## MATH 21D (SECTIONS 001), 2205 Haring, 8-9:40 MTWF ## Instructor: Dr. D. A. Kouba

Here is a brief introduction video from Kouba .

Here is a message from Kouba: "These are currently strange and unpredictable times. Due to unforeseen and difficult circumstances, UC Davis Chancellor May has cancelled all in-class lectures for Spring Quarter 2020. This means that all of us professors are being asked to move our Spring Quarter classes online on very short notice using untested methods, and that all of you students are being asked to make new and major adjustments to how you learn. I would ask that we all do our best to be patient and understanding, and work together with the goal of making your learning of the Math 21D material our top priority. With that in mind, my current plan is to post highly detailed classnotes for most lecture days; create, edit, and post short videos for most lecture days; and have occasional office hours using the video platform Zoom. Please understand that these changes are provisional and may change on short notice. Let's work together to make the best of a bad situation. Together let's turn these proverbial lemons into lemonade."

Last Updated: June 30, 2020

Text: Thomas' Calculus: Early Transcendentals (11th edition) by Weir, Hass

Office: 3135 MSB

Phone: (530) 754-0469

Office Hours: I will have random Zoom Office Hours.

Here are Basic Derivative Formulas From Math 21A and Trig Identities .

Here are Basic Integration Techniques From Math 21B .

Here are Trig Integration Formulas From Math 21B .

Here are Supplementary Class Handouts .

### This is an OPTIONAL EXTRA CREDIT (NOT Available during Summer 2020) SURVEY .

Click here for additional optional PRACTICE PROBLEMS with SOLUTIONS found at THE CALCULUS PAGE , a site that I created .

## HERE ARE HOMEWORK ASSIGNMENTS, LECTURE NOTES, AND SHORT VIDEOS

The following homework assignments (taken from the Text: Thomas' Calculus: Early Transcendentals (11th edition) by Weir, Hass) are subject to minor changes. Please report to me any typos that you find in the Solutions. Thank you.

## Math 21D Homework Solutions

SCANNED PROBLEMS for Sections 15.1-15.6

-- Here are notes and examples describing regions in the plane using Vertical and Horizontal Cross-Sections , and here is a video with three examples. Here is another video with two examples.
-- Here are notes on the definition of a Double Integral .
-- Here are worked out Examples of Double Integrals, and here is a video with two examples. Here is a second s video with three examples.
-- Some functions do not have closed form anti-derivatives (also called nonelementary integrals ). If you encounter these functions when doing Double Integrals, you will need to SWITCH the ORDER of INTEGRATION. Here are worked out examples of Double Integrals where we SWITCH the ORDER of INTEGRATION , and here is a video with one example. Here is a second video with one example.

• HW #1 ... (Section 15.1) ... p. 1079: 1, 2, 4, 5, 7, 9-11, 13, 14, 16, (SET UP BUT DO NOT EVALUATE INTEGRALS FOR THE NEXT 6 PROBLEMS.) 21, 23, 25, 26 (Change e^x to e^y), 28, 30, 31-33, 36, 38, (SET UP BOTH ORDERS OF INTEGRATION BUT DO NOT EVALUATE INTEGRALS FOR THE NEXT 2 PROBLEMS.) 39, 40

-- Here are Review Examples of Graphing Surfaces in 3D-Space.
-- Here is a video showing one example of a divergent Improper Double Integral. Here is a second video showing one example of a convergent Improper Double Integral.
-- Here is a video of an example showing how to find the Projection of the Intersection of Surfaces in 3D-Space onto the xy-plane.
-- Here are challenging, worked-out examples , which combine Double Integrals, Volume, and Graphing in 3D-Space.

• HW #2 ... (Section 15.1) ... p. 1079: (SET UP BUT DO NOT EVALUATE INTEGRALS FOR THE NEXT 7 PROBLEMS.) 41, 43-47, 51, 54, 59, 61 ..... and ..... Worksheet 1

-- Here are notes and examples for finding Area using Double Integrals.
-- Here are notes and examples for finding the Average Value of Function z=f(x, y) over Region R in the xy-Plane.
-- Here is a video showing an example of Average Value of a Function.

