MATH 21C (SECTIONS C01,C02), 212 Viehmeyer, 99:50 MWF, Thursday Discussions
Instructor: Dr. D. A. Kouba
Last Updated: December 10, 2017
Text: Thomas' Calculus: Early Transcendentals (13th edition) by Weir, Hass, Giordano
Office: 3135 MSB
Phone: (530) 6014444x2007
My Office Hours: Variable (Announced in class or sent via UC Davis email)
Here is a link showing the days and times for the Math 21ABCD Calculus Room , where TA's are available to answer your questions.
In addition to the Calculus Room these TA Office Hours are available:
TA OFFICE HOURS 

Monday 
57 p.m. 

Arturo Palomino 
Calculus Room 1118 MSB 


Tuesday 
4:305:30 p.m. 

Arturo Palomino 
1011 Ghausi 


Thursday 
46 p.m. 

Douglas Sherman 
3240 MSB 
Here are Exam ID #'s .
EXAM DATES :
 EXAM 1 MONDAY, October 16, 2017
 EXAM 2 WEDNESDAY, November 8, 2017
 EXAM 3 MONDAY, December 4, 2017
 FINAL EXAM  MONDAY, December 11, 2017, 810 a.m. in 212 Viehmeyer
The course will likely cover the following sections in our textbook : 10.110.10, 12.112.5, and 14.114.8.
Here you can find all Math 21C
Homework Solutions and
Exam Solutions .
Here is a copy of the Course Syllabus .
Here is a copy of the Schedule of Lectures .
Here are copies of Supplementary Class Handouts .
Here are copies of Basic Derivative Formulas From Math 21A and Trig Identities .
Here are copies of Basic Trig Integrals From Math 21B and Trig Identities ... and ...
Basic Integral Formulas ... and ... Basic Integration Techniques
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 ... ****
This is an OPTIONAL EXTRA CREDIT survey
Click here for additional optional PRACTICE PROBLEMS with SOLUTIONS found at
THE CALCULUS PAGE .
Here are some
TIPS for doing well on my exams.
The following homework assignments are subject to minor changes.
SCANNED PROBLEMS for Chapter 10 (Sections 10.110.6)
 HW #1 ... (Section 10.1) ... p. 581: 2, 3, 4, 8, 11, 12, 13, 16, 18, 19, 21, 2325, 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, 6669, 74, 7881, 83, 86, 87, 89, 92, 93, 9699, 108 (Assume the case where x>1.), 116, 118, 119, 121, 123, 124 ... and ...
Worksheet 1 ... Here is a link showing why the number e is a limit ... 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, 1618, 20, 21, 28, 3035, 39, 40, 42, 45, 5052, 54, 59, 60, 62, 64, 65, 67, 8388, 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, 1317, 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 PSeries Test .
 HW #4 ... (Sections 10.4) ... p. 603: 1, 2, 4, 59, 1123, 2631, 3341, 4345, 47, 48, 51, 52, 54, 5658, 60, 61 (optional), 66 (optional) ... Here are some notes on Comparison Tests and Limit Comparison Tests .
 HW #5 ... (Section 10.5) ... p. 609: 1, 39, 11, 1315, 17, 2024, 2631, 35, 37, 38, 39 (Change n to n.), 43, 4648, 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: 27, 911, 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, 6668 ... 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.
EXAM 1 is Monday, October 16, 2017. It 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 10.110.6 in the book which was presented in lecture notes through Friday, October 13, 2017. MOST of the exam questions will be like examples from lecture notes, homework problems, or discussion sheets.
TYPES OF QUESTIONS FOR EXAM 1 FOR FALL 2017 (THIS IS SUBJECT TO UNANNOUNCED CHANGES.)
 68  Determine convergence or divergence of series using various series tests
 1  Alternating series
 1  Epsilon,N Proof
 1 or 2  (*) or (*)(*) problem
 1 or 2  Others
 1  OPTIONAL EXTRA CREDIT
HERE ARE SOME RULES FOR EXAM 1.
 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 notes, books, or classmates may be used as resources for this exam. YOU MAY USE A CALCULATOR 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.
SOLUTIONS TO EXAM 1 ARE POSTED ON THIS WEBPAGE.
THE GRADING SCALE FOR FALL 2017 EXAM 1 IS :
A+ ...... 100108
A ...... 8899
A/B+ ...... 8587
B ...... 6684
C ...... 4365
D ...... 3342
F ...... 032
SCANNED PROBLEMS for Chapter 10 (Sections 10.710.10)
 HW #7 ... (Section 10.7) ... p. 624: 1a, 4a, 6a, 7a, 10a, 12a, 17a, 24a, 25a, 27a, 29a, 3336, 3842, 44, 46, 50a, 53, 54, 60
 HW #8 ... (Section 10.8) ... p. 630: 14, 6, 8, 10, 11 (Use Maclaurin series for e^x.), 12 (Use Maclaurin series for e^x.), 13 (Use Maclaurin series for 1/(1x).), 14 (Use Maclaurin series for 1/(1x).), 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/(1x).), 25, 27, 29, 33 (Use Maclaurin series for 1/(1x) 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, 912, 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 wellknown Maclaurin Series .
 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, 4153, 61, 62, 65
SCANNED PROBLEMS for Chapter 12 (Sections 12.112.5)
 HW #11 ... (Section 12.1) ... p. 707: 1, 3, 6, 8, 12, 13, 16, 17ab, 18abc, 20ac, 21b, 22ab, 2628, 3032, 34,3639, 42, 43, 47, 52, 55, 5860, 6266 ... Here are some notes on points and graphs in ThreeDimensional Space .
 HW #12 ... (Section 12.2) ... p. 716: 1, 4, 6, 7, 913, 16, 18, 21, 23, 25, 28, 29, 31, 33, 35, 38, 40, 41, 43, 4548, 49, 52 (See problem 51 first.) ... Here are some detailed notes on Vectors in 2Dspace and 3Dspace ... 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, 1618, 2225, 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 , the (Orthogonal) Projection , and an Alternate form of the dot product.
