Math 226 Calculus Spring 2016 Practice Exam 1. (1) (10 Points) Let the differentiable function y = f(x) have inverse function x = f 1 (y).
|
|
- Barry Walton
- 5 years ago
- Views:
Transcription
1 Math 6 Calculus Spring 016 Practice Exam 1 1) 10 Points) Let the differentiable function y = fx) have inverse function x = f 1 y). a) Write down the formula relating the derivatives f x) and f 1 ) y). No work needs to be shown.) Explain why fx) = x 5 + x has an inverse function. c) For y = fx) = x 5 + x, use your answer to part a) to find f 1 ) 3). Note that f1) = 3. ) 10 Points) Let y = x x for x > 0. a) Find dy. Find the equation of the tangent line to y = x x at the point x = 1.
2 Math 6 Calculus Spring 016 Practice Exam 1 3) 5 Points) Use the fact that x = e to find 1 + x x) 1 ) n. n 3n 4) 15 Points) Find each of these derivatives. a) d sin 1 x 3 ) d 1 3tan x) c) d log x 4 + x + 1)
3 Math 6 Calculus Spring 016 Practice Exam 1 5) 15 Points) Find each of these integrals. a) 1 + 9x e e xlnx)) 4 c) π/4 0 sinx) e cosx)
4 Math 6 Calculus Spring 016 Practice Exam 1 6) 0 Points) Evaluate the following its. If you use L Hospital s Rule, show where you use it and explain what type of it you are using it on. a) sinx 3 ) sin3x 3 ) x lnx) + c) 1 + x + x ) 1/x.
5 Math 6 Calculus Spring 016 Practice Exam 1 7) 10 Points) In the year 1980 your parents invested $10, 000 in a special bank account which earned interest compounded continuously. In the year 000 that account was worth $160, 000. Assuming no withdrawals, and the same interest rate, what will that account be worth in the year 00? 8) 15 Points) Evaluate the following integrals. a) 5 Points) x cosx) 10 Points) e x cosx)
6 Math 6 Calculus Spring 016 Practice Exam 1 Solutions 1) 10 Points) a) points) Write down the formula relating the derivatives f x) and f 1 ) y). No work needs to be shown.) The relationship between them is f 1 ) y) = 1 f x). 4 points) Explain why fx) = x 5 + x has an inverse function. Since fx) = x 5 + x, f x) = 5x 4 + > 0 since 5x 4 0, so fx) is an increasing function, which is then one-to-one, so it has an inverse. c) 4 points) For y = fx) = x 5 + x, use your answer to part a) to find f 1 ) 3). Note that f1) = 3. From part a) we know that f 1 ) 3) = 1 f 1) = ) + = 1 7. ) 10 Points) Let y = x x for x > 0. a) dy = d ex lnx) = e x lnx) d x lnx)) = xx lnx) + x 1 x ) = xx lnx) + 1). The equation of the tangent line to y = x x at the point x = 1 is y y 0 = sx x 0 ) where x 0 = 1 and y 0 = x x 0 0 = 11 = 1 and the slope of the tangent line s = 1 1 ln1) + 1) = 1 is the value of the derivative at x = 1 from part a). So the equation we want is y 1 = 1x 1), that is, y 1 = x 1 which simplifies to y = x. 3) 5 Points) Use the fact that x = e to find 1 + x x) 1 ) n. n 3n Let x = 3n so x/3 = n. Since x as n, the it equals ) n = x/3 = n 3n x x) x x) x ) 1/3 = e 1/3.
7 Math 6 Calculus Spring 016 Practice Exam 1 Solutions 4) 15 Points) Find each of these derivatives. a) d sin 1 x 3 ) = 3x 1 x3 ) = 3x 1 x 6. d 1 3tan x) = 3 tan 1 x) x ln3) c) d log x 4 + x + 1) = 4x 3 + x x 4 + x + 1) ln) 5) 15 Points) Find each of these integrals. a) 1 + 9x Using the substitution u = 3x so du = 3, we get du/3 1 + u = 1 3 tan 1 u) + C = 1 3 tan 1 3x) + C. e e xlnx)) 4 Using substitution u = lnx) we get du = x u = lne) = 1 to u = lne ) =, so e e and the bounds of the integral are from ] xlnx)) 4 = u 4 du = u 3 = ) 1 3 = 7 4. c) π/4 0 sinx) e cosx) Using substitution u = cosx) we get du = sinx) and the bounds of the integral are from u = cos0) = 1 to u = cosπ/4) = 1/, so 1/ ] u=1/ e u du = e u = e 1/ e 1 ) = e e 1/. 1 u=1 π/4 0 sinx) e cosx) =
8 Math 6 Calculus Spring 016 Practice Exam 1 Solutions 6) 0 Points) Evaluate the following its. If you use L Hospital s Rule, show where you use it and explain what type of it you are using it on. a) sinx 3 ) sin3x 3 ) 7 points) By L Hospital s Rule for a 0 0-type indeterminate form, sinx 3 ) sin3x 3 ) L H = 6x cosx 3 ) 9x cos3x 3 ) = 6)1) 9)1) = 3. x lnx) + 6 points) x lnx) is an indeterminate form of the type 0) ), which we + make into a lnx) L H 1/x x 3/ -type by writing it as + x 1/ = + 1 = = x 3/ + x x1/ = 0 + c) 1 + x + x ) 1/x. 7 points) Let y = 1 + x + x ) 1/x ln1 + x + x ) so lny) =. Then x ln1 + x + x ) lny) = x is a type 0 0 indeterminate form. L Hospital s Rule gives ln1 + x + x ) x L H = 1+4x 1+x+x 1 = 1 so y e 1 = e is the it we seek.
