MATH 18.01, FALL PROBLEM SET # 3A
|
|
- Adrian Taylor
- 5 years ago
- Views:
Transcription
1 MATH 18.01, FALL PROBLEM SET # 3A Professor: Jared Speck Due: by Friday 1:45pm on (in the boxes outside of Room during the day; stick it under the door if the room is locked; write your name, recitation instructor, and recitation meeting days/time on your homework) Supplementary Notes (including Exercises and Solutions) are available on the course web page: This is where to find the exercises labeled 1A, 1B, etc. You will need these to do the homework. Part I consists of exercises given and solved in the Supplementary Notes. It will be graded quickly, checking that all is there and the solutions not copied. Part II consists of problems for which solutions are not given; it is worth more points. Some of these problems are longer multi-part exercises given here because they do not fit conveniently into an exam or short-answer format. See the guidelines below for what collaboration is acceptable, and follow them. To encourage you to keep up with the lectures, both Part I and Part II tell you for each problem on which day you will have the needed background for it. You are encouraged to use graphing calculators, software, etc. to check your answers and to explore calculus. However, (unless otherwise indicated) we strongly discourage you from using these tools to solve problems, perform computations, graph functions, etc. An extremely important aspect of learning calculus is developing these skills. And you will not be allowed to use any such tools on the exams. Part I (15 points) Notation: The problems come from three sources: the Supplementary Notes, the Simmons book, and problems that are described in full detail inside of this pset. I refer to the former two sources using abbreviations such as the following ones: 2.1 = Section 2.1 of the Simmons textbook; Notes G = Section G of the Supplementary Notes; Notes 1A: 1a, 2 = Exercises 1a and 2 in the Exercise Section 1A of the Supplementary Notes; Section 2.4: 13 = Problem 13 in Section 2.4 of Simmons, etc. Lecture 7. (Thurs., Sept. 20) Linear and quadratic approximations. Read: Notes A. Homework: Notes 2A: 2, 3, 7, 11, 12ade. Lecture 8. (Tues., Sept. 25) Curve-sketching. Read: 4.1, 4.2. Homework: Notes 2B: 1, 2aeh, 4, 6ab, 7ab. 1
2 2 MATH 18.01, FALL PROBLEM SET # 3A Part II (40 points) Directions and Rules: Collaboration on problem sets is encouraged, but: i) Attempt each part of each problem yourself. Read each portion of the problem before asking for help. If you don t understand what is being asked, ask for help interpreting the problem and then make an honest attempt to solve it. ii) Write up each problem independently. On both Part I and II exercises you are expected to write the answer in your own words. You must show your work; bare solutions will receive very little credit. iii) Write on your problem set whom you consulted and the sources you used. If you fail to do so, you may be charged with plagiarism and subject to serious penalties. iv) It is illegal to consult materials from previous semesters. 0. (not until due date; 3 points) Write the names of all the people you consulted or with whom you collaborated and the resources you used, or say none or no consultation. This includes visits outside recitation to your recitation instructor. If you don t know a name, you must nevertheless identify the person, as in, tutor in Room 2-106, or the student next to me in recitation. Optional: note which of these people or resources, if any, were particularly helpful to you. This Problem 0 will be assigned with every problem set. Its purpose is to make sure that you acknowledge (to yourself as well as others) what kind of help you require and to encourage you to pay attention to how you learn best (with a tutor, in a group, alone). It will help us by letting us know what resources you use. 1. (Sept. 26.; quadratic approximations; = 8 points) Your electric guitar speaker has a continuous dial that varies from 0 to 11. Assume that the volume output v of the speaker is a twice differentiable function of the dial number d. Using your decibel meter, you have measured the volume output of your speaker at the following three settings: d = 8 d = 9 d = 10 v (in decibels) You are very curious to know the volume output when d = 11, but since you live in a dorm, you don t dare turn the dial to 11. Luckily, you are enrolled in and are therefore able to estimate the output using the following procedure. a) Estimate v (9) and v (10) by using difference quotients of the form f(x) f(x x) x for x = 1. We compute that v (9) v (10) v(9) v(8) = , 9 8 v(10) v(9) = b) Use part a) and a similar procedure to estimate v (10). We compute that
3 MATH 18.01, FALL PROBLEM SET # 3A 3 v (10) v (10) v (9) c) Derive a quadratic approximation of v(d) near d = 10. In your quadratic approximation, you can use the approximate values for v and v from parts a) and b). Then use your quadratic approximation to estimate v(11). We use the quadratic approximation formula to deduce that when d is near d 0, we have v(d) v(d 0 ) + v (d 0 )(d d 0 ) v (d 0 )(d d 0 ) 2 Setting d 0 = 10 and d = 11, we therefore have that (d 10) (.426)(d 10)2. v(d) (11 10) + 1 (.426)(11 10)2 2 = (Sept. 26.; limits; continuity; quadratic approximations; = 10 points) In this problem, you will investigate what happens to the surface area of a sphere when you slightly squash it. A perfect unit sphere can be visualized in the following way: consider the set of points in the (x, y) plane that satisfy the equation x 2 + y 2 = 1. This curve describes a sphere of radius 1 centered at the origin. Now imagine that this circle is continuously rotated about the y axis until it has spun a full 360 degrees. The resulting 3 d object that is traced out is a sphere of radius 1 in three-dimensional space centered at the origin. Now instead of the circle, consider the curve ( y ) 2 x 2 + = 1, a where a is some positive constant with 0 < a < 1. This curve describes an ellipse in the (x, y) plane centered at the origin. The horizontal axis of the ellipse has length 2 and the vertical axis has length 2a. Now imagine that this ellipse is rotated about the y axis in the manner described above. The resulting 3 d object that gets traced out is called an oblate ellipsoid, which means that it is a sphere that has been squashed in one direction (in this case the y axis is the squashed direction). This geometric object comes up when one studies the shape of the earth: the earth s rotation gives it a slightly squashed character (in reality the earth is quite bumpy, and therefore the oblate ellipsoid model of the earth may not be accurate enough for all applications). There is an important quantity associated to the oblate ellipsoid called the eccentricity E. It is 0 for a perfect sphere, and for a oblate ellipsoid, it measures how squashed it is: E = 1 a 2.