 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.), 2729, 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 RightHand Rule for cross products ... Here are notes on the Triple Scalar Product .
 HW #15 ... (Section 12.5) ... p. 738: 1, 4, 68, 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
EXAM 2 is Wednesday, November 8, 2017. It 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 10.710.10 and 12.112.5 in the book which was presented in lecture notes through Monday, November 6, 2017. MOST of the exam questions will be like examples from lecture notes, homework problems, or discussion sheets.
TYPES OF QUESTIONS FOR EXAM 2 FOR FALL 2017 (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
HERE ARE SOME RULES FOR EXAM 2.
 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 notes, books, or classmates may be used as resources for this exam. YOU MAY USE A CALCULATOR ON THIS EXAM.
 2.) You will be graded on proper use of derivative and integral notation.
 3.) Put units on answers where units are appropriate.
 4.) 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.
SOLUTIONS TO EXAM 2 ARE POSTED ON THIS WEBPAGE.
THE GRADING SCALE FOR FALL 2017 EXAM 2 IS :
A+ ...... 100  108
A ...... 90  99
A/B+ ...... 82  89
B ...... 64  81
C ...... 41  63
D ...... 31  40
F ...... 0  30
SCANNED PROBLEMS for Chapter 12 (Sections 12.6)
 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's 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 .
SCANNED PROBLEMS for Chapter 14 (Sections 14.114.6)
 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(4x^2y^2)
 HW #17 ... (Section 14.2) ... p. 807: 1, 4, 9, 12, 14, 16, 2024, 4150, ... and ... p. 876: 12, 16 ... and ... Worksheet 2 (with solutions ) ... Here is the statement for the Precise Limit of a Function of Two Variables and a worked out Example .
 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, 1921, 4146, 48, 5154, 58, 60 (Change x^2+y^2 to x^2+y^4.), 62, 65, 7577, 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, 4345, 47, 48, 49a, 51, 52 ... and .... these secondorder 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, 3235, 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
EXAM 3 is Monday, December 4, 2017. It will cover handouts, lecture notes, and examples from class, homework assignments 16 through 21, discussion sheets 7, 8, and 9 (EXCEPT problems 69), and material from sections 14.114.6 in the book which was presented in lecture notes through Friday, December 1, 2017. MOST of the exam questions will be like examples from lecture notes, homework problems, or discussion sheets.
TYPES OF QUESTIONS FOR EXAM 3 FOR FALL 2017 (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  Epsilon,Delta Proof
 1  Other
 1  OPTIONAL EXTRA CREDIT
HERE ARE SOME RULES FOR EXAM 3.
 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.
 00.) IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO COPY ANSWERS FROM ANOTHER STUDENT's EXAM.
 1.) No notes, books, or classmates may be used as resources for this exam. YOU MAY USE A CALCULATOR ON THIS EXAM.
 2.) You will be graded on proper use of derivative and integral notation.
 3.) Put units on answers where units are appropriate.
 4.) Do not use any shortcuts from the book when using the method of integration by parts.
 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.
SOLUTIONS TO EXAM 3 ARE POSTED ON THIS WEBPAGE.
THE GRADING SCALE FOR FALL 2017 EXAM 3 IS :
A+ ...... 100110
A ...... 9099
A/B+ ...... 8389
B ...... 6882
C ...... 4567
D ...... 3444
F ...... 033
SCANNED PROBLEMS for Chapter 14 (Sections 14.7. 14.8)
 HW #22 ... (Section 14.7) ... p. 855: 1, 5, 15, 16, 19, 22, 26, 2830 ... 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).), 5355, 5759 ... 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
The FINAL EXAM is Monday, December 11, 2017
810 a.m.
in 212 Viehmeyer
BRING A PICTURE ID TO THE EXAM
AND BE PREPARED TO SHOW IT TO KOUBA OR THE TEACHING ASSISTANT !!
The final exam will cover handouts, lecture notes, and examples from class, homework assignments 1 through 23 (Omit problems 31, 34, and 41 and omit HW #24.), and material from sections 10.110.10, 12.112.5, 14.114.7 (Omit Section 14.8.), and discusssion sheets 110.
TYPES OF QUESTIONS FOR THE FALL 2017 FINAL EXAM (THIS IS SUBJECT TO UNANNOUNCED CHANGES.). The following topics will NOT BE COVERED on this final exam  Taylor Error (Remainder), 3Dgraphing, epsilon/delta proofs, epsilon/N proofs, and 3Dlimits.
 1  Domain, Range
 2  Taylor Series
 1  Taylor Polynomial
 1  (*) or (*)(*)
 1  Chain Rule
 1  Absolute and Conditional Convergence
 1  Interval of Convergence
 2  Directional Derivatives
 1  Differential
 1  Find and Classify Critical Points
 3 or 4  Others
 1 or 2  OPTIONAL EXTRA CREDIT
HERE ARE SOME RULES FOR THE FINAL EXAM.
 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 notes, books, or classmates may be used as resources for this exam. YOU MAY USE A CALCULATOR 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.
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Your comments, questions, or suggestions can be sent via email to Kouba by
clicking on the following address :
kouba@math.ucdavis.edu .