9 Math 6 Calculus Spring 016 Practice Exam 1 Solutions 7) 10 Points) In the year 1980 your parents invested $10, 000 in a special bank account which earned interest compounded continuously. In the year 000 that account was worth $160, 000. Assuming no withdrawals, and the same interest rate, what will that account be worth in the year 00? For continuous compounding, the value of the account is P t) = 10000e It where I is the annual interest rate. Then we know = 10000e 0I so 16 = e 0I gives I = ln16) 0. Then in the year 00, when t = 40, the value of the account would be P 40) = 10000e 40I = 10000e ln16) = ) = ) =, 560, 000 dollars. 8) 15 Points) Evaluate the following integrals. a) 5 Points) x cosx) Using integration by parts with u = x and dv = cosx), we have du = and v = sinx), so x cosx) = x sinx) sinx) = x sinx) + cosx) + C 10 Points) e x cosx) Use integration by parts twice, the first time with u = e x and dv = cosx), so that du = e x and v = sinx), giving e x cosx) = e x sinx) e x sinx) For the second integration by parts on the remaining integral, use u = e x and dv = sinx), so that du = e x and v = cosx), giving [ ] e x cosx) = e x sinx) e x cosx) cosx)e x ) = e x sinx) + e x cosx) 4 e x cosx) Bringing the last term to the other side of the equation, and then dividing by 5 gives the answer e x cosx) = 1 [ ] e x sinx) + e x cosx) + C 5
Math 106 Fall 2014 Exam 2.1 October 31, ln(x) x 3 dx = 1. 2 x 2 ln(x) + = 1 2 x 2 ln(x) + 1. = 1 2 x 2 ln(x) 1 4 x 2 + C
Math 6 Fall 4 Exam. October 3, 4. The following questions have to do with the integral (a) Evaluate dx. Use integration by parts (x 3 dx = ) ( dx = ) x3 x dx = x x () dx = x + x x dx = x + x 3 dx dx =
More informationMath 180, Final Exam, Fall 2012 Problem 1 Solution
Math 80, Final Exam, Fall 0 Problem Solution. Find the derivatives of the following functions: (a) ln(ln(x)) (b) x 6 + sin(x) e x (c) tan(x ) + cot(x ) (a) We evaluate the derivative using the Chain Rule.
More informationSolutions to Exam 1, Math Solution. Because f(x) is one-to-one, we know the inverse function exists. Recall that (f 1 ) (a) =
Solutions to Exam, Math 56 The function f(x) e x + x 3 + x is one-to-one (there is no need to check this) What is (f ) ( + e )? Solution Because f(x) is one-to-one, we know the inverse function exists
More informationMath 131 Final Exam Spring 2016
Math 3 Final Exam Spring 06 Name: ID: multiple choice questions worth 5 points each. Exam is only out of 00 (so there is the possibility of getting more than 00%) Exam covers sections. through 5.4 No graphing
More informationA.P. Calculus BC Test Two Section One Multiple-Choice Calculators Allowed Time 40 minutes Number of Questions 15
A.P. Calculus BC Test Two Section One Multiple-Choice Calculators Allowed Time 40 minutes Number of Questions 15 The scoring for this section is determined by the formula [C (0.25 I)] 1.8 where C is the
More informationMA1021 Calculus I B Term, Sign:
MA1021 Calculus I B Term, 2014 Final Exam Print Name: Sign: Write up your solutions neatly and show all your work. 1. (28 pts) Compute each of the following derivatives: You do not have to simplify your
More informationUniversity Calculus I. Worksheet # 8 Mar b. sin tan e. sin 2 sin 1 5. b. tan. c. sec sin 1 ( x )) cos 1 ( x )) f. csc. c.
