Math 180, Final Exam, Fall 2012 Problem 1 Solution

Size: px
Start display at page:

Download "Math 180, Final Exam, Fall 2012 Problem 1 Solution"

Transcription

1 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. d dx ln(ln(x)) = ln(x) d dx ln(x) = ln(x) x (b) We evaluate the derivative using the Power and Product Rules. d ( x 6 + sin(x) e x) = 6x 5 + sin(x) e x + cos(x) e x dx (c) We evaluate the derivative using the Chain Rule. d ( tan(x ) + cot(x ) ) = sec (x ) x csc (x ) x dx

2 Math 80, Final Exam, Fall 0 Problem Solution. Let f(x) = 4x 3 48x + 3. (a) Find all local maxima and minima of f(x). (b) Find the absolute maximum and minimum of f(x) on [0, ]. (a) The function will attain local extreme values at its critical points, i.e. the values of x satisfying f (x) = 0. f (x) = 0 7x 48 = 0 4(3x ) = 0 x = 3 x = ± 3 To classify these points, we evaluate f (x) on either side of each critical point to determine how f changes sign. f ( ) = 4, f (0) = 48, f () = 4 Since f changes from positive to negative across x =, the value of f( ) is a 3 3 local maximum. Since f changes from negative to positive across x =, the value of f( ) is a 3 3 local minimum. (b) f is continuous on [0, ] so we are guaranteed absolute extrema at either the endpoints or at a critical point in the interior of the interval. The critical points were computed in part (a) and only x = lies in the given interval. The function values at x = 0,, 3 3 are f(0) = 3, f( ) = 3 + 3, f() =

3 Math 80, Final Exam, Fall 0 Problem 3 Solution 3. A cylindrical cup of height h and radius r has volume V = πr h and surface area πr +πrh. Among all such cups with volume V = π, find the one with minimal surface area. The constraint in this problem is that the volume is constant. That is, V = π πr h = π h = r The function we want to minimize is the surface area. Using the above equation, we can write the surface area as a function of r only. The critical points of f are: f(r) = πr + πrh f(r) = πr + πr ( r f(r) = π r + ), r > 0 r f (r) = 0 π (r r ) = 0 r = r r 3 = r = 3 The second derivative of f is f (r) = π ( + r ) 3 and is positive for all r > 0. Thus, the function is concave up on (0, ) and r = 3 corresponds to an absolute minimum of f. The corresponding height of the cup is h = r = /3

4 Math 80, Final Exam, Fall 0 Problem 4 Solution 4. Determine the following its (a) x 0 sin (x) x (b) x 0 + (ln(x)) (c) x sin (πx) x + (a) The value of the it is x 0 ( sin (x) = x x 0 ) sin(x) = = x (b) Since ln(x) as x 0 +, the value of the it is 0. (c) The given function is continuous at all x. Therefore, we may use substitution. x sin (πx) x + = sin (π) + = 0

5 Math 80, Final Exam, Fall 0 Problem 5 Solution 5. Evaluate the following definite integrals. (a) (b) π/ 0 ( x + + x) dx sin(x) cos(x) dx (a) Using the Fundamental Theorem of Calculus we obtain: ( x + [ + x) dx = 3 x3/ + ] ( + x)3/ 3 [ = 3 3/ + ] [ 3 33/ 3 + ] 3 3/ = 3 33/ 3 = 3 (3 3 ) (b) Our strategy here is to let u = cos(x), du = sin(x) dx. The its of integration change to x = cos(0) = 0 and x = cos(π/) =. Upon making these substitutions and using the Fundamental Theorem of Calculus we have π/ 0 sin(x) cos(x) dx = = = 3 0 u du [ 3 u3/ ] 0

6 Math 80, Final Exam, Fall 0 Problem 6 Solution 6. Let g(x) = x x +. (a) Using only the definition of the derivative, determine the value of g (). (b) Find the equation of the line tangent to the graph of g(x) at (, 3). (a) The value of g () is g () = x g(x) g() x = x (x x + ) x = x x x x = x x(x ) x = x x = (b) The derivative of g(x) is g (x) = x. Thus, the slope of the tangent line at (, 3) is g () = () = 3. Therefore, the equation of the tangent line is y 3 = 3(x )