• HW #3 ... (Sections 15.2) ... p. 1089: (SET UP BUT DO NOT EVALUATE INTEGRALS FOR THE FIRST 9 PROBLEMS.)(SET UP INTEGRALS USING BOTH VERTICAL AND HORIZONTAL CROSS-SECTIONS FOR PROBLEMS 2, 3, and 6.) 2, 3, 6, 7, 9, 11-14, 15a, 17 (Change region R to region R bounded by:y=2x, y=0, x=2.), 18 (Change function to 1/(x+1)y and change region R to region R described by: 0 <= x <=1, e^x <= y <= e.)

-- Here are notes and examples for finding the Moments, Centroids, Centers of Mass, Mass, and Moments of Inertia of Objects in 2D-Space.
-- Here is a development of the Moments of Inertia formula.
-- Here is a Summary of and an Example of the following Applications of Double Integrals-- Area, Mass, Moments, Centroids, Centers of Mass, and Moments of Inertia of Objects in 2D-Space.

• HW #4 ... (Section 15.2) ... p. 1089: (SET UP BUT DO NOT EVALUATE INTEGRALS FOR THE FIRST 11 PROBLEMS.) 19, (Ignore the x-axis as a boundary.) 21, 24, 26-28, 32-34, (Omit radius of gyration for problems 36 and 39.) 36, 39, 41, 44, (For problems 53 and 54 use Pappus' Formula on page 1075 and assume that the density of the regions is the constant "delta" mass units over area units.) 53, 54

-- Here is a brief Review of Polar Coordinates with some examples.
-- Here are notes and examples for Double Integrals using Polar Coordinates .
-- Here is a brief video explaining the "mystery" of why " dA = r dr d(theta) " in Polar Coordinates.
-- Here is a video showing an example of converting a Double Integral in Polar Coordinates to a Double Integral in Rectangular Coordinates.
-- Here is a video showing an example of converting a Double Integral in Rectangular Coordinates to a Double Integral in Polar Coordinates.

• HW #5 ... (Sections 15.3) ... p. 1097: 1, 3, 4 (should be dx dy not dy dx), 6, 8, 9, 12, 13, 16, 18-20, (SET UP BUT DO NOT EVALUATE INTEGRALS FOR THE NEXT 5 PROBLEMS.) 22-25, 27, 29, 32, 33, 37a, 40, 42

-- Here are notes and examples on how to Describe Solid Regions R in 3D-Space Using Projections.
-- Here is a video showing how to Describe Regions R in 3D-Space by first Projecting onto the xy-plane, yz-plane, or the xz-plane.
-- Here are notes and examples on the definition and evaluation of Triple Integrals using Rectangular Coordinates.

• HW #6 ... (Section 15.4) ... p. 1106: 7, 8, 11, 12, 16, 17, 19, 20, (SET UP BUT DO NOT EVALUATE INTEGRALS FOR THE NEXT 10 PROBLEMS.) 21, 23-25, 27, 29, 33, 34, 36, 38, 41, 42

-- Here are notes and examples using Triple Integrals and Rectangular Coordinates to find Volume, Mass, Moments, Center of Mass, Centroid, and Moment of Inertia.
-- Here is a video with six examples showing how to find the Distance Between Objects in 3D-Space. Here is a video with two more examples. This skill will be useful when finding the radius r for Moments of Inertia.

• HW #7 ... (Section 15.5) ... p. 1112: 2, 3 (Find the moment of inertia about a.) the x-axis and b.) the origin only.), 4a (Assume that the vertices are (0,0,0), (3,0,0), (0,2,0), and (0,0,1) and that the density at point (x,y,z) is xyz+1.) Find the centroid and moment of inertia about the z-axiz only.), 5 (Assume that the density at point (x,y,z) is x+y+z+2. Find the moment of inertia about the y-axis and center of mass only.), 7a (Assume that the density at point (x,y,z) is e^-(x^2+y^2+z^2).), 11 (Assume that the density at point (x,y,z) is z/(x+y+2). Find the moment of inertia only), 13, 15abc (On part c. find the moment of inertia about the z-axis and the moment of inertia about the origin only.), 17, 26acd

-- Here are notes and examples using Triple Integrals and Cylindrical Coordinates.
-- Here is a video showing conversion of a Triple Integral in Rectangular Coordinates to one in Cylindrical Coordinates.
-- Here is a video showing conversion of a Triple Integral in Cylindrical Coordinates to one in Rectangular Coordinates.