MATH 6 WINTER 06 University Calculus I Worksheet # 8 Mar. 06-0 The topic covered by this worksheet is: Derivative of Inverse Functions and the Inverse Trigonometric functions. SamplesolutionstoallproblemswillbeavailableonDL,
More informationMATH 1207 R02 MIDTERM EXAM 2 SOLUTION
MATH 7 R MIDTERM EXAM SOLUTION FALL 6 - MOON Name: Write your answer neatly and show steps. Except calculators, any electronic devices including laptops and cell phones are not allowed. () (5 pts) Find
More informationM GENERAL MATHEMATICS -2- Dr. Tariq A. AlFadhel 1 Solution of the First Mid-Term Exam First semester H
M - GENERAL MATHEMATICS -- Dr. Tariq A. AlFadhel Solution of the First Mid-Term Exam First semester 38-39 H 3 Q. Let A =, B = and C = 3 Compute (if possible) : A+B and BC A+B is impossible. ( ) BC = 3
More information3. (12 points) Find an equation for the line tangent to the graph of f(x) =
April 8, 2015 Name The total number of points available is 168 Throughout this test, show your work Throughout this test, you are expected to use calculus to solve problems Graphing calculator solutions
More informationSection 3.6 The chain rule 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 3.6 The chain rule 1 Lecture College of Science MATHS 101: Calculus I (University of Bahrain) Logarithmic Differentiation 1 / 1 Motivation Goal: We want to derive rules to find the derivative of
More informationMath 180, Exam 2, Practice Fall 2009 Problem 1 Solution. f(x) = arcsin(2x + 1) = sin 1 (3x + 1), lnx
Math 80, Exam, Practice Fall 009 Problem Solution. Differentiate the functions: (do not simplify) f(x) = x ln(x + ), f(x) = xe x f(x) = arcsin(x + ) = sin (3x + ), f(x) = e3x lnx Solution: For the first
More informationMath 152 Take Home Test 1
Math 5 Take Home Test Due Monday 5 th October (5 points) The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I
More informationPuzzle 1 Puzzle 2 Puzzle 3 Puzzle 4 Puzzle 5 /10 /10 /10 /10 /10
MATH-65 Puzzle Collection 6 Nov 8 :pm-:pm Name:... 3 :pm Wumaier :pm Njus 5 :pm Wumaier 6 :pm Njus 7 :pm Wumaier 8 :pm Njus This puzzle collection is closed book and closed notes. NO calculators are allowed
More informationSpring 2015 Sample Final Exam
Math 1151 Spring 2015 Sample Final Exam Final Exam on 4/30/14 Name (Print): Time Limit on Final: 105 Minutes Go on carmen.osu.edu to see where your final exam will be. NOTE: This exam is much longer than
More informationSET 1. (1) Solve for x: (a) e 2x = 5 3x
() Solve for x: (a) e x = 5 3x SET We take natural log on both sides: ln(e x ) = ln(5 3x ) x = 3 x ln(5) Now we take log base on both sides: log ( x ) = log (3 x ln 5) x = log (3 x ) + log (ln(5)) x x
More informationTHE USE OF A CALCULATOR, CELL PHONE, OR ANY OTHER ELECTRONIC DEVICE IS NOT PERMITTED IN THIS EXAMINATION.
MATH 110 FINAL EXAM SPRING 2008 FORM A NAME STUDENT NUMBER INSTRUCTOR SECTION NUMBER This examination will be machine processed by the University Testing Service. Use only a number 2 pencil on your scantron.