7 Math 80, Final Exam, Fall 0 Problem 7 Solution 7. For some number c, we define the function h(x) by h(x) = x + if x and by h(x) = + c if x <. 3 x (a) Determine x + h(x) and h(x). x (b) For which value or values of c does x h(x) exist? (c) For each of the values of c computed in part (b), determine whether or not h(x) is differentiable at x =. (a) The one-sided its are h(x) = + = 5 x + x h(x) = x 3 x + c = 3 + c = + c (b) The it exists when the one-sided its are the same. This occurs when + c = 5 c = 4 (c) The derivative of h(x) is x if x > and when x <. The derivative (3 x) approaches () = 4 as x + and (3 ) = as x. Since these its are not the same, the function is not differentiable at x =.

8 Math 80, Final Exam, Fall 0 Problem 8 Solution 8. Determine x f(x) for each of the following functions. (a) f(x) = x 3 (b) f(x) = x3 x x x (a) x x 3 = 0 since f(x) is a rational function where deg(p(x)) < deg(q(x)) (p(x) and q(x) are the numerator and denominator of f(x), respectively). (b) The it is computed as follows: x x 3 x x x = x x 3 x x x x = x = x 3/ x 3/ x x ( 0 0 )

9 Math 80, Final Exam, Fall 0 Problem 9 Solution 9. Let f(x) = x x 4. (a) Find all horizontal and vertical asymptotes of f(x). (b) Find the area of the region bounded by the x-axis, the line x = 3, the line x = 4, and the graph of f(x). (a) Since x + x x 4 = + we know that x = is a vertical asymptote of f(x). Furthermore, since x + x x 4 = + we know that x = is also a vertical asymptote. Since x ± x x 4 = 0 we know that y = 0 is a horizontal asymptote of f(x). (b) The area of the region is 4 x A = 3 x 4 dx [ ] 4 = ln(x 4) 3 = ln(4 4) ln(3 4) = ln() ln(5) = ln 5

MA1021 Calculus I B Term, Sign:

MA1021 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 information

Spring 2015 Sample Final Exam

Spring 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 information

a 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).

a 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 information

AB CALCULUS SEMESTER A REVIEW Show all work on separate paper. (b) lim. lim. (f) x a. for each of the following functions: (b) y = 3x 4 x + 2

AB CALCULUS SEMESTER A REVIEW Show all work on separate paper. (b) lim. lim. (f) x a. for each of the following functions: (b) y = 3x 4 x + 2 AB CALCULUS Page 1 of 6 NAME DATE 1. Evaluate each it: AB CALCULUS Show all work on separate paper. x 3 x 9 x 5x + 6 x 0 5x 3sin x x 7 x 3 x 3 5x (d) 5x 3 x +1 x x 4 (e) x x 9 3x 4 6x (f) h 0 sin( π 6

More information

Math 250 Skills Assessment Test

Math 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 information

Math 131 Final Exam Spring 2016

Math 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 information

Calculus I Exam 1 Review Fall 2016

Calculus I Exam 1 Review Fall 2016 Problem 1: Decide whether the following statements are true or false: (a) If f, g are differentiable, then d d x (f g) = f g. (b) If a function is continuous, then it is differentiable. (c) If a function

More information

Fall 2009 Math 113 Final Exam Solutions. f(x) = 1 + ex 1 e x?

Fall 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 information

Math 180, Exam 2, Practice Fall 2009 Problem 1 Solution. f(x) = arcsin(2x + 1) = sin 1 (3x + 1), lnx

Math 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 information

b n x n + b n 1 x n b 1 x + b 0

b n x n + b n 1 x n b 1 x + b 0 Math Partial Fractions Stewart 7.4 Integrating basic rational functions. For a function f(x), we have examined several algebraic methods for finding its indefinite integral (antiderivative) F (x) = f(x)

More information

MATH 10550, EXAM 2 SOLUTIONS. 1. Find an equation for the tangent line to. f(x) = sin x cos x. 2 which is the slope of the tangent line at

MATH 10550, EXAM 2 SOLUTIONS. 1. Find an equation for the tangent line to. f(x) = sin x cos x. 2 which is the slope of the tangent line at MATH 100, EXAM SOLUTIONS 1. Find an equation for the tangent line to at the point ( π 4, 0). f(x) = sin x cos x f (x) = cos(x) + sin(x) Thus, f ( π 4 ) = which is the slope of the tangent line at ( π 4,

