Calculus II Assignments

MTH 112: Calculus with Analytic Geometry II
Dr. Elmer K. Hayashi
Spring 2003
Assignments

Jan 15-17	Jan 20-24	Jan 27-31	Feb 3-7	Feb 10-14
Feb 17-21	Feb 24-28	Mar 3-7	Mar 17-21	Mar 24-28
Mar 31-Apr 4	Apr 7-Apr 11	Apr 14-17	Apr 21-25	Apr 28-30

Textbook: Stewart, Calculus, Fourth Edition
Graduate Teaching Assistant: To Be Named Later
Help Sessions, TBA.

Wed, 01/15/2003. Comparison Test for Integrals.: The first half of this semester, we will be learning how to approximate the sum of a series that has no simple formula. Analogously, we can approximate the value of an integral whose integrand has no simple formula. The trick is to compare the integral with one that we know how to integrate.
Review the definition of an improper integral of type 1 given on page 558. Read and learn the comparison test for improper integrals on pages 563-565.
Do problems 47-52, 69, 70 on pages 566-567 and problem 71 on page 570 to check your understanding.

Fri, 01/17/2003. Estimating the tail of an integral.: If an integral does not have an elementary formula, find an upper bound that does have an elementary formula to obtain an upper estimate for the value of the original integral.
Turn in the solutions to the following three problems next Wednesday.
Assignment 1 in PDF format
If you have not used Maple before, sign up for a Maple Tutorial session on the ground floor of Calloway Hall.

Mon, 01/20/2003. Martin Luther King, Jr. Holiday.: No Class

Tue, 01/21/2003. Infinite Sequences.: The limit of a sequence (if it exists) is a number (finite) to which the terms of the sequence get arbitrarily close for sufficiently large n. The limit of a sequence is infinite (and does not exist) if the terms of the sequence get arbitrarily large for sufficiently large n.
Study carefully pages 729-731.
Do problems 1, 3-14, and the odd problems 21-35 on page 736 to check understanding.

Wed, 01/22/2003. Monotone Sequence Theorem.: Every bounded, monotonic sequence is convergent. Once you know that a sequence converges, then you can apply the properties of convergent limits to help you find the value of the limit.
Read pages 733-734, and study example 11 on page 735.
Write up problems 60 and 62 on page 737 to turn in next Monday.
Do problems 52-58, 59, 61, and 63b on page 737 to check understanding.

Fri, 01/24/2003. Geometric Series.: Most infinite series do not have elementary formulas for their partial sums, and finding the sum of most convergent infinite series is difficult if not impossible. However, the geometric series is a notable exception, and we will rely on it greatly to make comparisons and estimations of the more intractable infinite series.
Learn the formulas and theorems in the red boxes on pages 740 and 743-744. Study examples 1-5 on pages 739-741 and example 8 on page 743.
Do odd problems 11-35 on page 745 to check understanding.

Mon, 01/27/2003. Telescoping Series.: Geometric series are the most important series whose sum is easily found when convergent. Another type of series whose sum can be found easily is the telescoping series. In a telescoping series, each term can be written as a difference with lots of cancellation occurring between terms so that a simple formula for each partial sum is obtained.
Review section 2.2.
Finish problems assigned on Friday, and also look at problems 3, 5, 7, 9, 49, 50, 63 on pges 745-747 to check understanding.

Tue, 01/28/2003. Integral Test.: Infinite series behave just like the corresponding improper integral provided that the integrand is a positive and decreasing function. This is really a comparison test comparing a series to an integral, just as we compared two integrals before and as we will compare two series later. The guiding principle is the same: if the larger converges, then the smaller must converge; if the smaller diverges, then the larger diverges. The reason you can use it both ways, is that by looking at it the right way you can make the integral either smaller or larger than the series.
Study examples 1 and 4 on pages 750-751.
Do problems 3, 5, 7, 11, 13, 15, 17, 19, 21 on page 754 to check understanding.

Wed, 01/29/2003. Series Approximation.: The proof of the integral test gives a way for using integrals to approximate the error made when a partial sum is used as an approximation for the sum of the entire series. Drawing an appropriate picture is a great help in determining the starting point to use for the integral.
Study examples 5 and 6 on pages 752-752, learn how to prove estimates in red rectangles on page 752.
Do 1, 25, 27, 31, 33, 35 on pages 754-755 to check understanding.
Write up problems 2, 22, 30 on pages 754-755 to turn in on Friday.