More information2. (12 points) Find an equation for the line tangent to the graph of f(x) =
November 23, 2010 Name The total number of points available is 153 Throughout this test, show your work Throughout this test, you are expected to use calculus to solve problems Graphing calculator solutions
More informationMath 222 Spring 2013 Exam 3 Review Problem Answers
. (a) By the Chain ule, Math Spring 3 Exam 3 eview Problem Answers w s w x x s + w y y s (y xy)() + (xy x )( ) (( s + 4t) (s 3t)( s + 4t)) ((s 3t)( s + 4t) (s 3t) ) 8s 94st + 3t (b) By the Chain ule, w
More information(e) 2 (f) 2. (c) + (d). Limits at Infinity. 2.5) 9-14,25-34,41-43,46-47,56-57, (c) (d) 2
Math 150A. Final Review Answers, Spring 2018. Limits. 2.2) 7-10, 21-24, 28-1, 6-8, 4-44. 1. Find the values, or state they do not exist. (a) (b) 1 (c) DNE (d) 1 (e) 2 (f) 2 (g) 2 (h) 4 2. lim f(x) = 2,
More informationIntegration by Parts
Calculus 2 Lia Vas Integration by Parts Using integration by parts one transforms an integral of a product of two functions into a simpler integral. Divide the initial function into two parts called u
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Spring 2019, WEEK 10 JoungDong Kim Week 10 Section 4.2, 4.3, 4.4 Mean Value Theorem, How Derivatives Affect the Shape of a Graph, Indeterminate Forms and L Hospital s
More informationReview for the Final Exam
Math 171 Review for the Final Exam 1 Find the limits (4 points each) (a) lim 4x 2 3; x x (b) lim ( x 2 x x 1 )x ; (c) lim( 1 1 ); x 1 ln x x 1 sin (x 2) (d) lim x 2 x 2 4 Solutions (a) The limit lim 4x
More informationSolutions to Exam 2, Math 10560
Solutions to Exam, Math 6. Which of the following expressions gives the partial fraction decomposition of the function x + x + f(x = (x (x (x +? Solution: Notice that (x is not an irreducile factor. If
More information7.1. Calculus of inverse functions. Text Section 7.1 Exercise:
Contents 7. Inverse functions 1 7.1. Calculus of inverse functions 2 7.2. Derivatives of exponential function 4 7.3. Logarithmic function 6 7.4. Derivatives of logarithmic functions 7 7.5. Exponential
More informationIf y = f (u) is a differentiable function of u and u = g(x) is a differentiable function of x then dy dx = dy. du du. If y = f (u) then y = f (u) u
Section 3 4B The Chain Rule If y = f (u) is a differentiable function of u and u = g(x) is a differentiable function of x then dy dx = dy du du dx or If y = f (u) then f (u) u The Chain Rule with the Power
More informationMath 226 Calculus Spring 2016 Exam 2V1
Math 6 Calculus Spring 6 Exam V () (35 Points) Evaluate the following integrals. (a) (7 Points) tan 5 (x) sec 3 (x) dx (b) (8 Points) cos 4 (x) dx Math 6 Calculus Spring 6 Exam V () (Continued) Evaluate
More informationIf y = f (u) is a differentiable function of u and u = g(x) is a differentiable function of x then dy dx = dy. du du. If y = f (u) then y = f (u) u
Section 3 4B Lecture The Chain Rule If y = f (u) is a differentiable function of u and u = g(x) is a differentiable function of x then dy dx = dy du du dx or If y = f (u) then y = f (u) u The Chain Rule
More informationMth Review Problems for Test 2 Stewart 8e Chapter 3. For Test #2 study these problems, the examples in your notes, and the homework.
For Test # study these problems, the examples in your notes, and the homework. Derivative Rules D [u n ] = nu n 1 du D [ln u] = du u D [log b u] = du u ln b D [e u ] = e u du D [a u ] = a u ln a du D [sin
More informationIF you participate fully in this boot camp, you will get full credit for the summer packet.
18_19 AP Calculus BC Summer Packet NOTE - Please mark July on your calendars. We will have a boot camp in my room from 8am 11am on this day. We will work together on the summer packet. Time permitting,
More informationLecture 5 - Logarithms, Slope of a Function, Derivatives
Lecture 5 - Logarithms, Slope of a Function, Derivatives 5. Logarithms Note the graph of e x This graph passes the horizontal line test, so f(x) = e x is one-to-one and therefore has an inverse function.
More informationMath 131 Exam 2 Spring 2016
Math 3 Exam Spring 06 Name: ID: 7 multiple choice questions worth 4.7 points each. hand graded questions worth 0 points each. 0. free points (so the total will be 00). Exam covers sections.7 through 3.0
More informationMATH 135 Calculus 1 Solutions/Answers for Exam 3 Practice Problems November 18, 2016
MATH 35 Calculus Solutions/Answers for Exam 3 Practice Problems November 8, 206 I. Find the indicated derivative(s) and simplify. (A) ( y = ln(x) x 7 4 ) x Solution: By the product rule and the derivative
More information2t t dt.. So the distance is (t2 +6) 3/2
Math 8, Solutions to Review for the Final Exam Question : The distance is 5 t t + dt To work that out, integrate by parts with u t +, so that t dt du The integral is t t + dt u du u 3/ (t +) 3/ So the
More informationFall 2009 Math 113 Final Exam Solutions. f(x) = 1 + ex 1 e x?