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

(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 information

MATH 1241 FINAL EXAM FALL 2012 Part I, No Calculators Allowed

MATH 1241 FINAL EXAM FALL 2012 Part I, No Calculators Allowed MATH 11 FINAL EXAM FALL 01 Part I, No Calculators Allowed 1. Evaluate the limit: lim x x x + x 1. (a) 0 (b) 0.5 0.5 1 Does not exist. Which of the following is the derivative of g(x) = x cos(3x + 1)? (a)

More information

Calculus I Review Solutions

Calculus 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 information

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).

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). 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).

More information

Calculus & Analytic Geometry I

Calculus & Analytic Geometry I TQS 124 Autumn 2008 Quinn Calculus & Analytic Geometry I The Derivative: Analytic Viewpoint Derivative of a Constant Function. For c a constant, the derivative of f(x) = c equals f (x) = Derivative of

More information

College of the Holy Cross MATH 133, Calculus With Fundamentals 1 Solutions for Final Examination Friday, December 15

College 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 information

f (x) = 2x x = 2x2 + 4x 6 x 0 = 2x 2 + 4x 6 = 2(x + 3)(x 1) x = 3 or x = 1.

f (x) = 2x x = 2x2 + 4x 6 x 0 = 2x 2 + 4x 6 = 2(x + 3)(x 1) x = 3 or x = 1. F16 MATH 15 Test November, 016 NAME: SOLUTIONS CRN: Use only methods from class. You must show work to receive credit. When using a theorem given in class, cite the theorem. Reminder: Calculators are not

More information

Calculus I Sample Exam #01

Calculus I Sample Exam #01 Calculus I Sample Exam #01 1. Sketch the graph of the function and define the domain and range. 1 a) f( x) 3 b) g( x) x 1 x c) hx ( ) x x 1 5x6 d) jx ( ) x x x 3 6 . Evaluate the following. a) 5 sin 6

More information

Math 229 Mock Final Exam Solution

Math 229 Mock Final Exam Solution Name: Math 229 Mock Final Exam Solution Disclaimer: This mock exam is for practice purposes only. No graphing calulators TI-89 is allowed on this test. Be sure that all of your work is shown and that it

More information

Topics and Concepts. 1. Limits

Topics and Concepts. 1. Limits Topics and Concepts 1. Limits (a) Evaluating its (Know: it exists if and only if the it from the left is the same as the it from the right) (b) Infinite its (give rise to vertical asymptotes) (c) Limits

More information

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents. Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

More information

Math 108, Solution of Midterm Exam 3

Math 108, Solution of Midterm Exam 3 Math 108, Solution of Midterm Exam 3 1 Find an equation of the tangent line to the curve x 3 +y 3 = xy at the point (1,1). Solution. Differentiating both sides of the given equation with respect to x,

More information

Math 131 Week-in-Review #11 (Final Exam Review: All previous sections as well as sections 5.5, 6.1, 6.5, and 6.7)

Math 131 Week-in-Review #11 (Final Exam Review: All previous sections as well as sections 5.5, 6.1, 6.5, and 6.7) Math 131 Week-in-Review #11 (Final Exam Review: All previous sections as well as sections 5.5, 6.1, 6.5, and 6.7) Note: This collection of questions is intended to be a brief overview of the exam material

More information

PRELIM 2 REVIEW QUESTIONS Math 1910 Section 205/209

PRELIM 2 REVIEW QUESTIONS Math 1910 Section 205/209 PRELIM 2 REVIEW QUESTIONS Math 9 Section 25/29 () Calculate the following integrals. (a) (b) x 2 dx SOLUTION: This is just the area under a semicircle of radius, so π/2. sin 2 (x) cos (x) dx SOLUTION:

More information

Old Math 220 Exams. David M. McClendon. Department of Mathematics Ferris State University

Old Math 220 Exams. David M. McClendon. Department of Mathematics Ferris State University Old Math 0 Exams David M. McClendon Department of Mathematics Ferris State University Last updated to include exams from Spring 05 Contents Contents General information about these exams 4 Exams from 0