• HW #8 ... (Section 15.6) ... p. 1124: 1, 6, 8, 10, (SET UP BUT DO NOT EVALUATE INTEGRALS FOR THE NEXT 15 PROBLEMS.) 11-13, 14 (Evaluate this one.), 16, 18, 19, 43, 44, 47, 53, 56, 63, 68, 73 (Do moment of inertia only.)

-- Here are notes and examples using Triple Integrals and Spherical Coordinates.
-- Here is an explanation of the Differential of Volume for Spherical Coordinates.
-- Here is a video showing two examples of graphing of a solid region R, which is described using Spherical Coordinates. Here is a second video showing one example.
-- Here is are Summaries and Conversions using Rectangular Coordinates, Cylindrical Coordinates, and Spherical Coordinates.
-- Here is a Summary of Applications of Triple Integrals.

• HW #9 ... (Section 15.6) ... p. 1124: 21, 24, 26, 27, 30, (SET UP BUT DO NOT EVALUATE INTEGRALS FOR THE NEXT 18 PROBLEMS. USE SPHERICAL COORDINATES ON ALL PROBLEMS EXCEPT 41ab.) 31-35, 37, 38, 40ab, 41abc, 42b (Assume density at point P=(x,y,z) is z.), 46, 52, 56, 65, 70 (Find x bar only.)
EXAM 1 is Monday, October 18, 2010. It will cover handouts, lecture notes, and examples from class, homework assignments 1 through 9, discussion sheets 1, 2, 3 and 4 (problem 1-7 only) and material from sections 15.1-15.6 in the book which was presented in lecture notes through Friday, October 15, 2010. 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 2010 (THIS IS SUBJECT TO UNANNOUNCED CHANGES.)

• 1 -- Describe Flat Region Using Rectangular Coordinates and Polar Coordinates
• 1 -- Describe Solid Region Using Rectangular, Cylindrical, or Spherical Coordinates
• 2 -- Evaluate Double Integrals
• 3 -- SET UP BUT DO NOT EVALUATE Triple Integrals in Rectangular, Cylindrical, or Spherical Coordinates for Volume, Mass, Average Value, Moment, Center of Mass, Centroid, or Moment of Inertia
• 1 -- Other
• 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 2010 EXAM 1 IS :

A+ ......

A ......

A-/B+ ......

B ......

C ......

D ......

F ...... We Can Do It !

SCANNED PROBLEMS for Sections 15.7, 13.1-13.4, 16.1

-- Here are notes and examples on Mappings from Plane to Plane.
-- Here are notes and examples on Change of Variables, the Jacobian, and Double Integrals.
-- Here is a video showing a Change of Variables for a Double Integral, use of the Jacobian, and application of the Change of Variable Theorem.

• HW #10 ... (Section 15.7) ... p. 1135: 1, 4, 6, 8-10, 12, 14, 15a, 16, 20, 22

-- Here are notes and examples about Vector Functions and their paths in 2D-Space and 3D-Space. These Vector Functions model position, direction of motion, speed of motion, and acceleration of motion along a path C.
-- Here are notes explaining the physical significance of the Acceleration Vector .
-- Here is an example of a Vector Function used to find the Velocity Vector, Acceleration Vector, Speed of motion, Acceleration of motion, and a Force related to this motion.

• HW #11 ... (Section 13.1) ... p. 916: 1, 3, 4, 6, 8, 10, 12, 13, 15, 17, 19-21, 24, 26, 31, 33, 34, 36, 37de, 38, 40, 41, 43, 45, 48

-- Here are notes and examples on the Integration of Vector Functions.
-- Here are notes and examples on Projectile Motion .
-- Here is a video showing an example of Projectile Motion.

• HW #12 ... (Section 13.2) ... p. 920: 1-5, 7, 11, 13, 17, 18, 26, 28

-- Here are notes and examples on Arc Length along a path C in 2D-Space or 3D-Space.
-- Here is a video where we find Arc Length s as a function of time t, and then write time t as a function of Arc Length s.
-- Here is an example where we find Arc Length s as a function of time t, and then write the Vector Position Function as a function of Arc Length s.
-- Here are notes and examples on the Unit Tangent Vector to a path C in 2D-Space or 3D-Space.

