## CALCULUS

MATH 21C FALL 2022

Lectures: in Medical Sciences Building C room 180 from 11:00 to 11:50 on MWF.
Office Hours:
Babson: 11:00 to 12:00 on Tuesdays via zoom 7150588313.
Deshmukh: 5pm to 6pm on Thursdays in the calc room.
Sections: on Tuesdays at the following times and locations by section: A01, A02, A03, A04, A05.
Text: Any calculus text, such as Thomas' Calculus: Early Transcendentals (13th edition) by Weir, Hass, Giordano.
Exams: There will be 480 pts from 3 midterms and a final with one midterm dropped and the final as half the grade or equal to one midterm- whichever is higher. Practice exams and content descriptions developed by Dr Kouba are linked below.
Grading: 0 < F < 160 < D- < 180 < D < 220 < D+ < 240 < C- < 260 < C < 300 < C+ < 320 < B- < 340 < B < 380 < B+ < 400 < A- < 420 < A < 460 < A+ < 480

Calculus Room: Math 21ABCD Calculus Room , where TAs are available to answer your questions.

You are expected to work hard and to try as many exercises as possible. This is the only way to learn mathematics. We are here to help. Please do not hesitate to come and see any of us if you have a question or problem.

Course Outline: The course covers sequences and series first and then multivariable calculus as per the syllabus. This is a rough outline of when topics will be covered and will be edited as the term progresses. The exam scheduling will not change. It is strongly suggested that you do the assigned problems. They will not be collected.

 Day Date Topics Homework Wednesday September 21 sequences HW#1: 1-35 Friday September 23 infinite series HW#1: 39-124, HW#2: 3-21, 50-59 Monday September 26 the integral test HW#2: 28-45, 60-92, HW#3: 1-24 Wednesday September 28 comparison tests HW#3: 28-58, HW#4 Friday September 30 ratio and root tests HW#5 Monday October 3 alternating series HW#6 Wednesday October 5 convergence review Friday October 7 Midterm I HW#1 - HW#6 Monday October 10 power series HW#7 Wednesday October 12 Taylor and Maclaurin series HW#8 Friday October 14 Taylor series converence HW#9 Monday October 17 binomial series HW#10 Wednesday October 19 Taylor applications HW#10 Friday October 21 vectors HW#11, HW#12 Monday October 24 dot and cross products HW#13, HW#14 Wednesday October 26 review Practice Midterm II 1-6, 10a. Friday October 28 Midterm II HW#7 - HW#14 Monday October 31 lines and planes HW#15 Wednesday November 2 multivariable functions HW#16.5 Friday November 4 limits and continuity HW#17 Monday November 7 partial derivatives HW#18 Wednesday November 9 the chain rule HW#19 Friday November 11 no class Monday November 14 directional derivatives HW#20 Wednesday November 16 review Practice Midterm III 1-6, 7c. Friday November 18 Midterm III HW#16.5 - HW#20 Monday November 21 tangent planes HW#21 Wednesday November 23 extrema and saddle points HW#22 Friday November 25 no class Monday November 28 Lagrange multipliers HW#23 Wednesday November 30 optimization review HW#24 Friday December 2 Review Wednesday December 7 Final:3:30 - 5:30 HW#1 - HW#24

The following homework assignments are subject to minor changes.
SCANNED PROBLEMS for Chapter 10 (Sections 10.1-10.6)
SCANNED PROBLEMS for Chapter 10 (Sections 10.7-10.10)
SCANNED PROBLEMS for Chapter 12 (Sections 12.1-12.5)
SCANNED PROBLEMS for Chapter 12 (Sections 12.6)
SCANNED PROBLEMS for Chapter 14 (Sections 14.1-14.6)
SCANNED PROBLEMS for Chapter 14 (Sections 14.7-14.8)

• HW #1 ... (Section 10.1) ... p. 581: 2, 3, 4, 8, 11, 12, 13, 16, 18, 19, 21, 23-25, 26 (Hint: Use greatest integer function.), 28 (Use Sandwich Theorem.), 32, 33, 35, 39 (Use Sandwich Theorem.), 40, 42, 43, 46 (Use Sandwich Theorem.), 48, 49, 52, 54, 56, 57, 59, 60, 63 (Use Sandwich Theorem.), 64, 66-69, 74, 78-81, 83, 86, 87, 89, 92, 93, 96-99, 108 (Assume the case where x>1.), 116, 118, 119, 121, 123, 124 ... and ... Worksheet 1 ... Here are some notes on the Formal Definition of the Limit of a Sequence and here is an example of a Factorial Sequence .

• HW #2 ... (Sections 10.2) ... p. 591: 3, 5, 7, 8, 12, 14, 16-18, 20, 21, 28, 30-35, 39, 40, 42, 45, 50-52, 54, 59, 60, 62, 64, 65, 67, 83-88, 89a, 90, 91, 92 ... Here are some notes on the Sequence of Partial Sums Test and the Geometric Series Test .

