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 -- Gradient
- 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:
Click here for additional optional PRACTICE PROBLEMS with SOLUTIONS found at
THE CALCULUS PAGE .
Dr Kouba's
Supplementary Class Handouts ,
Basic Derivative Formulas From Math 21A and Trig Identities ,
Basic Trig Integrals and Identities From Math 21B ,
Basic Integral Formulas and
Basic Integration Techniques.
The materials for this course were designed by Dr Kouba.