Exam 2 Solutions, Math March 17, ) = 1 2
|
|
- Patience Cooper
- 5 years ago
- Views:
Transcription
1 Eam Solutions, Math 56 March 7, 6. Use the trapezoidal rule with n = 3 to approimate (Note: The eact value of the integral is ln 5 +. (you do not need to verify this or use it in any way to complete this problem.)) Solution: We have = b a n = 3 = and f() = +. Thus, + = (f() + f() + f(3) + f()) ( ). 5. Which of the following is a direction field for the differential equation = y? (a) (b) (c) (d)
2 (e) Solution: When y =, =, thus on the line y =, we should have horizontal lines. That leaves only (a) as an answer. The graph of the direction field is shown below. 3. The length of the curve y = cos( ) from (, ) to (, cos()) is given by: Solution: We have = sin( ). Thus, Arclength = + ( sin( )) = + sin ( ).
3 . The improper integral e (ln()). Solution: Use the definition of improper integral and making the substitution u = ln with = du. Then t ln t e (ln ) t e (ln ) t u du t [ u ] ln t t ( ln t + ) =. 5. Using Euler s method with step size, which of the following gives an approimation for y(3), where y is the solution of with y() =? y = y Solution: Thus y(3) 5 n n y n n, y n ) ()() = = + () = 3 ()(3) = 6 = () = 5 6. Find n= ( ) n 3 n 5 n. Solution: Note that ( ) n 3 n = n n= This is a geometric series. The first term a is 3 and the ratio r is 5 5 Note that r = < hence our series converges. Now the sum is 5 in our case. 3 a r = = =
4 7. Evaluate the improper integral π tan (). Solution: Note that there is a trouble spot at π, since tan( π ) is undefined. Let us write Note that That gives us π tan () = tan () = π π tan () + sec () tan () t π π π tan (). = tan() + C. t tan () [tan() ] t t π tan(t) t t π = π. Hence π tan () diverges, therefore, π tan () also diverges. 8. The solution to the initial value problem satisfies the implicit equation = e ( + y ), y() = Solution: We can use the separation of variables: + y = e tan (y) = e = e e + C.
5 We have used integration by parts with u =, dv = e in the above integral. The initial condition gives us tan () = e e + C, or C =. Hence our solution satisfies the implicit equation tan (y) = e e. 9. Find the general solution to the differential equation y + 3y = + 5. Solution: We can put the equation in the following form: y + 3 y = + 5. This is a first-order linear differential equation. In order to solve it, we find the integrating factor. I() = e 3 = e 3 ln = e ln 3 = 3. Multplying across by I() we get Hence y = C 3. y y = (y 3 ) = ( + 5) y 3 = ( + 5 ) y 3 = C.. Consider the following sequences: (I) { ( ) n n n } n= (II) Which of the following statements is true? a) All three sequences converge. { ( ) n n n } n= b) Sequences I and II converge but sequence III diverges. (III) { e n n } n= 5
6 c) All three sequences diverge. d) Sequences I and III converge but sequence II diverges. e) Sequence I converges but sequences II and III diverge. { Solution: (I) ( ) n n } is an alternating sequence. It is a theorem that it n n= converges if and only if lim n n ( )n n n =. n n We can use L Hospital s rule to determine the above limit. n lim n n Hence sequence I converges. (II) ln (ln ) =. { } ( ) n n is an alternating sequence as well. As in (I), we consider n n= lim n n ( )n n Hence sequence (II) diverges. n n n n n (III) We write e n lim n n e. We have an indeterminate form so we can use LHospitals rule { e n Thus sequence III = n } n= e lim e =. is divergent. =. To sum up, the solution is: Sequence I converges but sequences II and III diverge. 6
7 . A tank initially contains liters of salt water with kilogram of dissolved salt. A well mied salt water solution containing kilograms of salt per liters is pumped into the tank at 5 liters per minute. The salt water in the tank is kept thoroughly mied and is drained at the rate of 5 liters per minute. (a) Let y = y(t) be the amount of salt in the tank at time t. Give a differential equation relating to y. dt Solution: We note that the rate of change of the amount of salt in the tank, denoted, is given by dt dt = rate in - rate out, where rate in denotes the rate at which the salt is entering the tank (in kg/min) and rate out denotes the rate at which the salt is eiting the tank (also in kg/min). Now, Similarly, rate in = (Conc. of Solution Entering Tank)(Flow Rate In) = kg L 5 L min = kg/min rate out = (Conc. of Solution Eiting Tank)(Flow Rate Out). The difference here is that the concentration of the solution eiting the tank is no longer constant. This concentration in the tank (assuming the tank is evenly mied) is given by C(t) = y(t), where V (t) is the volume in the tank at time t. Since the V (t) rate at which solution is entering the tank is the same at which it is eiting, we have dv/dt = and the volume of solution in the tank is constant at L. So C(t) = y(t) kg/l. Putting this together, we see that Thus, rate out = y(t)kg L dt 5 L min = y(t) kg/min. = rate in - rate out = y(t). 7