• HW #3 ... (Section 10.3) ... p. 598: 1, 3, 4, 6, 7, 13-17, 19, 22, 24, 28, 30, 32, 33, 35, 38, 41, 43, (Use (*)(*) from the Integral Test Handout for the following two problems) 49, 52, 58 ... and ... Problems Using Star and Double Star from the Integral Test Handout. Here are the Solutions ... Here are some notes on Equations (*) and (**), the Integral Test, and the P-Series Test .

• HW #4 ... (Sections 10.4) ... p. 603: 1, 2, 4, 5-9, 11-23, 26-31, 33-41, 43-45, 47, 48, 51, 52, 54, 56-58, 60, 61 (optional), 66 (optional) ... Here are some notes on Comparison Tests and Limit Comparison Tests .

• HW #5 ... (Section 10.5) ... p. 609: 1, 3-9, 11, 13-15, 17, 20-24, 26-31, 35, 37, 38, 39 (Change -n to n.), 43, 46-48, 51, 52, 54, 55, 57, 58, 60, 61, 66 ... Here are some notes on the Ratio Test and the Root Test .

• HW #6 ... (Section 10.6) ... p. 615: 2-7, 9-11, 13, 14, 16, 19, 20, 22, 25, 27, 28, 31, 33, 36, 40, 41, 42, 44, 47, 48, 50, 53, 54, 56, 58, 62, 63, 66-68 ... Here are handouts on the Alternating Series Test and the Absolute Convergence Test .

• Here is a Summary of tests for infinite series and this is a list of Subtle Facts about infinite series.

• HW #7 ... (Section 10.7) ... p. 624: 1a, 4a, 6a, 7a, 10a, 12a, 17a, 24a, 25a, 27a, 29a, 33-36, 38-42, 44, 46, 50a, 53, 54, 60

• HW #8 ... (Section 10.8) ... p. 630: 1-4, 6, 8, 10, 11 (Use Maclaurin series for e^x.), 12 (Use Maclaurin series for e^x.), 13 (Use Maclaurin series for 1/(1-x).), 14 (Use Maclaurin series for 1/(1-x).), 15 (Use Maclaurin series for sin x.), 17 (CHANGE THE FUNCTION TO: x^2 cos(x^3) and use Maclaurin series for cos x.), 21, 22 (Change x^2/(x+1) to x^2/(x^3+1) and use Maclaurin series for 1/(1-x).), 25, 27, 29, 33 (Use Maclaurin series for 1/(1-x) and for cos x.), 34 (Use Maclaurin series for e^x.), 36 (Use Maclaurin series for sin x.), 41b, 42b, 43b ... Here are some notes on Taylor Series and Taylor Remainder (Error) .

• HW #9 ... (Section 10.9) ... p. 637: 1, 4, 5, 6 (Change the function from cos(1/sqrt{2}*x^(2/3)) to cos( 1/sqrt{2}*x^(3/2)), 7, 9-12, 16, 17 (Do 2 ways : i. trig identity 1st, ii. series multiplication), 19, 20, 21 (Do 2 ways : i. differentiation 1st, ii. series multiplication), 22, 24 (Do 2 ways : i. trig identity 1st, ii. series multiplication), 25 (Change e^x + 1/(1+x) to (e^x)(1/(1+x))., 29, 31, 33, 37, 40 (Begin by finding the first 4 nonzero terms and the general formula for the Maclaurin Series for f(x)= sqrt(1+x).) 41, 43, 48 ... Here is a list of well-known Maclaurin Series and a way to estimate pi.

• HW #10 ... (Section 10.10) ... p. 645: 1, 2, 6, 7, 10, 12, 15, 17, 20, 22, 25, 28, (For 29, 32, 34, 38, and 39 also use L'Hopital's Rule to evaluate limits.) 29, 32, 34, 38, 39, 41-53, 61, 62, 65

• HW #11 ... (Section 12.1) ... p. 707: 1, 3, 6, 8, 12, 13, 16, 17ab, 18abc, 20ac, 21b, 22ab, 26-28, 30-32, 34,36-39, 42, 43, 47, 52, 55, 58-60, 62-66 ... Here are some notes on points and graphs in Three-Dimensional Space .

• HW #12 ... (Section 12.2) ... p. 716: 1, 4, 6, 7, 9-13, 16, 18, 21, 23, 25, 28, 29, 31, 33, 35, 38, 40, 41, 43, 45-48, 49, 52 (See problem 51 first.) ... Here are some detailed notes on Vectors in 2D-space and 3D-space ... Here are two worked out Examples using vectors; one is a hanging weight and the other is a plane flying into a headwind.