• HW #13 ... (Section 13.3) ... p. 935: 1, 3, 6, 7, 9, (For problems 12-14 write time t as a function of arc length s and write position vector function r bar as a a function of s.) 12-14, 17, 19, 20

-- Here are notes and examples on the Principal Unit Normal Vector to paths C in 2D-Space or 3D-Space.
-- Here are notes and examples on the Curvature and Circles of Curvature related to paths C in 2D-Space or 3D-Space.
-- In this video I explain why the definition of Curvature is a sensible one.
-- In this video we do an example where the numerical values of Curvature are computed.

• HW #14 ... (Section 13.4) ... p. 942: 2-5,9, 11, 13, 14, 17, 19, 21, 22

-- Here are notes and examples on Line Integrals Along a Path C in 2D-Space or 3D-Space.
-- Here are two Applications of Line Integrals.
-- Here is a long, detailed Example finding Mass, Length, Center of Mass, Centroid, and Moments of Inertia for a Bent Wire Along a Path C in 2D-Space.
-- In this video we use a Line Integral to find the Area of a curved wall.

• HW #15 ... (Section 16.1) ... p. 1147: 1-8, 10-13, 15-19, 22, 23, 24 (Find y bar only.), 25a , 27 (Omit radius of gyration.), 29 (Assume that density at point P=(x,y,z) is z+1. Find moment of inertia about the z-axis, moment of inertia about the origin, the z bar coordinate for center of mass, and the z bar coordinate for centroid.)
EXAM 2 is Friday, November 5, 2010. It will cover handouts, lecture notes, and examples from class, homework assignments 10 through 15, discussion sheets 4 (problems 8-12), 5, and 6, and material from sections 15.7, 13.1-13.4, and 16.1 in the book which was presented in lecture notes through Wednesday, November 3, 2010. 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 2010 (THIS IS SUBJECT TO UNANNOUNCED CHANGES.)

• 2 -- Projectile motion problems
• 1 -- Position vector, velocity vector, acceleration vector, unit tangent vector, principal unit normal vector, speed, and acceleration problems
• 2 -- Mapping, Jacobian, magnification problems
• 2 -- Arc length problems
• 2 -- Line Integrals
• 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 ARE POSTED ON THIS WEBPAGE.

THE GRADING SCALE FOR FALL 2010 EXAM 2 IS :

A+ ...... 100-110

A ...... 85-99

A-/B+ ...... 82-84

B ...... 67-81

C ...... 47-66

D ...... 31-46

F ...... 0-30 Bring It On !

SCANNED PROBLEMS for Sections 16.2-16.5

-- Here are notes and examples on Work done by a Vector Field F(bar) over a path C in 2D-Space or 3D-Space.
-- Here is a video using a Gradient Vector as the Force Vector to compute Work Along a Path C in 2D-Space using Line Integrals.

• HW #16 ... (Section 16.2) ... p. 1158: 1, 3-6, 7abc, 12abc, 13, 15, 17, 19-22, 31 (Do not plot horizontal and vertical components.), 32 (Do not plot horizontal and vertical components.), 35

-- Here are notes and examples on the Flow of a Vector Field F(bar) Along a path C in 2D-Space or 3D-Space.
-- Here are notes on the Flux of a Vector Field F(bar) Across a region R in the xy-Plane enclosed by path C.
-- Here are Flux and Flow examples of a Vector Field F(bar) Across and Along a path C in the xy-Plane.
-- In this video we find Flux Across a Closed path C.

• HW #17 ... (Section 16.2) ... p. 1158: 23a, 24, 26, 27, 29ab, 30, 37, 40, 41, 43

-- Here are several examples exploring the unique properties of the Work of a Gradient Vector Field F(bar) Across and Along a path C.
-- In this video we start with a Gradient Vector Function and work backwards to find the scalar function f from which it came.
-- Here are notes, theorems, and examples on Conservative Vector Fields .