4 4 MATH 18.01, FALL PROBLEM SET # 3A In vector calculus, you will develop the tools necessary to calculate the surface area of the oblate ellipsoid. Here, I will only provide you with the formula. The surface area S can be expressed in terms of E as follows: where S = 2π ( 1 + f(e) (1 E 2 ) ), f(e) = { tanh 1 (E) E if E 0, 1 if E = 0. In the above formula, tanh 1 (E) is the inverse of the hyperbolic tangent function. That is, y = tanh 1 (E) if and only if tanh(y) = E, where tanh(e) = sinh(e)/ cosh(e). a) Show that f(e) is continuous at E = 0. We first investigate lim y 0 tanh(y)/y. We use the linear approximations e y 1 + y and e y 1 y for y near 0. Therefore, sinh(y) = 1 2 (ey e y ) y, cosh(y) = 1 2 (ey + e y ) 1, and tanh(y) = sinh(y)/ cosh(y) = y + O(y 2 ). It follows that lim y 0 y + O(y 2 ) tanh(y)/y = lim tanh(y)/y = lim y 0 y 0 y = lim {1 + O(y)} = 1. y 0 We now use the above limit to investigate the limit of interest. We set y = tanh 1 (E), which implies that tanh(y) = E. We note that when E 0, it follows that y 0. Therefore, with the help of the above limit, we compute that Therefore, y lim E 0 tanh 1 (E)/E = lim y 0 tanh(y) = 1. Hence, f(e) is continuous at E = 0 by definition. b) Show that (d/de) tanh 1 (E) = 1/(1 E 2 ). lim f(e) = lim E 0 E 0 tanh 1 (E)/E = 1 = f(0). We set y = tanh 1 (E). Our goal is to compute dy/de. To this end, we consider the equation E = tanh(y). We take the derivative with respect to E of both sides and use the quotient rule, the chain rule, and the relation cosh 2 y sinh 2 (y) = 1 in order to deduce that
5 MATH 18.01, FALL PROBLEM SET # 3A 5 Therefore, 1 = d { dy d tanh(y) dy de = dy ( )} sinh(y) dy cosh(y) de = cosh(y) d d sinh(y) sinh(y) cosh(y) dy dy cosh 2 dy (y) de = cosh2 (y) sinh 2 (y) cosh 2 (y) 1 = cosh 2 (y) dy de. dy de dy de = cosh2 (y). We would like to express the right-hand side in terms of E. To this end, use first use the relation cosh 2 y sinh 2 (y) = 1 to deduce that that cosh 2 (y) = sinh2 (y). We can use the first and last = cosh2 y 1 tanh 2 (y) tanh 2 y equalities in this equation to solve for cosh 2 (y) in terms of tanh 2 (y) = E 2 : We conclude that as desired. cosh 2 (y) = 1 1 tanh 2 (y) = 1 1 E. 2 dy de = 1 1 E 2 c) Find the quadratic approximation to 1/(1 E 2 ) near E = 0. That is, find the constants A 0, A 1 and A 2 such that 1/(1 E 2 ) = A 0 + A 1 E + A 2 E 2 + O(E 3 ) near E = 0. In class, we saw that 1/(1 x) = (1 x) 1 = 1 + x + O(x 2 ) for x near 0. Applying this expansion with x = E 2, we deduce that 1/(1 E 2 ) = 1+E 2 +O(E 4 ). It follows that A 0 = 1, A 1 = 0, and A 2 = 1. d) Imagine that you have made a cubic approximation to tanh 1 (E) near E = 0. That is, assume that you have approximated tanh 1 (E) = B 0 + B 1 E + B 2 E 2 + B 3 E 3 + O(E 4 ) for constants B 0, B 1, B 2, B 3. Assume that B 0 +B 1 E +B 2 E 2 is the usual quadratic approximation to tanh 1 (E) near E = 0. Assume in addition that the derivative of your cubic approximation to tanh 1 (E) is precisely the quadratic approximation to 1/(1 E 2 ) = (d/de) tanh 1 (E) that you found in part b). Under these assumptions, find the constants B 0, B 1, B 2, B 3. We take the derivative of the cubic approximation and set it equal to the quadratic approximation, which was found in part c):
6 6 MATH 18.01, FALL PROBLEM SET # 3A d de (B 0 + B 1 E + B 2 E 2 + B 3 E 3 ) = B 1 + 2B 2 E + 3B 3 E 2 = 1 + E 2. We then equate powers of E to discover that B 1 = 1, B 2 = 0, B 3 = 1/3. The last thing to determine is B 0. By assumption, B 0 is the term one gets from the usual quadratic approximation of tanh 1 (E) near E = 0. That is, B 0 = tanh 1 (0) = 0. In total, the cubic approximation to tanh 1 (E) near E = 0 is tanh 1 (E) = E + (1/3)E 3 + O(E 4 ). e) Use part d) to find the quadratic approximation to f(e) near E = 0. We simply divide the cubic approximation from part d) by E to discover that f(e) = tanh 1 (E) E = E2 + O(E 3 ). f) Use part e) to find the quadratic approximation to S near E = 0. Remark: The quadratic approximation procedure outlined in this problem can be rigorously justified. It turns out that the quadratic approximations you are deriving are in some sense the best possible ones. These ideas fall under the umbrella of Taylor series, which we will discuss at the end of the course. Using part e), we see that Therefore, f(e) (1 E 2 ) = ( ) 3 E2 + O(E 3 ) ( 1 E 2 ) ) = E2 + O(E 3 ). S = 2π ( 1 + f(e)(1 E 2 ) ) ( = 2π ) 3 E2 + O(E 3 ) = 4π 4π 3 E2 + O(E 3 ). g) When E is positive but near 0, does your quadratic approximation predict that the surface area of the ellipsoid will be greater than or less than the surface area of the perfect sphere of radius 1 (the perfect sphere has E = 0)? When E is positive and near 0, the quadratic approximation part from f) predicts that S(E) < 4π because of the 4π 3 E2 term. 3. (Sept. 28.; = 10 points) Section 4.1: 18abc, 20ab