• HW #13 ... (Section 12.3) ... p. 724: 1, 4, 7, 10, 12, 13, 16-18, 22-25, 27, 41, 43, 44, 49 (Do not use results from problems 31 and 32 to do problem 49. Simply use basic facts about vectors and lines.) ... Here are brief handouts on Properties of Dot Product and (Orthogonal) Projection .

• HW #14 ... (Section 12.4) ... p. 730: 2, 3, 7, 9, 10, 14, 16, 19, 20, 23, 25 (See definition of torque on page 729.), 27-29, 31, 33, 36, 39, 42, 45, 46 (Use points (0,0,0), (-2,3,0) and (3,1,0).), 48, 50 ... Here is a brief explanation of the Right-Hand Rule for cross products ... Here are notes on the Triple Scalar Product .

• HW #15 ... (Section 12.5) ... p. 738: 1, 4, 6-8, 10, 21, 22, 24, 25, 28,30 (and find point of intersection), 31, 34, 37, 40, 43, 45, 47, 51, 55, 59, 65, 67, 69

• HW #16 ... (Sections 12.6) ... p. 744: 1, 4, 6, 7, 13, 15, (Foe next 5 problems use intercepts, traces, and/or the indicated values of z for level curves to create a topographical map for the surface.) 18 (Use z=-4, -2, 0, 2, 4), 22 (Use z=8, 7, 4, -1, -8), 25 (Use z=-2, -1, 0, 1, 2), 27 (Use z= -sqrt{8}, -sqrt{3}, 0, sqrt{3}, sqrt{8}), 29, 31, 39 ... Here is a 3D graphing example using Level Curves ... Here is an example where we create an equation for a Surface of Revolution .

• HW # 16.5 ... (Section 14.1) ... p. 799: 1d, 4c, 5, 7, 10, 11, (Determine and sketch domain for the following functions.) 18ab, 19ab, 23ab, 24ab, 25ab, and f(x,y)=ln(4-x^2-y^2)

• HW #17 ... (Section 14.2) ... p. 807: 1, 4, 9, 12, 14, 16, 20-24, 41-50, ... and ... p. 876: 12, 16 ... and ... Worksheet 2 ... Here is the statement for the Precise Limit of a Function of Two Variables and Examples .

• HW #18 ... (Section 14.3) ... p. 819: 2, 4, 5, 7, 10, 12, 13, 15 (Change ln(x+y) to ln(3x+y^2).), 16, 19-21, 41-46, 48, 51-54, 58, 60 (Change x^2+y^2 to x^2+y^4.), 62, 65, 75-77, 81, 84, 85

• HW #19 ... (Section 14.4) ... p. 828: 1 (Change the function to w=x^2+2y.), 3, 5, 6, 8, 9, 14, 15, 20, 24, 26, 28, 30, 32, 39, 40, 42, 43-45, 47, 48, 49a, 51, 52 ... and .... these second-order chain rule problems ... Here is a handout on Exact Change, the Differential, and the Chain Rule for z=f(x,y) ... Here is an explanation for finding the Second Partial Derivative using the Chain Rule.

• HW #20 ... (Section 14.5) ... p. 838: 2, 3, 6, 7, 10, 12, 13, 15, 16, 17, 19, 22, 29, 32-35, 36a ... Here are notes on the Directional Derivatives and Gradient Vectors for a function of two variables ... Here is an alternate form of the Differential for z=f(x,y) using a Directional Derivative ... Here is a handout discussing why Gradient Vectors are Normal to Level Curves.

• HW #21 ... (Section 14.6) ... p. 845: 1, 4, 9, 12, 14, 15, 20, 21, 23, 26b, 28a, 30b, 52, 56

• HW #22 ... (Section 14.7) ... p. 855: 1, 5, 15, 16, 19, 22, 26, 28-30 ... Here is a statement of the Second Derivative Test for partial derivatives to find and classify critical points for a function of two variables ... For those interested, here is a Proof of the Second Derivative Test.

• HW #23 ... (Section 14.7) ... p. 855: 31 (REWRITE PROBLEM: Use the triangle formed by the graphs of x=0, y=3, and y=x.), 34, 41 (For the remaining problems you need only find the critical points and extreme values. You need NOT verify that each corresponds to a maximum or minimum.) 50 (HINT: Start with a point (x,y,z) on the paraboloid. Use a projection vector to find the distance from (x,y,z) to the plane. Then find the critical point for the distance function.), 51 (First find the distance from point (x,y,z) on the plane to the point (0,0,0).), 53-55, 57-59 ... and ... the two problems below ...
I.) The material for the top and bottom of a rectangular box costs 3 cents per square foot, and that for the sides costs 2 cents per square foot. What are the cost and dimensions of the least expensive box that has a volume of 1 cubic foot ?
II.) Determine the dimensions and volume of the closed rectangular box of largest volume if the total surface area is to be 12 square meters.