8 (Units are still kg/min.) We also note that the tank initially contains kg of dissolved salt, giving us the initial condition y() =. So our differential equation becomes dt = y(t), y() =. (b) Give a formula for the amount of salt in the tank at time t. Solution: We may rewrite the equation above as dt = y(t), y() = and we see that this is an eample of a separable differential equation. Separating variables and integrating both sides with respect to t we see y(t) dt = y(t) dt dt = ln y(t) = t + C dt ln y(t) = t + C ( C = C) y(t) = e t+ C y(t) = ±e Ce t = Ae t (A = ±e C) y(t) = Ae t. We now plug in the initial condition to solve for the constant A. y() = = Ae = A A = Thus our formula for the amount of salt in the tank at time t is given by y(t) = e /t. 8
9 Remark: This differential equation is also an eample of a linear differential equation. We can see this by rewriting the equation as dt + y =. This means there are actually two different ways to approach this problem; we could have also used the method for solving linear differential equations. (c) Find the limit of y(t) as t goes to. Solution: lim y(t) t t e /t t = = kg. e (/)t. Find the arc length of the curve given by from = to =. f() = e + e Solution: Let L be the arc length of the curve given above from = to =. By the arc length formula, we know L = + (f ()). We first need to compute the derivative f (): f () = e + e ( ) = e e = (e e ). 9
10 Net, we will simplify the integrand, ( ) + (f ()) = + (e e ) ( ) = + (e + e ) e + + e = (e + e = ) = e + e. Plugging this back into our formula for arc length, we obtain L = + (f ()) e + e = [ ] e e = = e e = e e. e e
11 3. For each of the following improper integrals, determine whether the integral converges or diverges. To receive full credit for this problem, you must justify each answer and state clearly whether each integral converges or diverges. (a) + 3/ Solution: The p-test for improper integrals states that + converges if p > and diverges if p. p In this case, p = 3 < and this integral DIVERGES. Alternatively, you could try to evaluate this integral directly. + t 3/ t 3/ t t 3/ [ ] / t t [ t / ] t = Since this limit is not finite, the integral diverges. (b) + 3/ Solution: This integral CONVERGES by the p-test, since p = 3 >.
12 You can also show the integral converges by evaluating it directly: + t 3/ t 3/ [ ] / t t [ t / + ] t [ ] t t = (c) + 3/ + Solution: CONVERGES The Comparison Test for Improper Integrals states that if f and g are continuous functions with g() f() for all a. i) If a ii) If a f() is convergent, then g() is convergent. a g() is divergent, then f() is divergent. a Now, for all, we have 3/ + 3/ 3/ +. 3/ We also can see that both g() = and f() = are both continuous 3/ + 3/ functions on the interval [, ] (actually these are both continuous functions at all points ecept = ). Since converges (we showed this in part b), by the 3/ comparison test, so does +. 3/ + (d) + 3/
13 Solution: DIVERGES This actually is an improper integral for two reasons: i) the interval is infinite, ii) f() = is discontinuous at = 6. To find when f() is discontinuous in the 3/ interval [, ] we set the denominator equal to and solve for : So, 3/ = ( ) 3/ / = = or / = / = / = = = / = 6 3/ + 6 3/ Note that BOTH of these integrals must converge for the entire integral to be convergent. We will compute the first integral directly: 6 t 3/ t 6 3/ t t 6 3/ ( Use the rationalizing substitution u = /, du = / 3/, so = 3/ du = u 3 du.) t 6 t 6 t t / 8u 3 u 3 u du 8 u du [ 8 ln u ]t/ t 6 8 ln t/ + 8 ln t 6 = Here, we are using the fact that if t 6, t / approaches from the right and lim ln =. Since this integral diverges, the entire integral must diverge. + 3
14 Remark: We can alternatively show that the second integral 6 3/ diverges (you still only need to show that one of the two diverges). Choosing a test point in the interval (6, ), say = 8, we see that f(8) = (8 3/ 8/) = (7 8/) = ( 7/) = /7. Consequently, f() < for all on the entire interval (6, ) (otherwise, the intermediate value theorem says we must have f() = for some value of in this interval, and this never happens). Now, 6 Furthermore, when 6 3/ = 6 3/ 3/ 3/ 3/ Now, 6 = diverges and so, by the comparison test, so does 6. Thus, 6 3/ 6 3/ = also diverges. This also 6 3/ + is enough to show 3/ diverges.