More information

MATH 408N PRACTICE FINAL

MATH 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 information

Final Exam. Math 3 December 7, 2010

Final 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 information

Calculus 221 worksheet

Calculus 221 worksheet Calculus 221 worksheet Graphing A function has a global maximum at some a in its domain if f(x) f(a) for all other x in the domain of f. Global maxima are sometimes also called absolute maxima. A function

More information

MA Practice Exam #2 Solutions

MA Practice Exam #2 Solutions MA 123 - Practice Exam #2 Solutions Name: Instructions: For some of the questions, you must show all your work as indicated. No calculators, books or notes of any form are allowed. Note that the questions

More information

2. (12 points) Find an equation for the line tangent to the graph of f(x) =

2. (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 information

2015 Math Camp Calculus Exam Solution

2015 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 information

Math 241 Final Exam, Spring 2013

Math 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 information

Calculus Midterm Exam October 31, 2018

Calculus Midterm Exam October 31, 2018 Calculus Midterm Exam October 31, 018 1. Use l Hôpital s Rule to evaluate the it, if it exists. tan3x (a) (6 points) sinx tan3x = 0, sinx = 0, and both tan3x and sinx are differentiable near x = 0, tan3x

More information

3/1/2012: First hourly Practice A

3/1/2012: First hourly Practice A Math 1A: Introduction to functions and calculus Oliver Knill, Spring 2012 3/1/2012: First hourly Practice A Your Name: Start by writing your name in the above box. Try to answer each question on the same

More information

x x implies that f x f x.

x x implies that f x f x. Section 3.3 Intervals of Increase and Decrease and Extreme Values Let f be a function whose domain includes an interval I. We say that f is increasing on I if for every two numbers x 1, x 2 in I, x x implies

More information

c) xy 3 = cos(7x +5y), y 0 = y3 + 7 sin(7x +5y) 3xy sin(7x +5y) d) xe y = sin(xy), y 0 = ey + y cos(xy) x(e y cos(xy)) e) y = x ln(3x + 5), y 0

c) xy 3 = cos(7x +5y), y 0 = y3 + 7 sin(7x +5y) 3xy sin(7x +5y) d) xe y = sin(xy), y 0 = ey + y cos(xy) x(e y cos(xy)) e) y = x ln(3x + 5), y 0 Some Math 35 review problems With answers 2/6/2005 The following problems are based heavily on problems written by Professor Stephen Greenfield for his Math 35 class in spring 2005. His willingness to

More information

MATH 2053 Calculus I Review for the Final Exam

MATH 2053 Calculus I Review for the Final Exam MATH 05 Calculus I Review for the Final Exam (x+ x) 9 x 9 1. Find the limit: lim x 0. x. Find the limit: lim x + x x (x ).. Find lim x (x 5) = L, find such that f(x) L < 0.01 whenever 0 < x

More information

Math. 151, WebCalc Sections December Final Examination Solutions

Math. 151, WebCalc Sections December Final Examination Solutions Math. 5, WebCalc Sections 507 508 December 00 Final Examination Solutions Name: Section: Part I: Multiple Choice ( points each) There is no partial credit. You may not use a calculator.. Another word for

More information

In general, if we start with a function f and want to reverse the differentiation process, then we are finding an antiderivative of f.

In general, if we start with a function f and want to reverse the differentiation process, then we are finding an antiderivative of f. Math 1410 Worksheet #27: Section 4.9 Name: Our final application of derivatives is a prelude to what will come in later chapters. In many situations, it will be necessary to find a way to reverse the differentiation

More information

1.18 Multiple Choice Questions on Limits

1.18 Multiple Choice Questions on Limits 24 The AP CALCULUS PROBLEM BOOK 3x 4 2x + 33. lim x 7x 8x 5 =.8 Multiple Choice Questions on Limits A) B) C) 0 D) 3 7 E) 3 8 34. lim x 0 x = A) B) C) 0 D) E) does not exist 9x 2 35. lim x /3 3x = A) B)

More information

Suppose that f is continuous on [a, b] and differentiable on (a, b). Then

Suppose that f is continuous on [a, b] and differentiable on (a, b). Then Lectures 1/18 Derivatives and Graphs When we have a picture of the graph of a function f(x), we can make a picture of the derivative f (x) using the slopes of the tangents to the graph of f. In this section