• HW #24 ... (Section 14.8) ... p. 864: 1, 3, (Minimize distance squared.) 8, 14, (Minimize distance squared.) 21, 27, 30, 37, (Minimize distance squared.) 39

EXAM 1 will cover handouts, lecture notes, and examples from class, homework assignments 1 through 6, discussion sheets 1, 2, and 3 (except problem 3) and material from sections 10.1-10.6 in the book which was presented in lecture notes. MOST of the exam questions will be like examples from lecture notes, homework problems, or discussion sheets.

TYPES OF QUESTIONS FOR EXAM 1 (THIS IS SUBJECT TO UNANNOUNCED CHANGES.)

• 6-8 -- Determine convergence or divergence of series using various series tests
• 1 or 2 -- Alternating series
• 1 or 2 -- problem on integral test errors
• 1 or 2 -- Others
• 1 -- OPTIONAL EXTRA CREDIT

EXAM 2 will cover handouts, lecture notes, and examples from class, homework assignments 7 through 15, discussion sheets 3 (Problem 3 only), 4, 5, and 6, and material from sections 10.7-10.10 and 12.1-12.5 in the book which was presented in lecture notes. MOST of the exam questions will be like examples from lecture notes, homework problems, or discussion sheets.

TYPES OF QUESTIONS FOR EXAM 2 (THIS IS SUBJECT TO UNANNOUNCED CHANGES.)

• 1 -- Find interval of convergence for power series
• 2 -- Find 1st 3 nonzero terms of Taylor Series centered at x=a
• 1 -- Lagrange form of the Taylor remainder.
• 1 -- Use Taylor Polynomial to compute an estimate
• 5 -- Problems involving lines, planes, angles, normal vectors, parallel vectors, points of intersection, etc.
• 1 -- Other
• 1 -- OPTIONAL EXTRA CREDIT

EXAM 3 will cover handouts, lecture notes, and examples from class, homework assignments 16 through 21, discussion sheets 7, 8, and 9 (EXCEPT problems 6-9), and material from sections 14.1-14.6 in the book which was presented in lecture notes. MOST of the exam questions will be like examples from lecture notes, homework problems, or discussion sheets.

TYPES OF QUESTIONS FOR EXAM 3 (THIS IS SUBJECT TO UNANNOUNCED CHANGES.)

• 1 -- 3D Graphing (intercepts, traces, level curves)
• 1 -- Domain and Range
• 2 or 3 -- Limits
• 1 -- Compute various partial derivatives
• 1 or 2 -- Chain Rule
• 1 or 2 -- Directional Derivative
• 1 -- Other
• 1 -- OPTIONAL EXTRA CREDIT

The final exam will cover handouts, lecture notes, and examples from class, homework assignments 1 through 24, and material from sections 10.1-10.10, 12.1-12.5, 14.1-14.8, and discusssion sheets 1-10.

TYPES OF QUESTIONS FOR THE FINAL EXAM (THIS IS SUBJECT TO UNANNOUNCED CHANGES.). The following topics will NOT BE COVERED on this final exam -- 3D-graphing.

• 1 -- Domain, Range
• 2 -- Taylor Series
• 1 -- Taylor Polynomial
• 1 -- Integral test error bounds
• 1 -- Chain Rule
• 1 -- Absolute and Conditional Convergence
• 1 -- Interval of Convergence
• 2 -- Directional Derivatives
• 1 -- Find and Classify Critical Points
• 1 -- Lagrange Multipliers
• 2 or 3 -- Others
• 1 -- OPTIONAL EXTRA CREDIT

Here are Math 21C discussion sheets : Sheet 1 , Sheet 2 , Sheet 3 , Sheet 4 , Sheet 5 , Sheet 6 , Sheet 7 , Sheet 8 , Sheet 9 , Sheet 10 ,
Here are Math 21C Practice Exams ...

**** ... PRACTICE EXAM 1 ... and ... SOLUTIONS ... ****
**** ... PRACTICE EXAM 2 ... and ... SOLUTIONS ... ****
**** ... PRACTICE EXAM 3 ... and ... SOLUTIONS ... ****
**** ... PRACTICE FINAL ... and ... SOLUTIONS ... ****
Here are some TIPS for doing well on exams.

HERE ARE SOME RULES FOR THE EXAMS.

• 0.) IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK YOU FOR YOUR COOPERATION.
• 1.) No books, or classmates may be used as resources for this exam. YOU MAY USE ONE SHEET OF NOTES (both sides) ON THIS EXAM.
• 2.) You will be graded on proper use of limit notation.
• 3.) You will be graded on proper use of derivative and integral notation.
• 4.) Put units on answers where units are appropriate.
• 5.) Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will NOT receive full credit. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important.

Review and supplementary materials: