Name Class. 5. Find the particular solution to given the general solution y C cos x and the. x 2 y
|
|
- Moses Carter
- 6 years ago
- Views:
Transcription
1 10 Differential Equations Test Form A 1. Find the general solution to the first order differential equation: y 1 yy 0. 1 (a) (b) ln y 1 y ln y 1 C y y C y 1 C y 1 y C. Find the general solution to the first order differential equation: y d y 0. (a) (b) ln y y ln y y C C Cy y ln y C 3. In 1980 the population of a town was 1,000 and in 1990 it was 0,000. Assuming the population decreases continuously at a rate proportional to the eisting population, estimate the population in the year 010. (a) 17,619 (b) 18,000 19,048 18, A radioactive element has a half-life of 50 days. What percentage of the original sample is left after 60 days? (a) 43.53% (b) 49.56% 37.50% 5.00% y sin 5. Find the particular solution to given the general solution y C cos and the initial condition y 1. (a) cos (b) cos 1 cos 1 cos 6. Find the orthogonal trajectories for the family of curves y C 3. (a) y 3 K (b) 3y K 3ky 0 3y 7. The slope field for a differential equation is shown. Choose the equation that could be a particular solution to that differential equation. (a) y sin y (e) y ln (b) y 1 y e y
2 Test Bank Consider the differential equation with the initial condition y 0. Use Euler s method with h 0.01 to approimate y (a) (b) Find the general solution to the first-order differential equation tan y cos. d (a) y sec C (b) y ln y sec sin C ln y cos sin C y cos Ce sin L 10. The logistics differential equation produces y Find the value of b for the logistics 1 be kt. differential equation dp dt y y dt ky 1 y L 3P P given the initial condition 0, (a) 15 (b) (e) 79
3 104 Differential Equations Test Form B 1. Find the general solution to the first order differential equation: 4 y d 0. (a) y C 4 (b) 4y y y C y 4 C y 4 4 C. Find the general solution to the first order differential equation: y 3 y 3 d 0. (a) 3 4 8y 3 C (b) y 3 ln C y 3 3 y 3 ln 3 C y ln C 3. A certain type of bacteria increases continuously at a rate proportional to the number present. If there are 500 present at a given time and 1000 present hours later, how many will there be 5 hours from the initial time given? (a) 1750 (b) Choose the differential equation that matches the solution curves sketched in the slope field. 4 y (a) (e) y 1 y y y (b) y ln y Find the particular solution to given the general solution y sin A B and the initial conditions y 0, y. (a) sin 1 (b) sin sin 1 sin 1 6. Find the solution to the initial value problem e y e y with the initial condition y 0 0. (a) (b) y ln e 1 y ln e 1 y ln 1 e y sin y ln e 1
4 Test Bank Find the general solution of the differential equation: 1 tan y. d C (a) y (b) y C 1 1 cos y C 1 cos y C 1 8. Consider the differential equation with the initial condition y 0 1. Use Euler s method with h 0.1 to approimate y 0.3. (a) 1 (b) y y 9. Find the general solution to the first-order differential equation 1. d (a) y C 3 (b) y ln C 6y C y y ln 4 C 10. Solve the Bernoulli equation y y y5. (a) y Ce 1 4 (b) y Ce 1 4 y Ce 1 y 1 Ce 1
5 106 Differential Equations Test Form C 1. Find the particular solution to the differential equation given the general solution y Ce 3 and the initial condition y 1 0. (a) y 0e 3 3 (b) y 0e 3 y 0e y 0e y 3y. Find the general solution of the differential equation y y 0. (a) y ln C (b) y y C y C y C 3. Find the orthogonal trajectories for the family of curves y C 0. (a) (b) y ln y3 3 ln K Ky 4y K ln 0 y ln y K 4. A radioactive element has a half-life of 40 days. What percentage of the original sample is left after 48 days? (a) 49.56% (b) 43.53% 5.00% 37.50% 5. Determine which function is a solution to the differential equation 1 (a) (b) ln y e y y Determine whether the function f, y 3 y 4y y 3 is homogeneous, and if so, determine its degree. (a) Homogeneous; degree 1 (b) Homogeneous; degree Homogeneous; degree 3 Not homogeneous L 7. The logistics differential equation produces y Find the value of b for the logistics 1 be kt. dt ky 1 y L dp differential equation given the initial condition 0, 15. dt 7 P P (a) 14 (b)
6 Test Bank Consider the differential equation with the initial condition y 1 1. Use Euler s Method with h 0.1 to approimate y 1.3. y y (a) (b) Find the particular solution of the differential equation y y 4, y 0 4. (a) y 4 (b) y e y e y e 10. Solve the Bernoulli equation (a) (b) y y Ce Ce y 1 Ce 1 y y y 3. y 1 Ce 1
7 108 Differential Equations Test Form D 1. Find the general solution of the differential equation: cos y tan y 0. d. Find the general solution of the differential equation: y d y A certain type of bacteria increases continuously at a rate proportional to the number present. If there are 500 present at a given time and 1000 present hours later, how many hours (from the initial given time) will it take for the number of bacteria to be 500? Round your answer to decimal places. 4. Sketch a slope field for the differential equation at the points indicated on the graph below. d y y Verify that the equation y Ce is a solution to the differential equation y y Find the orthogonal trajectories of the family y C and sketch several members of each family. 7. Find the solution to the initial value problem: e y cos y y, y Consider the differential equation with the initial condition y 1 0. Use Euler s Method y y 1 with h 0.01 to approimate y Find the general solution to the first-order differential equation y y. 10. Find the particular solution of the linear differential equation y y sin sin, y.