7 MATH 18.01, FALL PROBLEM SET # 3A 7 18) We recall that f(x) = x m (1 x) n, where m, n are positive integers. a) If m is even and x is near 0, then f(x) is non-negative because x m is a square and (1 x) n is near 1. Since f(0) = 0, x = 0 must be a minimum. b) Similarly, if n is even and x is near 1, then f(x) is non-negative because (1 x) n is a square and x m is near 1. Since f(1) = 0, x = 1 must be a minimum. c) Using the product and chain rules, we compute that f (x) = mx m 1 (1 x) n n(1 x) n 1. To find the critical points of f, we set f (x) = 0, which yields the equation mx m 1 (1 x) n nx m (1 x) n 1 = 0. Assuming that x 0 and x 1, we may divide both sides by mx m 1 (1 x) n 1, thus arriving at the following equation: f (x) mx m 1 (1 x) n 1 = (1 x) n m x = 0. The above equation has only the solution x = m, which means that aside from the possibility of m+n x = 0 or x = 1, x = m is the only critical point of f. Note that the expression (1 x) n x is strictly m+n m positive for 0 < x < m and strictly negative for 1 > x > m. Therefore, the first equality in the m+n m+n previous equation implies that the same statement is also true for f (x). Hence, f(x) is increasing for 0 < x < m and decreasing for 1 > x > m. We conclude that x = m is a maximum. m+n m+n m+n 20) a) See the figure below. b) See the figure below. 4. (Sept. 28.; = 9 points) Section 4.2: 12, 16, 18 12) Set y = f(x) = 12/x 2 12/x. Note that there is a discontinuity at x = 0. We compute that y = f (x) = 24x x 2, y = f (x) = 72x 4 24x 3. To find the inflection points, we set y = 0 and multiply the equation by x 4 (for convenience) to derive the equation 72 24x = 0. The above equation has only the solution x = 3, which is the inflection point. For x < 3, we have that f (x) is positive, while for x > 3, we have that f (x) is negative. The graph of f(x) is therefore concave up for x < 3 and concave down for x > 3. See the figure below for a sketch. 16) The question is: Is it possible for a function f(x) defined for all x to have the properties i) f(x) > 0, ii) f (x) < 0, and iii) f (x) < 0? The answer is no.
8 8 MATH 18.01, FALL PROBLEM SET # 3A Figure 1. Section 4.1: 20 a) Here is an explanation. By assumption, f (0) is equal to some negative number N. Since f (x) < 0, f (x) is a decreasing function for all x. Therefore, f (x) < N holds for all x > 0. Let us give a geometric interpretation of this. Let L be the tangent line to the graph of f at x = 0. Then L has a (negative) slope equal to N and is above the x axis at x = 0, so L must intersect the x axis at some point x 0 > 0. Since f (x) < N holds for all x > 0, the graph of f is more negatively sloped than L for all x > 0. It follows that the graph of f lies below the line L for all x > 0. Therefore, the graph of f(x) must intersect the x axis at some x value in between x = 0 and x 0. This contradicts property i) and therefore shows that the three properties are not mutually compatible. 18) Note that x 2 + y 2 = a 2 describes a circle of radius a centered at the origin. We differentiate each side of the equation with respect to x and use the chain rule relation (d/dx)(y 2 ) = 2yy to obtain 2x + 2yy = 0. We then differentiate the above equation with respect to x to obtain
9 MATH 18.01, FALL PROBLEM SET # 3A 9 Figure 2. Section 4.1: 20 b) We now use the above equation to solve for y : 2 + 2(y ) 2 + 2yy = 0. y = (y ) 2 1 y y. Note that the above formula agrees with the graph of the circle: above the x axis, y is positive and the circle is concave down. This agrees with the above formula, which shows that y < 0 when y > 0. On the other hand, below the x axis, y is negative and the circle is concave up. This also agrees with the above formula, which shows that y > 0 when y < 0.