Fri, 01/31/2003. Comparison Test.: Some series with positive terms can be compared to a geometric series or a p-series. If a series with positive terms is smaller than a convergent series, then it must be convergent. If a series with positive terms is larger than a divergent series, then it must be divergent.
Review the geometric series test on page 740 and the p-series test on page 750. Learn the comparison test on page 756 and the limit comparison test on page 757.
On pages 759-760, do problems 1, 3, 5, 7, 9, 11, 13, 21, 23, 25, 27, 33, 35 to check understanding.

Mon, 02/03/2003. Alternating Series Test.: The alternating series test allows us to determine the convergence of a series whose terms are alternately positive and negative. The absolute value of the first term of a series that converges by the alternating series test gives an upper bound for the absolute value of the sum of the series.
Learn the theorems in the red rectangles on page 761 and 763. Study examples 1-3 on page 762-763.
Do problem 2, 3, 7, 9, 12, 13, 14, 15, 21, 23 on page 764 to check understanding.

Tue, 02/04/2003. Absolute Convergence.: A series is said to converge absolutely if the sum of the absolute values of the terms is convergent. A series is convergent if it is absolutely convergent. A series is conditionally convergent if the series converges but not absolutely.
Learn the definitions and theorem stated above and on pages 765-766.
On pages 770-771, do problems 1, 3, 7, 9, 13, 24 to check understanding.
Write up problems 26 and 30 on page 765 and problem 18 on page 771 to turn in next Friday.
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Wed, 02/05/2003. Ratio Test.: Geometric series contain exponential terms. Thus any series that contains exponential terms has a chance of behaving like a geometric series for large n. Factorials grow more rapidly than exponentials, so terms involving factorials will either be a lot smaller or a lot larger than terms containing exponentials. Thus the ratio test works well on series that contain exponentials and/or factorials, but usually fails otherwise.
Learn the ration test on page 767, and study example 4 on page 768.
Do problems 11, 15, 17, 19, 21, 23, 29, 31, and 33 on pages 770-771 to check understanding.

Fri, 02/07/2003. Root Test and Strategy for Testing Series.: The root test is most effective if the nth term of a series is an nth power. Otherwise the ratio test usually easier to apply.
Learn the root test on page 769. Study example 6 on page 769.
Do 22, 23, 25, 26 on page 771 to check understanding.
Review all the convergence and divergence tests that we have studied, and practice deciding which method to apply to each of the problems 1-34 on page 773.

Mon, 02/10/2003. Power Series.: A power series represents a function whose domain is the interval of convergence for the series. The radius of convergence is determined by checking for absolute convergence of the power series using the ratio test (or the root test). The endpoints of the interval must be tested separately.
Study example 4 on pages 777.
Do 3, 7, 9, 11, 13, 19 on page 778 to check understanding.

Tue, 02/11/2003. Review.: Here are links to a couple of old exams in PDF format.
First Exam, Spring 2001
First Exam, Fall 2001

Wed, 02/12/2003. First Hour Exam.: The first exam will cover sections 12.1-12.7 as well as the Comparison Test for Improper Integrals on page 564. A take home part will be due on Friday.

Fri, 02/14/2003. Power Series in (x-a).: A power series in (x-a) converges on an interval centered at x = a. The radius of convergence is R means the series converges absolutely if |x-a| is less than R and diverges if |x-a| is greater than R. Note that |x-a| is read and means "the distance between x and a." Convergence at the endpoints (at the points where |x-a| = R}, must always be checked separately. Thus the interval of convergence contains all values of x for which the power series is convergent, and may contain both endpoints, only one endpoint, or neither endpoint.
Study example 5 on page 777-778.
On page 778, do problems 17, 19, 21, 23, 25, 27, 29, 30 to check understanding.

Mon, 02/17/2003.: Class canceled by bad weather.