More information

Math 147 Exam II Practice Problems

Math 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 information

MATH 135 Calculus 1 Solutions/Answers for Exam 3 Practice Problems November 18, 2016

MATH 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 information

MATH 151, FALL SEMESTER 2011 COMMON EXAMINATION 3 - VERSION B - SOLUTIONS

MATH 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 information

Exam 1. (2x + 1) 2 9. lim. (rearranging) (x 1 implies x 1, thus x 1 0

Exam 1. (2x + 1) 2 9. lim. (rearranging) (x 1 implies x 1, thus x 1 0 Department of Mathematical Sciences Instructor: Daiva Pucinskaite Calculus I January 28, 2016 Name: Exam 1 1. Evaluate the it x 1 (2x + 1) 2 9. x 1 (2x + 1) 2 9 4x 2 + 4x + 1 9 = 4x 2 + 4x 8 = 4(x 1)(x

More information

Mth Review Problems for Test 2 Stewart 8e Chapter 3. For Test #2 study these problems, the examples in your notes, and the homework.

Mth 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 information

CALCULUS ASSESSMENT REVIEW

CALCULUS ASSESSMENT REVIEW CALCULUS ASSESSMENT REVIEW DEPARTMENT OF MATHEMATICS CHRISTOPHER NEWPORT UNIVERSITY 1. Introduction and Topics The purpose of these notes is to give an idea of what to expect on the Calculus Readiness

More information

Calculus Example Exam Solutions

Calculus Example Exam Solutions Calculus Example Exam Solutions. Limits (8 points, 6 each) Evaluate the following limits: p x 2 (a) lim x 4 We compute as follows: lim p x 2 x 4 p p x 2 x +2 x 4 p x +2 x 4 (x 4)( p x + 2) p x +2 = p 4+2

More information

Calculus I Practice Exam 2

Calculus I Practice Exam 2 Calculus I Practice Exam 2 Instructions: The exam is closed book, closed notes, although you may use a note sheet as in the previous exam. A calculator is allowed, but you must show all of your work. Your

More information

MATH 162 R E V I E W F I N A L E X A M FALL 2016

MATH 162 R E V I E W F I N A L E X A M FALL 2016 MATH 6 R E V I E W F I N A L E X A M FALL 06 BASICS Graphs. Be able to graph basic functions, such as polynomials (eg, f(x) = x 3 x, x + ax + b, x(x ) (x + ) 3, know about the effect of multiplicity of

More information

Math 180, Final Exam, Fall 2007 Problem 1 Solution

Math 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 information

Calculus I Practice Problems 8: Answers

Calculus I Practice Problems 8: Answers Calculus I Practice Problems : Answers. Let y x x. Find the intervals in which the function is increasing and decreasing, and where it is concave up and concave down. Sketch the graph. Answer. Differentiate

More information

Math 131 Week-in-Review #7 (Exam 2 Review: Sections , , and )

Math 131 Week-in-Review #7 (Exam 2 Review: Sections , , and ) Math 131 WIR, copyright Angie Allen 1 Math 131 Week-in-Review #7 (Exam 2 Review: Sections 2.6-2.8, 3.1-3.4, and 3.7-3.9) Note: This collection of questions is intended to be a brief overview of the exam

More information

AP Calculus Summer Prep

AP Calculus Summer Prep AP Calculus Summer Prep Topics from Algebra and Pre-Calculus (Solutions are on the Answer Key on the Last Pages) The purpose of this packet is to give you a review of basic skills. You are asked to have

More information

Math 261 Exam 3 - Practice Problems. 1. The graph of f is given below. Answer the following questions. (a) Find the intervals where f is increasing:

Math 261 Exam 3 - Practice Problems. 1. The graph of f is given below. Answer the following questions. (a) Find the intervals where f is increasing: Math 261 Exam - Practice Problems 1. The graph of f is given below. Answer the following questions. (a) Find the intervals where f is increasing: ( 6, 4), ( 1,1),(,5),(6, ) (b) Find the intervals where

More information

AP Calculus Multiple Choice Questions - Chapter 5

AP Calculus Multiple Choice Questions - Chapter 5 1 If f'(x) = (x - 2)(x - 3) 2 (x - 4) 3, then f has which of the following relative extrema? I. A relative maximum at x = 2 II. A relative minimum at x = 3 III. A relative maximum at x = 4 a I only b III