. What are the domain and range of the function Fall 9 Math 3 Final Exam Solutions f(x) = + ex e x? Answer: The function is well-defined everywhere except when the denominator is zero, which happens when
More informationWeBWorK, Problems 2 and 3
WeBWorK, Problems 2 and 3 7 dx 2. Evaluate x ln(6x) This can be done using integration by parts or substitution. (Most can not). However, it is much more easily done using substitution. This can be written
More informationSection 3.6 The chain rule 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 3.6 The chain rule 1 Lecture College of Science MATHS 101: Calculus I (University of Bahrain) Logarithmic Differentiation 1 / 23 Motivation Goal: We want to derive rules to find the derivative
More informationMath 250 Skills Assessment Test
Math 5 Skills Assessment Test Page Math 5 Skills Assessment Test The purpose of this test is purely diagnostic (before beginning your review, it will be helpful to assess both strengths and weaknesses).
More informationFinal Exam Review Quesitons
Final Exam Review Quesitons. Compute the following integrals. (a) x x 4 (x ) (x + 4) dx. The appropriate partial fraction form is which simplifies to x x 4 (x ) (x + 4) = A x + B (x ) + C x + 4 + Dx x
More informationMATH 162. Midterm Exam 1 - Solutions February 22, 2007
MATH 62 Midterm Exam - Solutions February 22, 27. (8 points) Evaluate the following integrals: (a) x sin(x 4 + 7) dx Solution: Let u = x 4 + 7, then du = 4x dx and x sin(x 4 + 7) dx = 4 sin(u) du = 4 [
More informationa x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x).
You should prepare the following topics for our final exam. () Pre-calculus. (2) Inverses. (3) Algebra of Limits. (4) Derivative Formulas and Rules. (5) Graphing Techniques. (6) Optimization (Maxima and
More informationMATH 151, FALL SEMESTER 2011 COMMON EXAMINATION 3 - VERSION B - SOLUTIONS
Name (print): Signature: MATH 5, FALL SEMESTER 0 COMMON EXAMINATION - VERSION B - SOLUTIONS Instructor s name: Section No: Part Multiple Choice ( questions, points each, No Calculators) Write your name,
More informationMath 147 Exam II Practice Problems
Math 147 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture, all homework problems, all lab
More informationMath 106: Review for Exam II - SOLUTIONS
Math 6: Review for Exam II - SOLUTIONS INTEGRATION TIPS Substitution: usually let u a function that s inside another function, especially if du (possibly off by a multiplying constant) is also present
More informationSection 5.5 More Integration Formula (The Substitution Method) 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 5.5 More Integration Formula (The Substitution Method) 2 Lectures College of Science MATHS : Calculus I (University of Bahrain) Integrals / 7 The Substitution Method Idea: To replace a relatively
More informationMATH 32 FALL 2013 FINAL EXAM SOLUTIONS. 1 cos( 2. is in the first quadrant, so its sine is positive. Finally, csc( π 8 ) = 2 2.
MATH FALL 01 FINAL EXAM SOLUTIONS (1) (1 points) Evalute the following (a) tan(0) Solution: tan(0) = 0. (b) csc( π 8 ) Solution: csc( π 8 ) = 1 sin( π 8 ) To find sin( π 8 ), we ll use the half angle formula:
More informationSolutions to Second Midterm(pineapple)
Math 125 Solutions to Second Midterm(pineapple) 1. Compute each of the derivatives below as indicated. 4 points (a) f(x) = 3x 8 5x 4 + 4x e 3. Solution: f (x) = 24x 7 20x + 4. Don t forget that e 3 is
More informationCore Mathematics 3 Differentiation
http://kumarmaths.weebly.com/ Core Mathematics Differentiation C differentiation Page Differentiation C Specifications. By the end of this unit you should be able to : Use chain rule to find the derivative
More informationSection 5.6. Integration By Parts. MATH 126 (Section 5.6) Integration By Parts The University of Kansas 1 / 10
Section 5.6 Integration By Parts MATH 126 (Section 5.6) Integration By Parts The University of Kansas 1 / 10 Integration By Parts Manipulating the Product Rule d dx (f (x) g(x)) = f (x) g (x) + f (x) g(x)
More informationWeek #6 - Taylor Series, Derivatives and Graphs Section 10.1
Week #6 - Taylor Series, Derivatives and Graphs Section 10.1 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 005 by John Wiley & Sons, Inc. This material is used by
More informationHonors Calculus II [ ] Midterm II
Honors Calculus II [3-00] Midterm II PRINT NAME: SOLUTIONS Q]...[0 points] Evaluate the following expressions and its. Since you don t have a calculator, square roots, fractions etc. are allowed in your
More informationApril 9, 2009 Name The problems count as marked. The total number of points available is 160. Throughout this test, show your work.