Tue, 02/18/2003. Geometric Power Series.: Some power series are geometric series. In this untypical situation, we can easily find the formula for the function represented by the power series by using the geometric series sum formula (see page 740). The radius of convergence of a geometric power series is given by the geometric series test. Differentiating or integrating the resulting series will lead to power series for other functions. An important and convenient fact is that the radius of convergence of the differentiated or integrated series is the same as the radius of convergence of the original series.
Study examples 1-3 on pages 779-780.
Do problems 3, 5, 7, 9 on pages 783-784 to check understanding.
Write up problems 6, 16, 30 on pages 783-784 to turn in on Friday.

Wed, 02/19/2003. Differentiation and Integration of Power Series.: A power series represents a function on its interval of convergence. The polynomial p_n equal to the partial sum of the series up to and including the nth power of x should give us a fairly good approximation of the function represented by the power series. The larger n is, the better we would expect the approximation to be. Thus differentiating and integrating a power series, we would expect should be done just like we differentiate and integrate polynomials. A definite integral can be approximated by approximating the infinite series that results using the same techniques we used before.
Study examples 5, 6, 7 on page 781-782.
Do problems 13, 15, 17, 19, 21, 23, 25, 27, 37, 39 on page 784 to check understanding.

Fri, 02/21/2003. Taylor and Maclaurin Series.: We can use tricks like we used in the previous section to find the Taylor series for functions like ln(1+x) or arctan(x). Alternatively, we can find the Taylor series by using Taylor's formula given on page 786.
Study examples 3, 5, 6, 7 on pages 789-791, and know by heart the series given at the top of page 792.
On pages 794-795, do 1-6, 21, 23, 25, 27 to check understanding.

Mon, 02/24/2003. Review of Taylor Series.: Taylor series may be found by using Taylor's series, but also may be derived using other known series. The remarkable fact is that no matter what correct method is used, the same Taylor's series will result. With a lot of practice, you will eventually be able to see which method is likely to be easier for a particular function.
Study examples 8 and 9 on pages 792-793.
On page 795, do 31, 33, 35, 37, 39, 41, 43 to check understanding.

Tue, 02/25/2003. Finding Limits using Taylor's Series.: A Taylor's series about x = 0 gives a good approximation for the function represented near x = 0. This series is useful for approximating definite integrals over intervals close to 0 and for evaluating limits involving the function as x approaches 0.
Study example 9 on page 793.
Do problems 45, 47, 53,55, 57 on page 795 to check understanding.
Write up problems 42, 46, 48, 54 on page 795 to turn in Friday.

Wed, 02/26/2003.: The Taylor polynomial of degree n about x = a for a function f can be used to approximate the value of the function for x near a. If |x-a| = d, and the absolute value of the (n+1)st derivative of f is less than M for |x-a| less than or equal d, then |error| is less than M*(d)^(n+1)/(n+1)!
Study examples 1 and 2 on pages 801-803.
Do 3, 5, 13, 15, 21, 23 on pages 806-807 to check understanding.

Fri, 02/28/2003. Taylor's Inequality.: To approximate a function at value x, we first find a nice value a (f(a) is easily computed number). Find the Taylor's series about x = a, choose a Taylor polynomial of degree n and use it to approximate your function. Then use Taylor's inequality to estimate the maximum error.
Review what we have talked about this week.
Write up problems 16, 24, 26, 28 on page 807 to turn in next Tuesday.

Mon,03/03/2003. Three Space.: Review rectangular coordinates in three space, the formula for the distance between two points in three space, and the equation of a sphere.
Study examples 1, 3, 4, and 5 on pages 818-820.
Do problems 1, 3, 8, 9, 10, 11, 13, 15, 17, 29, 30, 31, and 33 on page 821 to check understanding.

Tue, 03/04/2003. Vectors.: Vectors have magnitude and direction whereas scalars have magnitude only. Vectors may be thought of as arrows, points, ordered triples (in 3 space), or linear combinations of the unit vectors i, j, and k. Each way of visualizing a vector adds to the understanding of the abstract concept of vector.
Study examples 1-5 on pages 824-828.
Do problems 5, 7, 11, 17, 19, 21, 25, 29, 31, and 33 on page 829 to check understanding.

Wed, 03/05/2003. Dot Product and Angle between Vectors.: The dot product of two vectors yields information about the angle between the two vectors.
Study examples 2-5 on pages 832-834.
Do problems 5, 7, 9, 11, 17, 25, 27, 33, 35 on page 836 to check understanding.