More information

Section 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. 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 information

Aim: How do we prepare for AP Problems on limits, continuity and differentiability? f (x)

Aim: 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 information

UNIT 3: DERIVATIVES STUDY GUIDE

UNIT 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 information

Calculus I (Math 241) (In Progress)

Calculus I (Math 241) (In Progress) Calculus I (Math 241) (In Progress) The following is a collection of Calculus I (Math 241) problems. Students may expect that their final exam is comprised, more or less, of one problem from each section,

More information

DRAFT - Math 102 Lecture Note - Dr. Said Algarni

DRAFT - Math 102 Lecture Note - Dr. Said Algarni Math02 - Term72 - Guides and Exercises - DRAFT 7 Techniques of Integration A summery for the most important integrals that we have learned so far: 7. Integration by Parts The Product Rule states that if

More information

Name Date Period. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Name Date Period. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. AB Fall Final Exam Review 200-20 Name Date Period MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. ) The position of a particle

More information

MTH30 Review Sheet. y = g(x) BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE

MTH30 Review Sheet. y = g(x) BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH0 Review Sheet. Given the functions f and g described by the graphs below: y = f(x) y = g(x) (a)

More information

Test 3 Review. y f(a) = f (a)(x a) y = f (a)(x a) + f(a) L(x) = f (a)(x a) + f(a)

Test 3 Review. y f(a) = f (a)(x a) y = f (a)(x a) + f(a) L(x) = f (a)(x a) + f(a) MATH 2250 Calculus I Eric Perkerson Test 3 Review Sections Covered: 3.11, 4.1 4.6. Topics Covered: Linearization, Extreme Values, The Mean Value Theorem, Consequences of the Mean Value Theorem, Concavity

More information

MATH 408N PRACTICE FINAL

MATH 408N PRACTICE FINAL 05/05/2012 Bormashenko MATH 408N PRACTICE FINAL Name: TA session: Show your work for all the problems. Good luck! (1) Calculate the following limits, using whatever tools are appropriate. State which results

More information

A.P. Calculus Holiday Packet

A.P. Calculus Holiday Packet A.P. Calculus Holiday Packet Since this is a take-home, I cannot stop you from using calculators but you would be wise to use them sparingly. When you are asked questions about graphs of functions, do

More information

M152: Calculus II Midterm Exam Review

M152: Calculus II Midterm Exam Review M52: Calculus II Midterm Exam Review Chapter 4. 4.2 : Mean Value Theorem. - Know the statement and idea of Mean Value Theorem. - Know how to find values of c making the theorem true. - Realize the importance

More information

Chapter 5B - Rational Functions

Chapter 5B - Rational Functions Fry Texas A&M University Math 150 Chapter 5B Fall 2015 143 Chapter 5B - Rational Functions Definition: A rational function is The domain of a rational function is all real numbers, except those values

More information

Final practice, Math 31A - Lec 1, Fall 2013 Name and student ID: Question Points Score Total: 90

Final practice, Math 31A - Lec 1, Fall 2013 Name and student ID: Question Points Score Total: 90 Final practice, Math 31A - Lec 1, Fall 13 Name and student ID: Question Points Score 1 1 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 Total: 9 1. a) 4 points) Find all points x at which the function fx) x 4x + 3 + x

More information

MA 113 Calculus I Fall 2016 Exam Final Wednesday, December 14, True/False 1 T F 2 T F 3 T F 4 T F 5 T F. Name: Section:

MA 113 Calculus I Fall 2016 Exam Final Wednesday, December 14, True/False 1 T F 2 T F 3 T F 4 T F 5 T F. Name: Section: MA 113 Calculus I Fall 2016 Exam Final Wednesday, December 14, 2016 Name: Section: Last 4 digits of student ID #: This exam has five true/false questions (two points each), ten multiple choice questions

More information

Math 121 Test 3 - Review 1. Use differentials to approximate the following. Compare your answer to that of a calculator

Math 121 Test 3 - Review 1. Use differentials to approximate the following. Compare your answer to that of a calculator Math Test - Review Use differentials to approximate the following. Compare your answer to that of a calculator.. 99.. 8. 6. Consider the graph of the equation f(x) = x x a. Find f (x) and f (x). b. Find

More information

Name: Instructor: 1. a b c d e. 15. a b c d e. 2. a b c d e a b c d e. 16. a b c d e a b c d e. 4. a b c d e... 5.

