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
Monday	April 1	sequences intro	HW#1 2-35, 92-98.
Wednesday	April 3	sequences theorems	HW#1
Friday	April 5	series- geometric, nth term	HW#2
Monday	April 8	integral tests	HW#3
Wednesday	April 10	integral bounds and comparisons	HW#3,4
Friday	April 12	ratio test and alternating series	HW#5,6
Monday	April 15	power series	HW#7
Wednesday	April 17	convergence review	PRACTICE EXAM 1
Friday	April 19	Midterm I and solutions	HW#1 - HW#6
Monday	April 22	power, Taylor and Maclaurin series	HW#8
Wednesday	April 24	Taylor series	HW#9
Friday	April 26	Taylor series remainders	HW#9
Monday	April 29	vectors	HW#11, HW#12
Wednesday	May 1	dot and cross products	HW#13, HW#14
Friday	May 3	lines, planes and multivariable functions	HW#15, HW#16.5
Monday	May 6	limits and continuity	HW#17
Wednesday	May 8	review	practice midterm 2
Friday	May 10	Midterm II	HW#7 - HW#16.5
Monday	May 13	partial derivatives	HW#18
Wednesday	May 15	chain rules	HW#19
Friday	May 17	directional derivatives	HW#20
Monday	May 20	tangent planes	HW#21
Wednesday	May 22	extrema and saddle points	HW#22
Friday	May 24	Lagrange multipliers	HW#23
Monday	May 27	Memorial Day: no class
Wednesday	May 29	review	Practice Midterm III ???
Friday	May 31	Midterm III	HW#17 - HW#23
Monday	June 3	least squares fits	HW#24
Wednesday	June 5	review
Friday	June 7	Final: 8:00AM-10:00	HW#1 - HW#24

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 7, discussion sheets 1, 2, and 3 and material from sections 10.1-10.7 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 8, 9 and 11 through 17, discussion sheets 4, 5, 6, 7 and Problem 1 from sheet 8 along with material from sections 10.8, 10.9, 12.1-12.5, 14.1 and 14.2 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 points of convergence for a power series,
  • 1 or 2 -- Find 1st 3 nonzero terms of Taylor Series centered at x=a,
  • 1 -- Taylor polynomial remainder estimate,
  • 2 or 3 -- Problems involving lines, planes, angles, normal vectors, parallel vectors, points of intersection, distances etc.
  • 1 -- Vector problem involving some of dot products, cross products, forces, work,
  • 1 -- Domain and Range,
  • 1 or 2 -- Other,
  • 1 -- OPTIONAL EXTRA CREDIT.
  
  
  
  EXAM 3 will cover handouts, lecture notes, and examples from class, homework assignments 18 through 23, discussion sheets 8 (xcept problem 1), 9 and 10, along with material from sections 14.3-14.7 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.)

  • 2 or 3 -- Compute various partial derivatives
  • 1 or 2 -- Chain Rule
  • 1 or 2 -- Directional Derivative
  • 1 -- Differential
  • 1 -- Find and Classify Critical Points
  • 1 -- Lagrange multipliers
  • 1 or 2 -- Other
  • 1 -- OPTIONAL EXTRA CREDIT
  
  
  
  
  The final exam will cover handouts, lecture notes, and examples from class, homework assignments 1 through 9 and 11 through 24 along with material from sections 10.1-10.9, 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 -- Taylor Error (Remainder), 3D-graphing, and 3D-limits.
  
  

  • 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 -- Differential
  • 1 -- Find and Classify Critical Points
  • 3 or 4 -- Others
  • 1 or 2 -- 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 midterm 2 ... and ... SOLUTIONS ... ****
  **** ... PRACTICE EXAM 3 ... 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 key to getting a good score on this exam. Neatness and organization are also important.