Fri, 03/07/2003. Cross Product.: The cross product of two vectors yields a vector orthogonal to both vectors. The direction of the cross product is determined by the right hand rule.
Study examples 1, 3, and 4 on pages 839-842.
Do problems 7, 9, 11, 15, 23, 27, 31, 33 on pages 843-844 to check understanding.

Mon, 03/17/2003. Lines in Three Space.: A line in three space can be describe using a vector equation, or parametrically, or symmetrically. Learn these three forms of representation and how to convert from each to the others, see pages 846-847.
Do problems 3, 4, 7, 9, 11, 13, 15, 17 on page 852 to check understanding.

Tue, 03/18/2003. Planes.: A normal to a plane is a vector that is orthogonal to any vector from one point in the plane to another. A normal to a plane can be found by taking the cross product of two vectors "in the plane."
Study examples 4-7 on pages 849-850.
Do problems 19, 21, 23, 25, 27, 29, 31, 35, 37 on page 853 to check understanding.
Write up problems 14, 18, 28, 40 on page 853 to turn in next Monday.

Wed, 03/19/2003.

Fri, 03/21/2003.: No class. I will be leaving for a conference at Clemson around noon on Thursday.

Mon, 03/24/2003.: Do problems 1-7, 10-11, 15-25 on pages 867-868 for more practice for the exam on Wednesday.

Tue, 03/25/2003. Review.: Review for the exam. Here are links to a couple of old exams in PDF format.
Second Exam, Spring 2001
Second Exam, Fall 2001

Wed, 03/26/2003. Second Exam.: Take home part of exam is due on Friday.
The exam will cover power series, vector operations, and equations of lines and planes.

Fri, 03/28/2003. Functions of Two Variables.: The domain and range of a function of two variables can be determined by knowing the domain and range of the related functions of one variable, and understanding that lines and conics separate the plane into regions described by equations or inequalities.
Study examples 1, 2, 4-8 on pages 908-911.
Do problems 5-9, 11, 13, 15, 17, 21, 23, 25, 27, 30 on pages 918-919 to check understanding.

Mon, 03/31/2003. Level Curves.: A level curve is a set of points in the domain of a function of two variables on which the function has a fixed value. A graph showing different level curves of a function is called a contour map of the surface which is the graph of the function.
See Using plot3d in Maple to graph functions of two variables
and Plotting level curves in Maple
Study examples 9-12 on page 914-915.
Do problems 30, 33, 35, 41, 43, 51-56 on page 919-920 to check understanding.
Write up problems 34, 50, 60 on pages 919-920 to turn in on Friday.

Tue, 04/01/2003. Limits.: When a function can't be evaluated at (0,0), then we want to look at the limit of this function as (x,y) approaches (0,0) to understand the behavior of the function near (0,0). Looking at a contour map of a function of two variables is a good way to see if the limit of a function exists as you approach the origin, and if the limit does not exist, the contour map indicates what curves approaching the origin to use to prove that the limit does not exist.
See Contour Maps to Analyze Limits
Study examples 1-6 on pages 923-926.
Do problems 7, 9, 11, 13, 15, 21, 27 on page 928 to check understanding.

Wed, 04/02/2003. Limits and Continuity.: If a function is continuous at (a,b), then the limit as (x,y) approaches (a,b) is just the value of the function at (a,b). Knowing when a function is continuous at a point can save you a lot of work.
Study examples 6-9 on pages 926-927.
Look at problems 29, 33, 35, 36 on page 929 to check understanding.
Write up problems 6, 8, 12, 14 on page 928 to turn in next Monday.

Fri, 04/04/2003. Partial Differentiation: Study examples 2, 3, 4,and 7 on pages 932-937.
Do problems 5, 7, 9, 15, 19, 21, 33, 35, 37, 39, 56, 59 on page 940 to check understanding.

Mon, 04/07/2003. Linear Approximation.: If the partitial derivatives of a function f(x,y) are continuous at a point (a,b), then f(x,y) is differentiable at (a,b), and has a tangent plane at the point (a,b,f(a,b)). This tangent plane defines a linearization, L(x,y) which can be used to approximate f(x,y) for (x,y) close to (a,b), and the differential, d(x,y), that can be used to approximate the change, f(x,y)-f(a,b), for (x,y) close to (a,b).
Study example 1 on pages 943-944, and example 2 on page 946.
On page 950, do problems 1, 3, 5, 9, 13, 17 to check understanding.