Name: Instructor: 1. a b c d e. 15. a b c d e. 2. a b c d e a b c d e. 16. a b c d e a b c d e. 4. a b c d e... 5. Name: Instructor: Math 155, Practice Final Exam, December The Honor Code is in effect for this examination. All work is to be your own. No calculators. The exam lasts for 2 hours. Be sure that your name

More information

MTH Calculus with Analytic Geom I TEST 1

MTH Calculus with Analytic Geom I TEST 1 MTH 229-105 Calculus with Analytic Geom I TEST 1 Name Please write your solutions in a clear and precise manner. SHOW your work entirely. (1) Find the equation of a straight line perpendicular to the line

More information

8.7 Taylor s Inequality Math 2300 Section 005 Calculus II. f(x) = ln(1 + x) f(0) = 0

8.7 Taylor s Inequality Math 2300 Section 005 Calculus II. f(x) = ln(1 + x) f(0) = 0 8.7 Taylor s Inequality Math 00 Section 005 Calculus II Name: ANSWER KEY Taylor s Inequality: If f (n+) is continuous and f (n+) < M between the center a and some point x, then f(x) T n (x) M x a n+ (n

More information

MAC 2311 Exam 1 Review Fall Private-Appointment, one-on-one tutoring at Broward Hall

MAC 2311 Exam 1 Review Fall Private-Appointment, one-on-one tutoring at Broward Hall Fall 2016 This review, produced by the CLAS Teaching Center, contains a collection of questions which are representative of the type you may encounter on the exam. Other resources made available by the

More information

Exam Review Sheets Combined

Exam Review Sheets Combined Exam Review Sheets Combined Fall 2008 1 Fall 2007 Exam 1 1. For each part, if the statement is always true, circle the printed capital T. If the statement is sometimes false, circle the printed capital

More information

What is on today. 1 Linear approximation. MA 123 (Calculus I) Lecture 17: November 2, 2017 Section A2. Professor Jennifer Balakrishnan,

What is on today. 1 Linear approximation. MA 123 (Calculus I) Lecture 17: November 2, 2017 Section A2. Professor Jennifer Balakrishnan, Professor Jennifer Balakrishnan, jbala@bu.edu What is on today 1 Linear approximation 1 1.1 Linear approximation and concavity....................... 2 1.2 Change in y....................................

More information

2. 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).

2. 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 information

Final Exam SOLUTIONS MAT 131 Fall 2011

Final Exam SOLUTIONS MAT 131 Fall 2011 1. Compute the following its. (a) Final Exam SOLUTIONS MAT 131 Fall 11 x + 1 x 1 x 1 The numerator is always positive, whereas the denominator is negative for numbers slightly smaller than 1. Also, as

More information

Section 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. 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 information

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents. Math120 - Precalculus. Final Review Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents. (a) 5

More information

Chapter 2: Differentiation

Chapter 2: Differentiation Chapter 2: Differentiation Winter 2016 Department of Mathematics Hong Kong Baptist University 1 / 75 2.1 Tangent Lines and Their Slopes This section deals with the problem of finding a straight line L

More information

Calculus III: Practice Final

Calculus III: Practice Final Calculus III: Practice Final Name: Circle one: Section 6 Section 7. Read the problems carefully. Show your work unless asked otherwise. Partial credit will be given for incomplete work. The exam contains

More information

Exam 3 MATH Calculus I

Exam 3 MATH Calculus I Trinity College December 03, 2015 MATH 131-01 Calculus I By signing below, you attest that you have neither given nor received help of any kind on this exam. Signature: Printed Name: Instructions: Show

More information

SOLUTIONS FOR PRACTICE FINAL EXAM

SOLUTIONS FOR PRACTICE FINAL EXAM SOLUTIONS FOR PRACTICE FINAL EXAM ANDREW J. BLUMBERG. Solutions () Short answer questions: (a) State the mean value theorem. Proof. The mean value theorem says that if f is continuous on (a, b) and differentiable