10 10 MATH 18.01, FALL PROBLEM SET # 3A Figure 3. Section 4.2: 12
MATH 18.01, FALL PROBLEM SET # 2
MATH 18.01, FALL 2012 - PROBLEM SET # 2 Professor: Jared Speck Due: by Thursday 4:00pm on 9-20-12 (in the boxes outside of Room 2-255 during the day; stick it under the door if the room is locked; write
More informationMATH 18.01, FALL PROBLEM SET # 8
MATH 18.01, FALL 01 - PROBLEM SET # 8 Professor: Jared Speck Due: by 1:45pm on Tuesday 11-7-1 (in the boxes outside of Room -55 during the day; stick it under the door if the room is locked; write your
More information18.02 Multivariable Calculus Fall 2007
MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.02 Problem Set 1 Due Thursday
More informationWeek 12: Optimisation and Course Review.
Week 12: Optimisation and Course Review. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway November 21-22, 2016 Assignments. Problem
More information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This midterm is a sample midterm. This means: The sample midterm contains problems that are of similar,
More informationMath 121: Final Exam Review Sheet
Exam Information Math 11: Final Exam Review Sheet The Final Exam will be given on Thursday, March 1 from 10:30 am 1:30 pm. The exam is cumulative and will cover chapters 1.1-1.3, 1.5, 1.6,.1-.6, 3.1-3.6,
More informationMath 41 First Exam October 12, 2010
Math 41 First Exam October 12, 2010 Name: SUID#: Circle your section: Olena Bormashenko Ulrik Buchholtz John Jiang Michael Lipnowski Jonathan Lee 03 (11-11:50am) 07 (10-10:50am) 02 (1:15-2:05pm) 04 (1:15-2:05pm)
More information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This is a midterm from a previous semester. This means: This midterm contains problems that are of
More informationWeekly Activities Ma 110
Weekly Activities Ma 110 Fall 2008 As of October 27, 2008 We give detailed suggestions of what to learn during each week. This includes a reading assignment as well as a brief description of the main points
More informationDisclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.
Disclaimer: This is meant to help you start studying. It is not necessarily a complete list of everything you need to know. The MTH 132 final exam mainly consists of standard response questions where students
More information1. Which one of the following points is a singular point of. f(x) = (x 1) 2/3? f(x) = 3x 3 4x 2 5x + 6? (C)
Math 1120 Calculus Test 3 November 4, 1 Name In the first 10 problems, each part counts 5 points (total 50 points) and the final three problems count 20 points each Multiple choice section Circle the correct
More informationMath 115 Syllabus (Spring 2017 Edition) By: Elementary Courses Committee Textbook: Intermediate Algebra by Aufmann & Lockwood, 9th Edition
Math 115 Syllabus (Spring 2017 Edition) By: Elementary Courses Committee Textbook: Intermediate Algebra by Aufmann & Lockwood, 9th Edition Students have the options of either purchasing the loose-leaf
More informationMath 111, Introduction to the Calculus, Fall 2011 Midterm I Practice Exam 1 Solutions
Math 111, Introduction to the Calculus, Fall 2011 Midterm I Practice Exam 1 Solutions For each question, there is a model solution (showing you the level of detail I expect on the exam) and then below
More informationThe First Derivative Test
The First Derivative Test We have already looked at this test in the last section even though we did not put a name to the process we were using. We use a y number line to test the sign of the first derivative
More informationSolutions to Math 41 First Exam October 18, 2012
Solutions to Math 4 First Exam October 8, 202. (2 points) Find each of the following its, with justification. If the it does not exist, explain why. If there is an infinite it, then explain whether it
More informationSection K MATH 211 Homework Due Friday, 8/30/96 Professor J. Beachy Average: 15.1 / 20. ), and f(a + 1).
Section K MATH 211 Homework Due Friday, 8/30/96 Professor J. Beachy Average: 15.1 / 20 # 18, page 18: If f(x) = x2 x 2 1, find f( 1 2 ), f( 1 2 ), and f(a + 1). # 22, page 18: When a solution of acetylcholine
More informationAnswer Key. Calculus I Math 141 Fall 2003 Professor Ben Richert. Exam 2
Answer Key Calculus I Math 141 Fall 2003 Professor Ben Richert Exam 2 November 18, 2003 Please do all your work in this booklet and show all the steps. Calculators and note-cards are not allowed. Problem
More informationSCIENCE PROGRAM CALCULUS III
SCIENCE PROGRAM CALCULUS III Discipline: Mathematics Semester: Winter 2005 Course Code: 201-DDB-05 Instructor: Objectives: 00UV, 00UU Office: Ponderation: 3-2-3 Tel.: 457-6610 Credits: 2 2/3 Local: Course
More informationUniversity of Connecticut Department of Mathematics
University of Connecticut Department of Mathematics Math 1131 Sample Exam 2 Fall 2015 Name: Instructor Name: Section: TA Name: Discussion Section: This sample exam is just a guide to prepare for the actual
More informationMon 3 Nov Tuesday 4 Nov: Quiz 8 ( ) Friday 7 Nov: Exam 2!!! Today: 4.5 Wednesday: REVIEW. In class Covers
Mon 3 Nov 2014 Tuesday 4 Nov: Quiz 8 (4.2-4.4) Friday 7 Nov: Exam 2!!! In class Covers 3.9-4.5 Today: 4.5 Wednesday: REVIEW Linear Approximation and Differentials In section 4.5, you see the pictures on
More informationFriday 09/15/2017 Midterm I 50 minutes
Fa 17: MATH 2924 040 Differential and Integral Calculus II Noel Brady Friday 09/15/2017 Midterm I 50 minutes Name: Student ID: Instructions. 1. Attempt all questions. 2. Do not write on back of exam sheets.