April 9, 009 Name The problems count as marked The total number of points available is 160 Throughout this test, show your work 1 (15 points) Consider the cubic curve f(x) = x 3 + 3x 36x + 17 (a) Build
More informationMath 113 (Calculus 2) Exam 4
Math 3 (Calculus ) Exam 4 November 0 November, 009 Sections 0, 3 7 Name Student ID Section Instructor In some cases a series may be seen to converge or diverge for more than one reason. For such problems
More informationCalculus I: Practice Midterm II
Calculus I: Practice Mierm II April 3, 2015 Name: Write your solutions in the space provided. Continue on the back for more space. Show your work unless asked otherwise. Partial credit will be given for
More informationEXAM 3 MAT 167 Calculus I Spring is a composite function of two functions y = e u and u = 4 x + x 2. By the. dy dx = dy du = e u x + 2x.
EXAM MAT 67 Calculus I Spring 20 Name: Section: I Each answer must include either supporting work or an explanation of your reasoning. These elements are considered to be the main part of each answer and
More informationMath 106: Review for Final Exam, Part II - SOLUTIONS. (x x 0 ) 2 = !
Math 06: Review for Final Exam, Part II - SOLUTIONS. Use a second-degree Taylor polynomial to estimate 8. We choose f(x) x and x 0 7 because 7 is the perfect cube closest to 8. f(x) x /3 f(7) 3 f (x) 3
More informationSection 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) 1 / 28 Indefinite Integral Given a function f, if F is a function such that
More informationFinal Exam. Math 3 December 7, 2010
Final Exam Math 3 December 7, 200 Name: On this final examination for Math 3 in Fall 200, I will work individually, neither giving nor receiving help, guided by the Dartmouth Academic Honor Principle.
More informationCollege of the Holy Cross MATH 133, Calculus With Fundamentals 1 Solutions for Final Examination Friday, December 15
College of the Holy Cross MATH 33, Calculus With Fundamentals Solutions for Final Examination Friday, December 5 I. The graph y = f(x) is given in light blue. Match each equation with one of the numbered
More informationMath 116 Second Midterm March 20, 2013
Math 6 Second Mierm March, 3 Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has 3 pages including this cover. There are 8 problems. Note that the
More informationChapter 7: Techniques of Integration
Chapter 7: Techniques of Integration MATH 206-01: Calculus II Department of Mathematics University of Louisville last corrected September 14, 2013 1 / 43 Chapter 7: Techniques of Integration 7.1. Integration
More informationExam 3 Solutions. Multiple Choice Questions
MA 4 Exam 3 Solutions Fall 26 Exam 3 Solutions Multiple Choice Questions. The average value of the function f (x) = x + sin(x) on the interval [, 2π] is: A. 2π 2 2π B. π 2π 2 + 2π 4π 2 2π 4π 2 + 2π 2.
More informationMath 110 Final Exam General Review. Edward Yu
Math 110 Final Exam General Review Edward Yu Da Game Plan Solving Limits Regular limits Indeterminate Form Approach Infinities One sided limits/discontinuity Derivatives Power Rule Product/Quotient Rule
More informationMATH 408N PRACTICE FINAL
2/03/20 Bormashenko MATH 408N PRACTICE FINAL Show your work for all the problems. Good luck! () Let f(x) = ex e x. (a) [5 pts] State the domain and range of f(x). Name: TA session: Since e x is defined
More informationMultiple Choice Answers. MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March Question
MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March 2018 Name: Section: Last 4 digits of student ID #: This exam has 12 multiple choice questions (five points each) and 4 free response questions (ten
More informationVirginia Tech Math 1226 : Past CTE problems
Virginia Tech Math 16 : Past CTE problems 1. It requires 1 in-pounds of work to stretch a spring from its natural length of 1 in to a length of 1 in. How much additional work (in inch-pounds) is done in
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 information7.1 Day 1: Differential Equations & Initial Value Problems (30L)
A P 7.1 Day 1: Differential Equations & Initial Value Problems (30L) Calculus 30 & 30L I CAN SOLVE DIFFERENTIAL EQUATIONS AND INITIAL VALUE PROBLEMS VIDEO LINKS: a) http://bit.ly/2bxsc6r b) http://bit.ly/2sshyyh
More informationSolutions to Math 41 Final Exam December 10, 2012
Solutions to Math 4 Final Exam December,. ( points) Find each of the following limits, with justification. If there is an infinite limit, then explain whether it is or. x ln(t + ) dt (a) lim x x (5 points)
More informationFinal Exam Solutions
Final Exam Solutions Laurence Field Math, Section March, Name: Solutions Instructions: This exam has 8 questions for a total of points. The value of each part of each question is stated. The time allowed
More informationMATH 104 Practice Problems for Exam 2
. Find the area between: MATH 4 Practice Problems for Exam (a) x =, y = / + x, y = x/ Answer: ln( + ) 4 (b) y = e x, y = xe x, x = Answer: e6 4 7 4 (c) y = x and the x axis, for x 4. x Answer: ln 5. Calculate
More information2. Find the midpoint of the segment that joins the points (5, 1) and (3, 5). 6. Find an equation of the line with slope 7 that passes through (4, 1).