More information

Calculus I: Practice Midterm II

Calculus 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 information

Math 152 Take Home Test 1

Math 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 information

AP Calculus. Analyzing a Function Based on its Derivatives

AP Calculus. Analyzing a Function Based on its Derivatives AP Calculus Analyzing a Function Based on its Derivatives Student Handout 016 017 EDITION Click on the following link or scan the QR code to complete the evaluation for the Study Session https://www.surveymonkey.com/r/s_sss

More information

You are being asked to create your own AP CALCULUS Survival kit. For the survival kit you will need:

You are being asked to create your own AP CALCULUS Survival kit. For the survival kit you will need: You are being asked to create your own AP CALCULUS Survival kit. For the survival kit you will need: Index cards Ring (so that you can put all of your flash cards together) Hole punch (to punch holes in

More information

MA 113 Calculus I Fall 2012 Exam 3 13 November Multiple Choice Answers. Question

MA 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 information

Math 112 (Calculus I) Midterm Exam 3 KEY

Math 112 (Calculus I) Midterm Exam 3 KEY Math 11 (Calculus I) Midterm Exam KEY Multiple Choice. Fill in the answer to each problem on your computer scored answer sheet. Make sure your name, section and instructor are on that sheet. 1. Which of

More information

MATH 151, FALL 2017 COMMON EXAM III - VERSION B

MATH 151, FALL 2017 COMMON EXAM III - VERSION B MATH 151, FALL 2017 COMMON EXAM III - VERSION B LAST NAME(print): FIRST NAME(print): INSTRUCTOR: SECTION NUMBER: DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited. 2. TURN OFF cell

More information

Final Exam Solutions

Final 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 information

Dr. Sophie Marques. MAM1020S Tutorial 8 August Divide. 1. 6x 2 + x 15 by 3x + 5. Solution: Do a long division show your work.

Dr. Sophie Marques. MAM1020S Tutorial 8 August Divide. 1. 6x 2 + x 15 by 3x + 5. Solution: Do a long division show your work. Dr. Sophie Marques MAM100S Tutorial 8 August 017 1. Divide 1. 6x + x 15 by 3x + 5. 6x + x 15 = (x 3)(3x + 5) + 0. 1a 4 17a 3 + 9a + 7a 6 by 3a 1a 4 17a 3 + 9a + 7a 6 = (4a 3 3a + a + 3)(3a ) + 0 3. 1a

More information

Name: Instructor: Exam 3 Solutions. Multiple Choice. 3x + 2 x ) 3x 3 + 2x 2 + 5x + 2 3x 3 3x 2x 2 + 2x + 2 2x 2 2 2x.

Name: Instructor: Exam 3 Solutions. Multiple Choice. 3x + 2 x ) 3x 3 + 2x 2 + 5x + 2 3x 3 3x 2x 2 + 2x + 2 2x 2 2 2x. . Exam 3 Solutions Multiple Choice.(6 pts.) Find the equation of the slant asymptote to the function We have so the slant asymptote is y = 3x +. f(x) = 3x3 + x + 5x + x + 3x + x + ) 3x 3 + x + 5x + 3x

More information

2. Which of the following is an equation of the line tangent to the graph of f(x) = x 4 + 2x 2 at the point where

2. Which of the following is an equation of the line tangent to the graph of f(x) = x 4 + 2x 2 at the point where AP Review Chapter Name: Date: Per: 1. The radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. In terms of the circumference C, what is the rate of change of the area of the

More information

MATH 1241 Common Final Exam Fall 2010

MATH 1241 Common Final Exam Fall 2010 MATH 1241 Common Final Exam Fall 2010 Please print the following information: Name: Instructor: Student ID: Section/Time: The MATH 1241 Final Exam consists of three parts. You have three hours for the

More information

THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK. Summer Examination 2009.

THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK. Summer Examination 2009. OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK Summer Examination 2009 First Engineering MA008 Calculus and Linear Algebra

More information

APPM 1350 Final Exam Fall 2017

APPM 1350 Final Exam Fall 2017 APPM 350 Final Exam Fall 207. (26 pts) Evaluate the following. (a) Let g(x) cos 3 (π 2x). Find g (π/3). (b) Let y ( x) x. Find y (4). (c) lim r 0 e /r ln(r) + (a) (9 pt) g (x) 3 cos 2 (π 2x)( sin(π 2x))(

More information