More informationMs. York s AP Calculus AB Class Room #: Phone #: Conferences: 11:30 1:35 (A day) 8:00 9:45 (B day)
Ms. York s AP Calculus AB Class Room #: 303 E-mail: hyork3@houstonisd.org Phone #: 937-239-3836 Conferences: 11:30 1:35 (A day) 8:00 9:45 (B day) Course Outline By successfully completing this course,
More informationMIDTERM 1. Name-Surname: 15 pts 20 pts 15 pts 10 pts 10 pts 10 pts 15 pts 20 pts 115 pts Total. Overall 115 points.
Name-Surname: Student No: Grade: 15 pts 20 pts 15 pts 10 pts 10 pts 10 pts 15 pts 20 pts 115 pts 1 2 3 4 5 6 7 8 Total Overall 115 points. Do as much as you can. Write your answers to all of the questions.
More informationSyllabus for BC Calculus
Syllabus for BC Calculus Course Overview My students enter BC Calculus form an Honors Precalculus course which is extremely rigorous and we have 90 minutes per day for 180 days, so our calculus course
More informationExam 2 Solutions October 12, 2006
Math 44 Fall 006 Sections and P. Achar Exam Solutions October, 006 Total points: 00 Time limit: 80 minutes No calculators, books, notes, or other aids are permitted. You must show your work and justify
More informationMATH 152 FINAL EXAMINATION Spring Semester 2014
Math 15 Final Eam Spring 1 MATH 15 FINAL EXAMINATION Spring Semester 1 NAME: RAW SCORE: Maimum raw score possible is 8. INSTRUCTOR: SECTION NUMBER: MAKE and MODEL of CALCULATOR USED: Answers are to be
More information18.01 Final Exam. 8. 3pm Hancock Total: /250
18.01 Final Exam Name: Please circle the number of your recitation. 1. 10am Tyomkin 2. 10am Kilic 3. 12pm Coskun 4. 1pm Coskun 5. 2pm Hancock Problem 1: /25 Problem 6: /25 Problem 2: /25 Problem 7: /25
More informationSCIENCE PROGRAM CALCULUS III
SCIENCE PROGRAM CALCULUS III Discipline: Mathematics Semester: Winter 2002 Course Code: 201-DDB-05 Instructor: R.A.G. Seely Objectives: 00UV, 00UU Office: H 204 Ponderation: 3-2-3 Tel.: 457-6610 Credits:
More informationReview for Final Exam, MATH , Fall 2010
Review for Final Exam, MATH 170-002, Fall 2010 The test will be on Wednesday December 15 in ILC 404 (usual class room), 8:00 a.m - 10:00 a.m. Please bring a non-graphing calculator for the test. No other
More informationGeometry and Motion, MA 134 Week 1
Geometry and Motion, MA 134 Week 1 Mario J. Micallef Spring, 2007 Warning. These handouts are not intended to be complete lecture notes. They should be supplemented by your own notes and, importantly,
More informationSOLUTIONS 1 (27) 2 (18) 3 (18) 4 (15) 5 (22) TOTAL (100) PROBLEM NUMBER SCORE MIDTERM 2. Form A. Recitation Instructor : Recitation Time :
Math 5 March 8, 206 Form A Page of 8 Name : OSU Name.# : Lecturer:: Recitation Instructor : SOLUTIONS Recitation Time : SHOW ALL WORK in problems, 2, and 3. Incorrect answers with work shown may receive
More informationCollege Algebra: Midterm Review
College Algebra: A Missive from the Math Department Learning College Algebra takes effort on your part as the student. Here are some hints for studying that you may find useful. Work Problems If you do,
More informationMathematics 1 Lecture Notes Chapter 1 Algebra Review
Mathematics 1 Lecture Notes Chapter 1 Algebra Review c Trinity College 1 A note to the students from the lecturer: This course will be moving rather quickly, and it will be in your own best interests to
More informationMATH 341, Section 001 FALL 2014 Introduction to the Language and Practice of Mathematics
MATH 341, Section 001 FALL 2014 Introduction to the Language and Practice of Mathematics Class Meetings: MW 9:30-10:45 am in EMS E424A, September 3 to December 10 [Thanksgiving break November 26 30; final
More informationMath 41 First Exam October 15, 2013
Math 41 First Exam October 15, 2013 Name: SUID#: Circle your section: Valentin Buciumas Jafar Jafarov Jesse Madnick Alexandra Musat Amy Pang 02 (1:15-2:05pm) 08 (10-10:50am) 03 (11-11:50am) 06 (9-9:50am)
More information171, Calculus 1. Summer 1, CRN 50248, Section 001. Time: MTWR, 6:30 p.m. 8:30 p.m. Room: BR-43. CRN 50248, Section 002
171, Calculus 1 Summer 1, 018 CRN 5048, Section 001 Time: MTWR, 6:0 p.m. 8:0 p.m. Room: BR-4 CRN 5048, Section 00 Time: MTWR, 11:0 a.m. 1:0 p.m. Room: BR-4 CONTENTS Syllabus Reviews for tests 1 Review
More informationMath 1190/01 Calculus I Fall 2007
Math 1190/01 Calculus I Fall 2007 Instructor: Dr. Sean Ellermeyer, SC 24, (770) 423-6129 (Math dept: (770) 423-6327), email: sellerme@kennesaw.edu, Web Site: http://math.kennesaw.edu/~sellerme. Time and
More informationINTRODUCTION TO DIFFERENTIATION
INTRODUCTION TO DIFFERENTIATION GRADIENT OF A CURVE We have looked at the process needed for finding the gradient of a curve (or the rate of change of a curve). We have defined the gradient of a curve
More informationLecture 10. (2) Functions of two variables. Partial derivatives. Dan Nichols February 27, 2018
Lecture 10 Partial derivatives Dan Nichols nichols@math.umass.edu MATH 233, Spring 2018 University of Massachusetts February 27, 2018 Last time: functions of two variables f(x, y) x and y are the independent
More information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This is a midterm from a previous semester. This means: This midterm contains problems that are of
More informationTake-Home Exam 1: pick up on Thursday, June 8, return Monday,
SYLLABUS FOR 18.089 1. Overview This course is a review of calculus. We will start with a week-long review of single variable calculus, and move on for the remaining five weeks to multivariable calculus.