8. Set up the integral to determine the force on the side of a fish tank that has a length of 4 ft and a heght of 2 ft if the tank is full.
. Determine the volume of the solid formed by rotating the region bounded by y = 2 and y = 2 for 2 about the -ais. 2. Determine the volume of the solid formed by rotating the region bounded by the -ais
More informationThe answers below are not comprehensive and are meant to indicate the correct way to solve the problem. sin
Math : Practice Final Answer Key Name: The answers below are not comprehensive and are meant to indicate the correct way to solve the problem. Problem : Consider the definite integral I = 5 sin ( ) d.
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals 8. Basic Integration Rules In this section we will review various integration strategies. Strategies: I. Separate
More information8.a: Integrating Factors in Differential Equations. y = 5y + t (2)
8.a: Integrating Factors in Differential Equations 0.0.1 Basics of Integrating Factors Until now we have dealt with separable differential equations. Net we will focus on a more specific type of differential
More informationMath 181, Exam 2, Fall 2014 Problem 1 Solution. sin 3 (x) cos(x) dx.
Math 8, Eam 2, Fall 24 Problem Solution. Integrals, Part I (Trigonometric integrals: 6 points). Evaluate the integral: sin 3 () cos() d. Solution: We begin by rewriting sin 3 () as Then, after using the
More informationPractice Final Exam Solutions
Important Notice: To prepare for the final exam, study past exams and practice exams, and homeworks, quizzes, and worksheets, not just this practice final. A topic not being on the practice final does
More information(a) x cos 3x dx We apply integration by parts. Take u = x, so that dv = cos 3x dx, v = 1 sin 3x, du = dx. Thus
Math 128 Midterm Examination 2 October 21, 28 Name 6 problems, 112 (oops) points. Instructions: Show all work partial credit will be given, and Answers without work are worth credit without points. You
More informationMat104 Fall 2002, Improper Integrals From Old Exams
Mat4 Fall 22, Improper Integrals From Old Eams For the following integrals, state whether they are convergent or divergent, and give your reasons. () (2) (3) (4) (5) converges. Break it up as 3 + 2 3 +
More information4x x dx. 3 3 x2 dx = x3 ln(x 2 )
Problem. a) Compute the definite integral 4 + d This can be done by a u-substitution. Take u = +, so that du = d, which menas that 4 d = du. Notice that u() = and u() = 6, so our integral becomes 6 u du
More informationLesson Objectives: we will learn:
Lesson Objectives: Setting the Stage: Lesson 66 Improper Integrals HL Math - Santowski we will learn: How to solve definite integrals where the interval is infinite and where the function has an infinite
More informationDepartment of Mathematical Sciences. Math 226 Calculus Spring 2016 Exam 2V2 DO NOT TURN OVER THIS PAGE UNTIL INSTRUCTED TO DO SO
Department of Mathematical Sciences Math 6 Calculus Spring 6 Eam V DO NOT TURN OVER THIS PAGE UNTIL INSTRUCTED TO DO SO NAME (Printed): INSTRUCTOR: SECTION NO.: When instructed, turn over this cover page
More informationPractice Midterm 1 Solutions Written by Victoria Kala July 10, 2017
Practice Midterm 1 Solutions Written by Victoria Kala July 10, 2017 1. Use the slope field plotter link in Gauchospace to check your solution. 2. (a) Not linear because of the y 2 sin x term (b) Not linear
More informationPractice Final Exam Solutions
Important Notice: To prepare for the final exam, one should study the past exams and practice midterms (and homeworks, quizzes, and worksheets), not just this practice final. A topic not being on the practice
More informationFINAL EXAM CALCULUS 2. Name PRACTICE EXAM
FINAL EXAM CALCULUS 2 MATH 2300 FALL 208 Name PRACTICE EXAM Please answer all of the questions, and show your work. You must explain your answers to get credit. You will be graded on the clarity of your
More informationMath Exam 1a. c) lim tan( 3x. 2) Calculate the derivatives of the following. DON'T SIMPLIFY! d) s = t t 3t
Math 111 - Eam 1a 1) Evaluate the following limits: 7 3 1 4 36 a) lim b) lim 5 1 3 6 + 4 c) lim tan( 3 ) + d) lim ( ) 100 1+ h 1 h 0 h ) Calculate the derivatives of the following. DON'T SIMPLIFY! a) y
More informationMA 114 Worksheet #01: Integration by parts
Fall 8 MA 4 Worksheet Thursday, 3 August 8 MA 4 Worksheet #: Integration by parts. For each of the following integrals, determine if it is best evaluated by integration by parts or by substitution. If
More informationIndeterminate Forms and L Hospital s Rule
APPLICATIONS OF DIFFERENTIATION Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at certain points. INDETERMINATE FORM TYPE
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 informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.4 Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at