Math 129: Pre-Calculus Spring 2018 Practice Problems for Final Exam Name (Print): 1. Find the distance between the points (6, 2) and ( 4, 5). 2. Find the midpoint of the segment that joins the points (5,
More informationPartial Derivatives for Math 229. Our puropose here is to explain how one computes partial derivatives. We will not attempt
Partial Derivatives for Math 229 Our puropose here is to explain how one computes partial derivatives. We will not attempt to explain how they arise or why one would use them; that is left to other courses
More informationMath RE - Calculus II Antiderivatives and the Indefinite Integral Page 1 of 5
Math 201-203-RE - Calculus II Antiderivatives and the Indefinite Integral Page 1 of 5 What is the Antiderivative? In a derivative problem, a function f(x) is given and you find the derivative f (x) using
More informationMath 180, Lowman, Summer 2008, Old Exam Problems 1 Limit Problems
Math 180, Lowman, Summer 2008, Old Exam Problems 1 Limit Problems 1. Find the limit of f(x) = (sin x) x x 3 as x 0. 2. Use L Hopital s Rule to calculate lim x 2 x 3 2x 2 x+2 x 2 4. 3. Given the function
More informationMATH 32 FALL 2012 FINAL EXAM - PRACTICE EXAM SOLUTIONS
MATH 2 FALL 2012 FINAL EXAM - PRACTICE EXAM SOLUTIONS (1) ( points) Solve the equation x 1 =. Solution: Since x 1 =, x 1 = or x 1 =. Solving for x, x = 4 or x = 2. (2) In the triangle below, let a = 4,
More informationUNIT 3: DERIVATIVES STUDY GUIDE
Calculus I UNIT 3: Derivatives REVIEW Name: Date: UNIT 3: DERIVATIVES STUDY GUIDE Section 1: Section 2: Limit Definition (Derivative as the Slope of the Tangent Line) Calculating Rates of Change (Average
More informationMath 1552: Integral Calculus Final Exam Study Guide, Spring 2018
Math 55: Integral Calculus Final Exam Study Guide, Spring 08 PART : Concept Review (Note: concepts may be tested on the exam in the form of true/false or short-answer questions.). Complete each statement
More informationNO CALCULATORS. NO BOOKS. NO NOTES. TURN OFF YOUR CELL PHONES AND PUT THEM AWAY.
FINAL EXAM-MATH 3 FALL TERM, R. Blute & A. Novruzi Name(Print LEGIBLY) I.D. Number Instructions- This final examination consists of multiple choice questions worth 3 points each. Your answers to the multiple
More informationAim: How do we prepare for AP Problems on limits, continuity and differentiability? f (x)
Name AP Calculus Date Supplemental Review 1 Aim: How do we prepare for AP Problems on limits, continuity and differentiability? Do Now: Use the graph of f(x) to evaluate each of the following: 1. lim x
More informationy = x 3 and y = 2x 2 x. 2x 2 x = x 3 x 3 2x 2 + x = 0 x(x 2 2x + 1) = 0 x(x 1) 2 = 0 x = 0 and x = (x 3 (2x 2 x)) dx
Millersville University Name Answer Key Mathematics Department MATH 2, Calculus II, Final Examination May 4, 2, 8:AM-:AM Please answer the following questions. Your answers will be evaluated on their correctness,
More informationMath 180, Final Exam, Fall 2007 Problem 1 Solution
Problem Solution. Differentiate with respect to x. Write your answers showing the use of the appropriate techniques. Do not simplify. (a) x 27 x 2/3 (b) (x 2 2x + 2)e x (c) ln(x 2 + 4) (a) Use the Power
More informationIntegrated Calculus II Exam 1 Solutions 2/6/4
Integrated Calculus II Exam Solutions /6/ Question Determine the following integrals: te t dt. We integrate by parts: u = t, du = dt, dv = e t dt, v = dv = e t dt = e t, te t dt = udv = uv vdu = te t (
More informationSolutions to Problem Set 2
Solutions to Problem Set Part I/Part II Part I(0 points) (a) ( points) p. 97, Section 3.3, Problem 44 (b) ( points) p. 107, Section 3.5, Problem 15 (c) ( points) p. 110, Section 3.6, Problem 3(c) (d) (
More informationMultiple Choice Answers. MA 114 Calculus II Spring 2013 Final Exam 1 May Question
MA 114 Calculus II Spring 2013 Final Exam 1 May 2013 Name: Section: Last 4 digits of student ID #: This exam has six multiple choice questions (six points each) and five free response questions with points
More informationMath 106: Review for Exam II - SOLUTIONS
Math 6: Review for Exam II - SOLUTIONS INTEGRATION TIPS Substitution: usually let u a function that s inside another function, especially if du (possibly off by a multiplying constant) is also present
More information***** Sorry - Solutions will not be posted *****
***** Sorry - Solutions will not be posted ***** FINAL EXAMINATION MATA32 - Calculus for Management I Examiners: R. Grinnell E. Moore Date: December 11, 2007 X. Jiang T. Pham Duration: 3 hours Provide
More informationHAND IN PART. Prof. Girardi Math 142 Spring Exam 1. NAME: key
HAND IN PART Prof. Girardi Math 4 Spring 4..4 Exam MARK BOX problem points 7 % NAME: key PIN: INSTRUCTIONS The mark box above indicates the problems along with their points. Check that your copy of the
More informationName: Answer Key No calculators. Show your work! 1. (21 points) All answers should either be,, a (finite) real number, or DNE ( does not exist ).