Tue, 04/08/2003. Maxima and Minima.: The discriminant, involving the second partial derivatives, enable you to classify some points as relative extremum points or as saddle points. If the discriminant is 0 at a critical point, you must analyze the function near the critical point by examining its formula or by looking at its graph to determine its behavior.
Study examples 1-3 on pages 974-975.
Do problems 3, 5, 7, 9, 19 on pages 981-982 to check understanding.

Wed, 04/09/2003. Maxima and Minima.: In an application problem, identify the formula for the quantity that you want to maximize or minimize, and use any other constraints to reduce the formula to a function of two variables. Then find the critical values, and use the discriminant to classify each critical value.
Study examples 5 and 6 on pages 948-949, and example 6 on page 978.
On page 951 do 32, 34 and on page 982 do 46 to turn in next Monday.

Fri, 04/11/2003. Double and Iterated Integrals.: The volume under the graph of a function of two variables is given by the double integral of the function over the rectangle which is the base. By Fubini's Theorem, a double integral can be evaluated as an iterated integral.
Study examples 1-3 on pages 1004-1005, and examples 1-2 on pages 1010-1012.
Do problems 5, 9, 11, 15 on pages 1008-1009 and problems 3, 11, 15, 16 on pages 1014-1015 to check understanding.
See Numerical Approximation of Double Integrals

Mon, 04/14/2003. Type I and Type II Integrals.: Functions of two variables can be integrated over regions of Type I and Type II. If a region is of both types, then the order of integration can be reversed by carefully analyzing the region.
Study example 3 on pages 1012-1013, and examples 1 and 3 on pages 1017-1019.
Do problems 27 and 29 on page 1015, and problems 1, 3, 5, 9, 18, 19, 33, 35, 37, 39 on page 1022 to check understanding.
See Drawing Type I and Type II regions in Maple

Tue, 04/15/2003. Properties of Double Integrals.: If a region is not of Type I or Type II, but can be divided up into disjoint regions, each of which is Type I or Type II, then we can evaluate a double integral over the region by adding the double integrals over the disjoint regions. Since larger functions have larger double integrals over a region, estimates for a double integral can be found using maximum and minimum values of the function.
Study examples 5 and 6 on pages 1020-1021.
On page 1022, do 21, 23, 25, 41, 43, 45, 47, 50 to check understanding.
See Graphing Surfaces over Circular Regions

Wed, 04/16/2003. Polar Coordinates.: When the domain over which a double integral is to be evaluated is bounded by circles with center at the origin and rays emanating from the origin, then polar coordinates is usually the right choice.
Study examples 1-2 on pages 1025-1026.
On page 1028, do problems 1, 3, 6, 7, 9, 11, 19, 23 to check understanding.

Fri, 04/18/2003. Good Friday Holiday: No Class

Mon, 04/21/2003. Review of Double Integrals.

Tue, 04/22/2003. Review.: Review 15.1-15.4, 15.7, 16.1-16.4 for exam.
See Fall 2001, Third Exam
and Spring 2001, Third Exam

Wed, 04/23/2003. Third Exam.: There will be a take home part due on Friday.

Fri, 04/25/2003. More general regions in polar coordinates.: There are regions in polar coordinates that analogous to Type I and Type II regions in rectangular coordinates, and they are dealt with in the same way.
On page 1028, do 15, 17, 21, 25, 26, 27, 29, 31, 32, 33 to check understanding.
You may do at most two extra credit assignments for the semester.
Extra credit assignment 1: Attend the Gentry Lecture.
Extra credit assignment 2: Do problem 5 on page 1072.
Extra credit assignment 3: Read pages 1032-1033 on Moment of Inertia, study example 4 on page 1033, and do problem 12 on page 1038.
Extra credit assignment 4: Attend the senior seminar on Monday, April 28, 2003 at 3:30 p.m. in Calloway 3 (no credit if you already attended the Gentry Lecture).

Mon, 04/28/2003.

Tue, 04/29/2003.

Wed, 04/30/2003.

Wednesday, 05/07/2003. Final Examination.: 9:00 a.m.-12:00 p.m.
Calloway 20.

Created 01/02/2003. Last modified 04/25/2003. Email to ekh@wfu.edu