series to apply this formula. Students should know the term–by–term theorems. Students should be able to use the remainder theorems to estimate the error and to compute the number of terms needed to reach a given precision. Optional: Binomial Series
12	11.10	Taylor and MacLaurin Series; Test 3 (11.3-11.9)	There is less material this week and the extra time can be used for review and/or catch up.
13	11.11	Applications of Taylor Polynomials	The entire point of Taylor Series is the applications so some should be mentioned. Note that in the Fall, this is Thanksgiving week.
14	8.1, 9.3, 9.5	Arc Length; Separable Equations; Integrating Factors	9.3 and 9.5 are differential equations. The solution techniques (rather than modeling or drawing direction fields) should be emphasized. Optional: Drawing direction fields; applications
15	10.1, 10.2, 10.3	Curves Defined by Parametric Equations; Calculus with Parametric Curves; Polar Coordinates	Feel Free to do section 8.1 in this week as well. As stated here, the last three weeks might seem odd. But there is a practical problem in that the MWF and TR classes have different numbers of class days (e.g. re-defined days). So each individual instructor will have to ensure that the extra time in Week 12 is used for "catch up" and that the material from weeks 13-15 is covered appropriately.