More informationMATH MIDTERM 4 - SOME REVIEW PROBLEMS WITH SOLUTIONS Calculus, Fall 2017 Professor: Jared Speck. Problem 1. Approximate the integral
MATH 8. - MIDTERM 4 - SOME REVIEW PROBLEMS WITH SOLUTIONS 8. Calculus, Fall 7 Professor: Jared Speck Problem. Approimate the integral 4 d using first Simpson s rule with two equal intervals and then the
More informationFinal Examination Solutions
Math. 5, Sections 5 53 (Fulling) 7 December Final Examination Solutions Test Forms A and B were the same except for the order of the multiple-choice responses. This key is based on Form A. Name: Section:
More informationMath 2260 Exam #2 Solutions. Answer: The plan is to use integration by parts with u = 2x and dv = cos(3x) dx: dv = cos(3x) dx
Math 6 Eam # Solutions. Evaluate the indefinite integral cos( d. Answer: The plan is to use integration by parts with u = and dv = cos( d: u = du = d dv = cos( d v = sin(. Then the above integral is equal
More information( ) ( ). ( ) " d#. ( ) " cos (%) " d%
Math 22 Fall 2008 Solutions to Homework #6 Problems from Pages 404-407 (Section 76) 6 We will use the technique of Separation of Variables to solve the differential equation: dy d" = ey # sin 2 (") y #
More informationMath 2250 Final Exam Practice Problem Solutions. f(x) = ln x x. 1 x. lim. lim. x x = lim. = lim 2
Math 5 Final Eam Practice Problem Solutions. What are the domain and range of the function f() = ln? Answer: is only defined for, and ln is only defined for >. Hence, the domain of the function is >. Notice
More informationFINAL EXAM CALCULUS 2. Name PRACTICE EXAM SOLUTIONS
FINAL EXAM CALCULUS MATH 00 FALL 08 Name PRACTICE EXAM SOLUTIONS Please answer all of the questions, and show your work. You must explain your answers to get credit. You will be graded on the clarity of
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.4 Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at
More information1 Exponential Functions Limit Derivative Integral... 5
Contents Eponential Functions 3. Limit................................................. 3. Derivative.............................................. 4.3 Integral................................................
More informationMath 122 Fall Handout 15: Review Problems for the Cumulative Final Exam
Math 122 Fall 2008 Handout 15: Review Problems for the Cumulative Final Exam The topics that will be covered on Final Exam are as follows. Integration formulas. U-substitution. Integration by parts. Integration
More informationANOTHER FIVE QUESTIONS:
No peaking!!!!! See if you can do the following: f 5 tan 6 sin 7 cos 8 sin 9 cos 5 e e ln ln @ @ Epress sin Power Series Epansion: d as a Power Series: Estimate sin Estimate MACLAURIN SERIES ANOTHER FIVE
More informationMath 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 information1 DL3. Infinite Limits and Limits at Infinity
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 78 Mark Sparks 01 Infinite Limits and Limits at Infinity In our graphical analysis of its, we have already seen both an infinite
More informationIn this note we will evaluate the limits of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0. f(x)
L Hôpital s Rule In this note we will evaluate the its of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0 f() Suppose a f() = 0 and a g() = 0. Then a g() the indeterminate
More informationWorksheet Week 7 Section
Worksheet Week 7 Section 8.. 8.4. This worksheet is for improvement of your mathematical writing skill. Writing using correct mathematical epression and steps is really important part of doing math. Please
More informationdy dx and so we can rewrite the equation as If we now integrate both sides of this equation, we get xy x 2 C Integrating both sides, we would have
LINEAR DIFFERENTIAL EQUATIONS A first-der linear differential equation is one that can be put into the fm 1 d Py Q where P and Q are continuous functions on a given interval. This type of equation occurs
More informationEx. Find the derivative. Do not leave negative exponents or complex fractions in your answers.
CALCULUS AB THE SECOND FUNDAMENTAL THEOREM OF CALCULUS AND REVIEW E. Find the derivative. Do not leave negative eponents or comple fractions in your answers. 4 (a) y 4 e 5 f sin (b) sec (c) g 5 (d) y 4
More informationTest 2 - Answer Key Version A
MATH 8 Student s Printed Name: Instructor: CUID: Section: Fall 27 8., 8.2,. -.4 Instructions: You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook,
More informationSection 7.4 #1, 5, 6, 8, 12, 13, 44, 53; Section 7.5 #7, 10, 11, 20, 22; Section 7.7 #1, 4, 10, 15, 22, 44
Math B Prof. Audrey Terras HW #4 Solutions Due Tuesday, Oct. 9 Section 7.4 #, 5, 6, 8,, 3, 44, 53; Section 7.5 #7,,,, ; Section 7.7 #, 4,, 5,, 44 7.4. Since 5 = 5 )5 + ), start with So, 5 = A 5 + B 5 +.