• HW #18 ... (Section 16.3) ... p. 1168: 1-10, 13, 14, 16, 17, 19, 20, 24, 25, 37

-- Here are notes and examples on Green's Theorem's 1 and 2 , which show the connection between Line Integrals on a Loop C and Double Integrals in the xy-Plane over a Closed Region R.
-- These notes will help explain why the definition of Divergence is a sensible one.
-- These notes will help explain why the definition of k-Component of Curl , also called Circulation Density, is a sensible one.
-- Here are notes and examples on Green's Theorem's 3 , which is applied to a Region R Enclosed by Two Loops C1 and C2. There is also a discussion about how to use a Line Integral on a Loop C to find the Area of the Enclosed Region R.

• HW #19 ... (Section 16.4) ... p. 1179: 1, 2, 4, 5, 7, 8, 10-13, 15-20, 22, 24, 26, 29, 30, 33

-- Here are notes and examples on Surface Integrals of Scalar Functions defined on Surfaces in 3D-Space.
-- Here is a summary of Applications of Surface Integrals.
-- Here is a video where we find the Area of a Surface S in 3D-Space using a Surface Integral.
-- Here is an example and explanation about how to compute Flux Across a Surface S in 3D-Space using a Surface Integral.
-- Here is a video showing the detailed Evaluation of a Surface Integral.

• HW #20 ... (Section 16.5) ... p. 1190: 1, 3, 5, 6, 7 (Assume that c=3.), 12, 13 (Assume that a=1.), 15, 17, 19-21, 24-27, 33 (Find x bar only. Find z bar first and use symmetry of region to conclude that x bar = y bar = z bar.), 35 (Find z bar only and the moment of inertia. Omit radius of gyration.)
EXAM 3 is Monday, November 22, 2010. It will cover handouts, lecture notes, and examples from class, homework assignments 16 through 20, discussion sheets 7, 8, and 9, and material from sections 16.2-16.5 in the book which was presented in lecture notes through Friday, November 19, 2010. 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 2010 (THIS IS SUBJECT TO UNANNOUNCED CHANGES.)

• 1 or 2 -- Work, Flow, or Circulation Problems
• 2 -- Conservative Vector Field, Path Independence Problems
• 2 -- Green's Theorem Problems
• 4 -- Surface Integral Problems
• 2 -- 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 2010 EXAM 3 IS :

A+ ...... 100-110

A ...... 85-99

A-/B+ ...... 80-84

B ...... 63-79

C ...... 40-62

D ...... 30-39

F ...... 0-29 You can get there !

SCANNED PROBLEMS for Sections 16.6-16.8

-- Here are notes and examples on Parametrizing Surfaces S in 3D-Space.
-- Here is a video showing examples of how to Parametrize a Surface S in 3D-Space.
-- Here is a video showing examples of how to Parametrize a Surface S in 3D-Space.
-- Here is another video showing an example of a Surface Integral over a Parametrized Surface S in 3D-Space with special attention given to the Diffential of Area, dA.

• HW #21 ... (Section 16.6) ... p. 1199: 1, 4, 5, 7, 9, 11, 14a, 17, 20, 24, 27, 30, 34, 35, 37, 39, 40, 41

-- Here are notes and examples on Curl and Stoke's Theorem . Let's get stoked !

• HW #22 ... (Section 16.7) ... p. 1209: 1, 3, 6, 7, 8, 13, 15, 17

-- Here are notes and examples on the Divergence Theorem .

• HW #23 ... (Section 16.8) ... p. 1220: 5, 6c, 7, 8, 14, 16, 26

The FINAL EXAM is Monday, December 6, 2010,

10:30 a.m. - 12:30 p.m.

in 212 Veihmeyer

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 21, and material from sections 15.1-15.7, 13.1-13.4, and 16.1-16.8, and discusssion sheets 1-10.

TYPES OF QUESTIONS FOR THE FALL 2010 FINAL EXAM (THIS IS SUBJECT TO UNANNOUNCED CHANGES.).

• 1 -- Double Integral (Rectangular or Polar Coordinates)
• 1 -- Triple Integral (Rectangular, Cylindrical, or Spherical Coordinates)
• 2 -- Green's Theorems Problems
• 2 -- Surface Integrals
• 1 -- Stoke's Theorem Problem
• 1 -- Divergence Theorem Problem
• 1 -- 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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