More informationJune Stone Bridge Math Department. Dear Advanced Placement Calculus BC Student,
Stone Bridge Math Department June 06 Dear Advanced Placement Calculus BC Student, Congratulations on your wisdom in taking the BC course. I know you will find it rewarding and a great way to spend your
More informationMath 2 Variable Manipulation Part 7 Absolute Value & Inequalities
Math 2 Variable Manipulation Part 7 Absolute Value & Inequalities 1 MATH 1 REVIEW SOLVING AN ABSOLUTE VALUE EQUATION Absolute value is a measure of distance; how far a number is from zero. In practice,
More informationChapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 1 Chapter 1A -- Real Numbers Math Symbols: iff or Example: Let A = {2, 4, 6, 8, 10, 12, 14, 16,...} and let B = {3, 6, 9, 12, 15, 18,
More informationCalculus and Parametric Equations
Calculus and Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Given a pair a parametric equations x = f (t) y = g(t) for a t b we know how
More informationMath 164-1: Optimization Instructor: Alpár R. Mészáros
Math 164-1: Optimization Instructor: Alpár R. Mészáros First Midterm, April 20, 2016 Name (use a pen): Student ID (use a pen): Signature (use a pen): Rules: Duration of the exam: 50 minutes. By writing
More informationA.P. Calculus BC Test Three Section Two Free-Response No Calculators Time 45 minutes Number of Questions 3
A.P. Calculus BC Test Three Section Two Free-Response No Calculators Time 45 minutes Number of Questions 3 Each of the three questions is worth 9 points. The maximum possible points earned on this section
More informationModule 2: Reflecting on One s Problems
MATH55 Module : Reflecting on One s Problems Main Math concepts: Translations, Reflections, Graphs of Equations, Symmetry Auxiliary ideas: Working with quadratics, Mobius maps, Calculus, Inverses I. Transformations
More informationLAKELAND COMMUNITY COLLEGE COURSE OUTLINE FORM
LAKELAND COMMUNITY COLLEGE COURSE OUTLINE FORM ORIGINATION DATE: 8/2/99 APPROVAL DATE: 3/22/12 LAST MODIFICATION DATE: 3/28/12 EFFECTIVE TERM/YEAR: FALL/ 12 COURSE ID: COURSE TITLE: MATH2500 Calculus and
More informationMath 115 Practice for Exam 2
Math 115 Practice for Exam Generated October 30, 017 Name: SOLUTIONS Instructor: Section Number: 1. This exam has 5 questions. Note that the problems are not of equal difficulty, so you may want to skip
More information2015 Math Camp Calculus Exam Solution
015 Math Camp Calculus Exam Solution Problem 1: x = x x +5 4+5 = 9 = 3 1. lim We also accepted ±3, even though it is not according to the prevailing convention 1. x x 4 x+4 =. lim 4 4+4 = 4 0 = 4 0 = We
More information1 The Derivative and Differrentiability
1 The Derivative and Differrentiability 1.1 Derivatives and rate of change Exercise 1 Find the equation of the tangent line to f (x) = x 2 at the point (1, 1). Exercise 2 Suppose that a ball is dropped
More informationReview Guideline for Final
Review Guideline for Final Here is the outline of the required skills for the final exam. Please read it carefully and find some corresponding homework problems in the corresponding sections to practice.
More informationSuccessful completion of the core function transformations unit. Algebra manipulation skills with squares and square roots.
Extension A: Circles and Ellipses Algebra ; Pre-Calculus Time required: 35 50 min. Learning Objectives Math Objectives Students will write the general forms of Cartesian equations for circles and ellipses,
More informationMA 113 Calculus I Fall 2012 Exam 3 13 November Multiple Choice Answers. Question
MA 113 Calculus I Fall 2012 Exam 3 13 November 2012 Name: Section: Last 4 digits of student ID #: This exam has ten multiple choice questions (five points each) and five free response questions (ten points
More informationMath 41 Second Exam November 4, 2010
Math 41 Second Exam November 4, 2010 Name: SUID#: Circle your section: Olena Bormashenko Ulrik Buchholtz John Jiang Michael Lipnowski Jonathan Lee 03 (11-11:50am) 07 (10-10:50am) 02 (1:15-2:05pm) 04 (1:15-2:05pm)
More informationAP Calculus Worksheet: Chapter 2 Review Part I
AP Calculus Worksheet: Chapter 2 Review Part I 1. Given y = f(x), what is the average rate of change of f on the interval [a, b]? What is the graphical interpretation of your answer? 2. The derivative
More informationMath 51 Second Exam May 18, 2017
Math 51 Second Exam May 18, 2017 Name: SUNet ID: ID #: Complete the following problems. In order to receive full credit, please show all of your work and justify your answers. You do not need to simplify
More informationMath 115 Practice for Exam 3
Math 115 Practice for Exam 3 Generated November 17, 2017 Name: SOLUTIONS Instructor: Section Number: 1. This exam has 6 questions. Note that the problems are not of equal difficulty, so you may want to
More informationSept , 17, 23, 29, 37, 41, 45, 47, , 5, 13, 17, 19, 29, 33. Exam Sept 26. Covers Sept 30-Oct 4.