More informationDaily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 584 Mark Sparks 2012
The Second Fundamental Theorem of Calculus Functions Defined by Integrals Given the functions, f(t), below, use F( ) f ( t) dt to find F() and F () in terms of.. f(t) = 4t t. f(t) = cos t Given the functions,
More informationMath 162: Calculus IIA
Math 62: Calculus IIA Final Exam ANSWERS December 9, 26 Part A. (5 points) Evaluate the integral x 4 x 2 dx Substitute x 2 cos θ: x 8 cos dx θ ( 2 sin θ) dθ 4 x 2 2 sin θ 8 cos θ dθ 8 cos 2 θ cos θ dθ
More informationMath 181, Exam 2, Study Guide 2 Problem 1 Solution. 1 + dx. 1 + (cos x)2 dx. 1 + cos2 xdx. = π ( 1 + cos π 2
Math 8, Exam, Study Guide Problem Solution. Use the trapezoid rule with n to estimate the arc-length of the curve y sin x between x and x π. Solution: The arclength is: L b a π π + ( ) dy + (cos x) + cos
More informationWorksheet 9. Math 1B, GSI: Andrew Hanlon. 1 Ce 3t 1/3 1 = Ce 3t. 4 Ce 3t 1/ =
Worksheet 9 Math B, GSI: Andrew Hanlon. Show that for each of the following pairs of differential equations and functions that the function is a solution of a differential equation. (a) y 2 y + y 2 ; Ce
More informationThe stationary points will be the solutions of quadratic equation x
Calculus 1 171 Review In Problems (1) (4) consider the function f ( ) ( ) e. 1. Find the critical (stationary) points; establish their character (relative minimum, relative maimum, or neither); find intervals
More informationGraphing Rational Functions
Graphing Rational Functions Let s use all of the material we have developed to graph some rational functions EXAMPLE 37 Graph y = f () = +3 3 lude both vertical and horizontal asymptotes SOLUTION First
More informationMAT01B1: Separable Differential Equations
MAT01B1: Separable Differential Equations Dr Craig 3 October 2018 My details: acraig@uj.ac.za Consulting hours: Tomorrow 14h40 15h25 Friday 11h20 12h55 Office C-Ring 508 https://andrewcraigmaths.wordpress.com/
More informationVANDERBILT UNIVERSITY. MATH 2610 ORDINARY DIFFERENTIAL EQUATIONS Practice for test 1 solutions
VANDERBILT UNIVERSITY MATH 2610 ORDINARY DIFFERENTIAL EQUATIONS Practice for test 1 solutions The first test will cover all material discussed up to (including) section 4.5. Important: The solutions below
More informationMATH 104 Practice Problems for Exam 2
. Find the area between: MATH 4 Practice Problems for Eam (a) =, y = / +, y = / (b) y = e, y = e, = y = and the ais, for 4.. Calculate the volume obtained by rotating: (a) The region in problem a around
More informationLimits at Infinity. Use algebraic techniques to help with indeterminate forms of ± Use substitutions to evaluate limits of compositions of functions.
SUGGESTED REFERENCE MATERIAL: Limits at Infinity As you work through the problems listed below, you should reference Chapter. of the recommended textbook (or the equivalent chapter in your alternative
More informationSEE and DISCUSS the pictures on pages in your text. Key picture:
Math 6 Notes 1.1 A PREVIEW OF CALCULUS There are main problems in calculus: 1. Finding a tangent line to a curve though a point on the curve.. Finding the area under a curve on some interval. SEE and DISCUSS
More informationMath 1B Final Exam, Solution. Prof. Mina Aganagic Lecture 2, Spring (6 points) Use substitution and integration by parts to find:
Math B Final Eam, Solution Prof. Mina Aganagic Lecture 2, Spring 20 The eam is closed book, apart from a sheet of notes 8. Calculators are not allowed. It is your responsibility to write your answers clearly..
More informationWork the following on notebook paper. No calculator. Find the derivative. Do not leave negative exponents or complex fractions in your answers.