Mat - Final Exam August 3 rd, Name: Answer Key No calculators. Sow your work!. points) All answers sould eiter be,, a finite) real number, or DNE does not exist ). a) Use te grap of te function to evaluate
More informationIntermediate Algebra Chapter 12 Review
Intermediate Algebra Chapter 1 Review Set up a Table of Coordinates and graph the given functions. Find the y-intercept. Label at least three points on the graph. Your graph must have the correct shape.
More information(c) Find the equation of the degree 3 polynomial that has the same y-value, slope, curvature, and third derivative as ln(x + 1) at x = 0.
Chapter 7 Challenge problems Example. (a) Find the equation of the tangent line for ln(x + ) at x = 0. (b) Find the equation of the parabola that is tangent to ln(x + ) at x = 0 (i.e. the parabola has
More informationChapter 6: Exponential and Logarithmic Functions
Section 6.1: Algebra and Composition of Functions #1-9: Let f(x) = 2x + 3 and g(x) = 3 x. Find each function. 1) (f + g)(x) 2) (g f)(x) 3) (f/g)(x) 4) ( )( ) 5) ( g/f)(x) 6) ( )( ) 7) ( )( ) 8) (g+f)(x)
More informationCalculus I Review Solutions
Calculus I Review Solutions. Compare and contrast the three Value Theorems of the course. When you would typically use each. The three value theorems are the Intermediate, Mean and Extreme value theorems.
More informationNAME: DATE: CLASS: AP CALCULUS AB SUMMER MATH 2018
NAME: DATE: CLASS: AP CALCULUS AB SUMMER MATH 2018 A] Refer to your pre-calculus notebook, the internet, or the sheets/links provided for assistance. B] Do not wait until the last minute to complete this
More informationFall 2013 Hour Exam 2 11/08/13 Time Limit: 50 Minutes
Math 8 Fall Hour Exam /8/ Time Limit: 5 Minutes Name (Print): This exam contains 9 pages (including this cover page) and 7 problems. Check to see if any pages are missing. Enter all requested information
More informationSECTION A. f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes.
SECTION A 1. State the maximal domain and range of the function f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes. 2. By evaluating f(0),
More informationMethods of Calculus (MAC 2233) Spring 2010 Sample Final Exam
Methods of Calculus (MAC ) Spring 1 Sample Final Eam The following questions are meant to check your readiness for the final eam, as well as to give you an idea of its level of difficulty. There may be
More informationMath 261 Calculus I. Test 1 Study Guide. Name. Decide whether the limit exists. If it exists, find its value. 1) lim x 1. f(x) 2) lim x -1/2 f(x)
Math 261 Calculus I Test 1 Study Guide Name Decide whether the it exists. If it exists, find its value. 1) x 1 f(x) 2) x -1/2 f(x) Complete the table and use the result to find the indicated it. 3) If
More informationx 2 y = 1 2. Problem 2. Compute the Taylor series (at the base point 0) for the function 1 (1 x) 3.
MATH 8.0 - FINAL EXAM - SOME REVIEW PROBLEMS WITH SOLUTIONS 8.0 Calculus, Fall 207 Professor: Jared Speck Problem. Consider the following curve in the plane: x 2 y = 2. Let a be a number. The portion of
More informationMath 120 Final Exam Practice Problems, Form: A
Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,
More information