MATH 23, FALL 2013 Text: Calculus, Early Transcendentals or Multivariable Calculus, 7th edition, Stewart, Brooks/Cole. We will cover chapters 12 through 16, so the multivariable volume will be fine. WebAssign
More informationMATH 120 THIRD UNIT TEST
MATH 0 THIRD UNIT TEST Friday, April 4, 009. NAME: Circle the recitation Tuesday, Thursday Tuesday, Thursday section you attend MORNING AFTERNOON A B Instructions:. Do not separate the pages of the exam.
More informationMcGILL UNIVERSITY FACULTY OF SCIENCE FINAL EXAMINATION MATHEMATICS CALCULUS 1
McGILL UNIVERSITY FACULTY OF SCIENCE FINAL EXAMINATION VERSION 1 MATHEMATICS 140 2008 09 CALCULUS 1 EXAMINER: Professor W. G. Brown DATE: Sunday, December 07th, 2008 ASSOCIATE EXAMINER: Dr. D. Serbin TIME:
More information2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY
2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY The following are topics that you will use in Geometry and should be retained throughout the summer. Please use this practice to review the topics you
More informationNotes about changes to Approved Syllabus # 43080v2
Notes about changes to Approved Syllabus # 43080v2 1. An update to the syllabus was necessary because of a county wide adoption of new textbooks for AP Calculus. 2. No changes were made to the Course Outline
More informationFor those of you who are taking Calculus AB concurrently with AP Physics, I have developed a
AP Physics C: Mechanics Greetings, For those of you who are taking Calculus AB concurrently with AP Physics, I have developed a brief introduction to Calculus that gives you an operational knowledge of
More informationMath 116 Second Midterm November 13, 2017
On my honor, as a student, I have neither given nor received unauthorized aid on this academic work. Initials: Do not write in this area Your Initials Only: Math 6 Second Midterm November 3, 7 Your U-M
More informationThe University of British Columbia Final Examination - December 11, 2013 Mathematics 104/184 Time: 2.5 hours. LAST Name.
The University of British Columbia Final Examination - December 11, 2013 Mathematics 104/184 Time: 2.5 hours LAST Name First Name Signature Student Number MATH 104 or MATH 184 (Circle one) Section Number:
More informationMathematics 131 Final Exam 02 May 2013
Mathematics 3 Final Exam 0 May 03 Directions: This exam should consist of twelve multiple choice questions and four handgraded questions. Multiple choice questions are worth five points apiece. The first
More informationAP Calculus AB - Course Outline
By successfully completing this course, you will be able to: a. Work with functions represented in a variety of ways and understand the connections among these representations. b. Understand the meaning
More informationMath 115 Second Midterm March 25, 2010
Math 115 Second Midterm March 25, 2010 Name: EXAM SOLUTIONS Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 9 pages including this cover. There are 8 problems.
More informationUniversity of Connecticut Department of Mathematics
University of Connecticut Department of Mathematics Math 1131 Sample Exam 2 Fall 2014 Name: Instructor Name: Section: TA Name: Discussion Section: This sample exam is just a guide to prepare for the actual
More informationLecture 20: Further graphing
Lecture 20: Further graphing Nathan Pflueger 25 October 2013 1 Introduction This lecture does not introduce any new material. We revisit the techniques from lecture 12, which give ways to determine the
More informationCalculus with business applications, Lehigh U, Lecture 03 notes Summer
Calculus with business applications, Lehigh U, Lecture 03 notes Summer 01 1 Lines and quadratics 1. Polynomials, functions in which each term is a positive integer power of the independent variable include
More informationMATH 100 and MATH 180 Learning Objectives Session 2010W Term 1 (Sep Dec 2010)
Course Prerequisites MATH 100 and MATH 180 Learning Objectives Session 2010W Term 1 (Sep Dec 2010) As a prerequisite to this course, students are required to have a reasonable mastery of precalculus mathematics
More informationMath 113 Winter 2005 Key
Name Student Number Section Number Instructor Math Winter 005 Key Departmental Final Exam Instructions: The time limit is hours. Problem consists of short answer questions. Problems through are multiple
More informationMAT 146. Semester Exam Part II 100 points (Part II: 50 points) Calculator Used Impact on Course Grade: approximately 30% Score
MAT 146 Semester Exam Part II Name 100 points (Part II: 50 points) Calculator Used Impact on Course Grade: approximately 30% Score Questions (17) through (26) are each worth 5 points. See the grading rubric
More informationMA 125 CALCULUS I FALL 2006 December 08, 2006 FINAL EXAM. Name (Print last name first):... Instructor:... Section:... PART I
CALCULUS I, FINAL EXAM 1 MA 125 CALCULUS I FALL 2006 December 08, 2006 FINAL EXAM Name (Print last name first):............................................. Student ID Number:...........................