ALULUS B WORKSHEET ON 8. & REVIEW Find the derivative. Do not leave negative eponents or comple fractions in your answers. sec. f 8 7. f e. y ln tan. y cos tan. f 7. f cos. y 7 8. y log 7 Evaluate the
More informationMATH1120 Calculus II Solution to Supplementary Exercises on Improper Integrals Alex Fok November 2, 2013
() Solution : MATH Calculus II Solution to Supplementary Eercises on Improper Integrals Ale Fok November, 3 b ( + )( + tan ) ( + )( + tan ) +tan b du u ln + tan b ( = ln + π ) (Let u = + tan. Then du =
More informationMathematics 132 Calculus for Physical and Life Sciences 2 Exam 3 Review Sheet April 15, 2008
Mathematics 32 Calculus for Physical and Life Sciences 2 Eam 3 Review Sheet April 5, 2008 Sample Eam Questions - Solutions This list is much longer than the actual eam will be (to give you some idea of
More informationMATH 31B: MIDTERM 2 REVIEW. sin 2 x = 1 cos(2x) dx = x 2 sin(2x) 4. + C = x 2. dx = x sin(2x) + C = x sin x cos x
MATH 3B: MIDTERM REVIEW JOE HUGHES. Evaluate sin x and cos x. Solution: Recall the identities cos x = + cos(x) Using these formulas gives cos(x) sin x =. Trigonometric Integrals = x sin(x) sin x = cos(x)
More informationPTF #AB 07 Average Rate of Change
The average rate of change of f( ) over the interval following: 1. y dy d. f() b f() a b a PTF #AB 07 Average Rate of Change ab, can be written as any of the. Slope of the secant line through the points
More informationMath Applied Differential Equations
Math 256 - Applied Differential Equations Notes Basic Definitions and Concepts A differential equation is an equation that involves one or more of the derivatives (first derivative, second derivative,
More informationMath 2250 Final Exam Practice Problem Solutions. f(x) = ln x x. 1 x. lim. lim. x x = lim. = lim 2
Math 5 Final Eam Practice Problem Solutions. What are the domain and range of the function f() = ln? Answer: is only defined for, and ln is only defined for >. Hence, the domain of the function is >. Notice
More informationSection 2.5. Evaluating Limits Algebraically
Section 2.5 Evaluating Limits Algebraically (1) Determinate and Indeterminate Forms (2) Limit Calculation Techniques (A) Direct Substitution (B) Simplification (C) Conjugation (D) The Squeeze Theorem (3)
More informationContinuous functions. Limits of non-rational functions. Squeeze Theorem. Calculator issues. Applications of limits
Calculus Lia Vas Continuous functions. Limits of non-rational functions. Squeeze Theorem. Calculator issues. Applications of limits Continuous Functions. Recall that we referred to a function f() as a
More informationAPPM 1360 Final Exam Spring 2016
APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan
More information1. By the Product Rule, in conjunction with the Chain Rule, we compute the derivative as follows: and. So the slopes of the tangent lines to the curve
MAT 11 Solutions TH Eam 3 1. By the Product Rule, in conjunction with the Chain Rule, we compute the derivative as follows: Therefore, d 5 5 d d 5 5 d 1 5 1 3 51 5 5 and 5 5 5 ( ) 3 d 1 3 5 ( ) So the
More informationf(x) g(x) = [f (x)g(x) dx + f(x)g (x)dx
Chapter 7 is concerned with all the integrals that can t be evaluated with simple antidifferentiation. Chart of Integrals on Page 463 7.1 Integration by Parts Like with the Chain Rule substitutions with
More informationSolutions to Math 41 Final Exam December 9, 2013
Solutions to Math 4 Final Eam December 9,. points In each part below, use the method of your choice, but show the steps in your computations. a Find f if: f = arctane csc 5 + log 5 points Using the Chain
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 information( + ) 3. AP Calculus BC Chapter 6 AP Exam Problems. Antiderivatives. + + x + C. 2. If the second derivative of f is given by f ( x) = 2x cosx
Chapter 6 AP Eam Problems Antiderivatives. ( ) + d = ( + ) + 5 + + 5 ( + ) 6 ( + ). If the second derivative of f is given by f ( ) = cos, which of the following could be f( )? + cos + cos + + cos + sin
More informationMath 266, Midterm Exam 1
Math 266, Midterm Exam 1 February 19th 2016 Name: Ground Rules: 1. Calculator is NOT allowed. 2. Show your work for every problem unless otherwise stated (partial credits are available). 3. You may use
More informationRational Functions. A rational function is a function that is a ratio of 2 polynomials (in reduced form), e.g.
Rational Functions A rational function is a function that is a ratio of polynomials (in reduced form), e.g. f() = p( ) q( ) where p() and q() are polynomials The function is defined when the denominator
More informationAP Calculus Review Assignment Answer Sheet 1. Name: Date: Per. Harton Spring Break Packet 2015
AP Calculus Review Assignment Answer Sheet 1 Name: Date: Per. Harton Spring Break Packet 015 This is an AP Calc Review packet. As we get closer to the eam, it is time to start reviewing old concepts. Use
More information18.01 Final Answers. 1. (1a) By the product rule, (x 3 e x ) = 3x 2 e x + x 3 e x = e x (3x 2 + x 3 ). (1b) If f(x) = sin(2x), then
8. Final Answers. (a) By the product rule, ( e ) = e + e = e ( + ). (b) If f() = sin(), then f (7) () = 8 cos() since: f () () = cos() f () () = 4 sin() f () () = 8 cos() f (4) () = 6 sin() f (5) () =
More informationOld Math 330 Exams. David M. McClendon. Department of Mathematics Ferris State University
Old Math 330 Exams David M. McClendon Department of Mathematics Ferris State University Last updated to include exams from Fall 07 Contents Contents General information about these exams 3 Exams from Fall
More informationImproper Integrals. Goals:
Week #0: Improper Integrals Goals: Improper Integrals Improper Integrals - Introduction - Improper Integrals So far in our study of integration, we have dealt with functions that were always continuous
More informationIntegration Techniques for the BC exam
Integration Techniques for the B eam For the B eam, students need to: determine antiderivatives of the basic functions calculate antiderivatives of functions using u-substitution use algebraic manipulation
More informationIf you need more room, use the backs of the pages and indicate that you have done so.
Math 125 Final Exam Winter 2018 Your Name Your Signature Student ID # Quiz Section Professor s Name TA s Name Turn off and stow away all cell phones, watches, pagers, music players, and other similar devices.
More informationChapter 8 Indeterminate Forms and Improper Integrals Math Class Notes
Chapter 8 Indeterminate Forms and Improper Integrals Math 1220-004 Class Notes Section 8.1: Indeterminate Forms of Type 0 0 Fact: The it of quotient is equal to the quotient of the its. (book page 68)
More informationMath 2300 Calculus II University of Colorado
Math 3 Calculus II University of Colorado Spring Final eam review problems: ANSWER KEY. Find f (, ) for f(, y) = esin( y) ( + y ) 3/.. Consider the solid region W situated above the region apple apple,
More informationChapter 2 Section 3. Partial Derivatives
Chapter Section 3 Partial Derivatives Deinition. Let be a unction o two variables and. The partial derivative o with respect to is the unction, denoted b D1 1 such that its value at an point (,) in the
More information8.4 Integration of Rational Functions by Partial Fractions Lets use the following example as motivation: Ex: Consider I = x+5
Math 2-08 Rahman Week6 8.4 Integration of Rational Functions by Partial Fractions Lets use the following eample as motivation: E: Consider I = +5 2 + 2 d. Solution: Notice we can easily factor the denominator
More information1. Evaluate the integrals. a. (9 pts) x e x/2 dx. Solution: Using integration by parts, let u = x du = dx and dv = e x/2 dx v = 2e x/2.
MATH 8 Test -SOLUTIONS Spring 4. Evaluate the integrals. a. (9 pts) e / Solution: Using integration y parts, let u = du = and dv = e / v = e /. Then e / = e / e / e / = e / + e / = e / 4e / + c MATH 8
More informationPart I: Multiple Choice Mark the correct answer on the bubble sheet provided. n=1. a) None b) 1 c) 2 d) 3 e) 1, 2 f) 1, 3 g) 2, 3 h) 1, 2, 3
Math (Calculus II) Final Eam Form A Fall 22 RED KEY Part I: Multiple Choice Mark the correct answer on the bubble sheet provided.. Which of the following series converge absolutely? ) ( ) n 2) n 2 n (
More informationAnswer Key. ( 1) n (2x+3) n. n n=1. (2x+3) n. = lim. < 1 or 2x+3 < 4. ( 1) ( 1) 2n n
Math Midterm Eam #3 December, 3 Answer Key. [5 Points] Find the Interval and Radius of Convergence for the following power series. Analyze carefully and with full justification. Use Ratio Test. L lim a
More informationMAT12X Intermediate Algebra
MAT12X Intermediate Algebra Workshop 3 Rational Functions LEARNING CENTER Overview Workshop III Rational Functions General Form Domain and Vertical Asymptotes Range and Horizontal Asymptotes Inverse Variation
More informationSpring 2011 solutions. We solve this via integration by parts with u = x 2 du = 2xdx. This is another integration by parts with u = x du = dx and
Math - 8 Rahman Final Eam Practice Problems () We use disks to solve this, Spring solutions V π (e ) d π e d. We solve this via integration by parts with u du d and dv e d v e /, V π e π e d. This is another
More informationMATH 101 Midterm Examination Spring 2009
MATH Midterm Eamination Spring 9 Date: May 5, 9 Time: 7 minutes Surname: (Please, print!) Given name(s): Signature: Instructions. This is a closed book eam: No books, no notes, no calculators are allowed!.