More informationMath 115 Second Midterm November 12, 2018
EXAM SOLUTIONS Math 5 Second Midterm November, 08. Do not open this eam until you are told to do so.. Do not write your name anywhere on this eam. 3. This eam has 3 pages including this cover. There are
More informationMath 1310 Final Exam
Math 1310 Final Exam December 11, 2014 NAME: INSTRUCTOR: Write neatly and show all your work in the space provided below each question. You may use the back of the exam pages if you need additional space
More informationAP CALCULUS AB Study Guide for Midterm Exam 2017
AP CALCULUS AB Study Guide for Midterm Exam 2017 CHAPTER 1: PRECALCULUS REVIEW 1.1 Real Numbers, Functions and Graphs - Write absolute value as a piece-wise function - Write and interpret open and closed
More informationSpring /06/2009
MA 123 Elementary Calculus FINAL EXAM Spring 2009 05/06/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or
More informationMath 106 Answers to Exam 3a Fall 2015
Math 6 Answers to Exam 3a Fall 5.. Consider the curve given parametrically by x(t) = cos(t), y(t) = (t 3 ) 3, for t from π to π. (a) (6 points) Find all the points (x, y) where the graph has either a vertical
More information1.4 Techniques of Integration
.4 Techniques of Integration Recall the following strategy for evaluating definite integrals, which arose from the Fundamental Theorem of Calculus (see Section.3). To calculate b a f(x) dx. Find a function
More informationCHAPTER 1. DIFFERENTIATION 18. As x 1, f(x). At last! We are now in a position to sketch the curve; see Figure 1.4.
CHAPTER. DIFFERENTIATION 8 and similarly for x, As x +, fx), As x, fx). At last! We are now in a position to sketch the curve; see Figure.4. Figure.4: A sketch of the function y = fx) =/x ). Observe the
More informationSince x + we get x² + 2x = 4, or simplifying it, x² = 4. Therefore, x² + = 4 2 = 2. Ans. (C)
SAT II - Math Level 2 Test #01 Solution 1. x + = 2, then x² + = Since x + = 2, by squaring both side of the equation, (A) - (B) 0 (C) 2 (D) 4 (E) -2 we get x² + 2x 1 + 1 = 4, or simplifying it, x² + 2
More informationMTH 173 Calculus with Analytic Geometry I and MTH 174 Calculus with Analytic Geometry II
MTH 173 Calculus with Analytic Geometry I and MTH 174 Calculus with Analytic Geometry II Instructor: David H. Pleacher Home Phone: 869-4883 School Phone: 662-3471 Room: 212 E-Mail Address: Pleacher.David@wps.k12.va.us
More informationMath 1131 Final Exam Review Spring 2013
University of Connecticut Department of Mathematics Math 1131 Final Exam Review Spring 2013 Name: Instructor Name: TA Name: 4 th February 2010 Section: Discussion Section: Read This First! Please read
More informationAP Calculus AB Course Description and Syllabus
AP Calculus AB Course Description and Syllabus Course Objective: This course is designed to prepare the students for the AP Exam in May. Students will learn to use graphical, numerical, verbal and analytical
More information1S11: Calculus for students in Science
1S11: Calculus for students in Science Dr. Vladimir Dotsenko TCD Lecture 21 Dr. Vladimir Dotsenko (TCD) 1S11: Calculus for students in Science Lecture 21 1 / 1 An important announcement There will be no
More informationMath 241 Final Exam, Spring 2013
Math 241 Final Exam, Spring 2013 Name: Section number: Instructor: Read all of the following information before starting the exam. Question Points Score 1 5 2 5 3 12 4 10 5 17 6 15 7 6 8 12 9 12 10 14
More informationParabolas and lines
Parabolas and lines Study the diagram at the right. I have drawn the graph y = x. The vertical line x = 1 is drawn and a number of secants to the parabola are drawn, all centred at x=1. By this I mean
More informationChemistry 883 Computational Quantum Chemistry
Chemistry 883 Computational Quantum Chemistry Instructor Contact Information Professor Benjamin G. Levine levine@chemistry.msu.edu 215 Chemistry Building 517-353-1113 Office Hours Tuesday 9:00-11:00 am
More informationMATH 1190 Exam 4 (Version 2) Solutions December 1, 2006 S. F. Ellermeyer Name
MATH 90 Exam 4 (Version ) Solutions December, 006 S. F. Ellermeyer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of presentation.
More informationMath 115 Second Midterm November 12, 2013
Math 5 Second Midterm November 2, 203 Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so. 2. This exam has 9 pages including this cover. There are 0 problems.
More informationFinal Exam Review Exercise Set A, Math 1551, Fall 2017
Final Exam Review Exercise Set A, Math 1551, Fall 2017 This review set gives a list of topics that we explored throughout this course, as well as a few practice problems at the end of the document. A complete
More informationReplacing the a in the definition of the derivative of the function f at a with a variable x, gives the derivative function f (x).
Definition of The Derivative Function Definition (The Derivative Function) Replacing the a in the definition of the derivative of the function f at a with a variable x, gives the derivative function f
More informationPage Points Score Total: 100
Math 1130 Spring 2019 Sample Exam 1c 1/31/19 Name (Print): Username.#: Lecturer: Rec. Instructor: Rec. Time: This exam contains 8 pages (including this cover page) and 7 problems. Check to see if any pages
More informationMAT 122 Homework 7 Solutions
MAT 1 Homework 7 Solutions Section 3.3, Problem 4 For the function w = (t + 1) 100, we take the inside function to be z = t + 1 and the outside function to be z 100. The derivative of the inside function
More information4.3 How derivatives affect the shape of a graph. The first derivative test and the second derivative test.
Chapter 4: Applications of Differentiation In this chapter we will cover: 41 Maximum and minimum values The critical points method for finding extrema 43 How derivatives affect the shape of a graph The
More informationWithout fully opening the exam, check that you have pages 1 through 11.
MTH 33 Solutions to Final Exam May, 8 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show
More information