More informationSolutions to Math 41 Exam 2 November 10, 2011
Solutions to Math 41 Eam November 10, 011 1. (1 points) Find each of the following its, with justification. If the it does not eist, eplain why. If there is an infinite it, then eplain whether it is or.
More informationSample Questions, Exam 1 Math 244 Spring 2007
Sample Questions, Exam Math 244 Spring 2007 Remember, on the exam you may use a calculator, but NOT one that can perform symbolic manipulation (remembering derivative and integral formulas are a part of
More informationAnswer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26.
Answer Key 969 BC 97 BC. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E.
More informationMATH 1207 R02 FINAL SOLUTION
MATH 7 R FINAL SOLUTION SPRING 6 - MOON Write your answer neatly and show steps. Except calculators, any electronic devices including laptops and cell phones are not allowed. () Let f(x) = x cos x. (a)
More informationMath 1000 Final Exam Review Solutions. (x + 3)(x 2) = lim. = lim x 2 = 3 2 = 5. (x + 1) 1 x( x ) = lim. = lim. f f(1 + h) f(1) (1) = lim
Math Final Eam Review Solutions { + 3 if < Consider f() Find the following limits: (a) lim f() + + (b) lim f() + 3 3 (c) lim f() does not eist Find each of the following limits: + 6 (a) lim 3 + 3 (b) lim
More informationf(r) = (r 1/2 r 1/2 ) 3 u = (ln t) ln t ln u = (ln t)(ln (ln t)) t(ln t) g (t) = t
Math 4, Autumn 006 Final Exam Solutions Page of 9. [ points total] Calculate the derivatives of the following functions. You need not simplfy your answers. (a) [4 points] y = 5x 7 sin(3x) + e + ln x. y
More informationDirections: Please read questions carefully. It is recommended that you do the Short Answer Section prior to doing the Multiple Choice.
AP Calculus AB SUMMER ASSIGNMENT Multiple Choice Section Directions: Please read questions carefully It is recommended that you do the Short Answer Section prior to doing the Multiple Choice Show all work
More informationMath 113 Fall 2005 key Departmental Final Exam
Math 3 Fall 5 key Departmental Final Exam Part I: Short Answer and Multiple Choice Questions Do not show your work for problems in this part.. Fill in the blanks with the correct answer. (a) The integral
More informationMath 1500 Fall 2010 Final Exam Review Solutions
Math 500 Fall 00 Final Eam Review Solutions. Verify that the function f() = 4 + on the interval [, 5] satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that
More informationAssignment 16 Assigned Weds Oct 11
Assignment 6 Assigned Weds Oct Section 8, Problem 3 a, a 3, a 3 5, a 4 7 Section 8, Problem 4 a, a 3, a 3, a 4 3 Section 8, Problem 9 a, a, a 3, a 4 4, a 5 8, a 6 6, a 7 3, a 8 64, a 9 8, a 0 56 Section
More informationUNIT 3. Rational Functions Limits at Infinity (Horizontal and Slant Asymptotes) Infinite Limits (Vertical Asymptotes) Graphing Rational Functions
UNIT 3 Rational Functions Limits at Infinity (Horizontal and Slant Asymptotes) Infinite Limits (Vertical Asymptotes) Graphing Rational Functions Recall From Unit Rational Functions f() is a rational function
More informationMATH 251 Examination I October 5, 2017 FORM A. Name: Student Number: Section:
MATH 251 Examination I October 5, 2017 FORM A Name: Student Number: Section: This exam has 13 questions for a total of 100 points. Show all your work! In order to obtain full credit for partial credit
More informationMath 116 Second Midterm November 14, 2012
Math 6 Second Midterm November 4, Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has pages including this cover. There are 8 problems. Note that
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 informationChapter 29 BC Calculus Practice Test
Chapter 9 BC Calculus Practice Test The Eam AP Calculus BC Eam SECTION I: Multiple-Choice Questions DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time hour and 5 minutes Number
More informationAP Calculus AB Sample Exam Questions Course and Exam Description Effective Fall 2016
P alculus Sample Eam Questions ourse and Eam escription Effective Fall 6 Section I, Part ( graphing calculator may not be used) Multiple hoice Questions.. 3.. 5. lim f( g( )) f(lim g( )) f() 3 7 sin lim
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 information