8 Test Bank 109 Test Form E 1. Solve the differential equation: y y.. Find the general solution of the differential equation: 3. Find the particular solution of the differential equation 500 y that satisfies the initial d condition y 0 7. y e y. 4. The rate of change of y with respect to is inversely proportional to the square root of y. a. Write a differential equation for the given statement. b. Solve the differential equation in part a. L 5. The logistics differential equation produces y Find the logistics equation that for 1 be kt. dp dt 3P 5 dt ky 1 y L P that satisfies the initial conditions 0, Find the orthogonal trajectories of the family 4y C and sketch several members of each family. 7. Solve the homogenous differential equation: y y d Consider the differential equation with the initial condition y 1 1. Use Euler s Method y 1 y with h 0.1 to approimate y Solve, by any appropriate method, the first-order differential equation y y tan tan. 10. Find the particular solution to the first-order linear differential equation y y 1 with the initial condition y
( + ) 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 informationAP Exam Practice Questions for Chapter 6
AP Eam Practice Questions for Chapter 6 AP Eam Practice Questions for Chapter 6. To find which graph is a slope field for, 5 evaluate the derivative at selected points. At ( 0, ),.. 3., 0,. 5 At ( ) At
More informationChapter 6: Messy Integrals
Chapter 6: Messy Integrals Review: Solve the following integrals x 4 sec x tan x 0 0 Find the average value of 3 1 x 3 3 Evaluate 4 3 3 ( x 1), then find the area of ( x 1) 4 Section 6.1: Slope Fields
More informationdx. Ans: y = tan x + x2 + 5x + C
Chapter 7 Differential Equations and Mathematical Modeling If you know one value of a function, and the rate of change (derivative) of the function, then yu can figure out many things about the function.
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Chapter Practice Test Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the general solution to the eact differential equation. ) dy dt =
More informationAP Exam Practice Questions for Chapter 5
AP Eam Practice Questions for Chapter 5 AP Eam Practice Questions for Chapter 5 d. To find which graph is a slope field for, 5 evaluate the derivative at selected points. d At ( 0, ),. d At (, 0 ),. 5
More informationExam 1 Review: Questions and Answers. Part I. Finding solutions of a given differential equation.
Exam 1 Review: Questions and Answers Part I. Finding solutions of a given differential equation. 1. Find the real numbers r such that y = e x is a solution of y y 30y = 0. Answer: r = 6, 5 2. Find the
More informationCalculus II Practice Test Questions for Chapter , 9.6, Page 1 of 9
Calculus II Practice Test Questions for Chapter 9.1 9.4, 9.6, 10.1 10.4 Page 1 of 9 This is in no way an inclusive set of problems there can be other types of problems on the actual test. To prepare for
More information3. Identify and find the general solution of each of the following first order differential equations.
Final Exam MATH 33, Sample Questions. Fall 6. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C is the general solution of a differential
More informationSolutionbank Edexcel AS and A Level Modular Mathematics
Page of Exercise A, Question The curve C, with equation y = x ln x, x > 0, has a stationary point P. Find, in terms of e, the coordinates of P. (7) y = x ln x, x > 0 Differentiate as a product: = x + x
More informationName Class. (a) (b) (c) 4 t4 3 C
Chapter 4 Test Bank 77 Test Form A Chapter 4 Name Class Date Section. Evaluate the integral: t dt. t C (a) (b) 4 t4 C t C C t. Evaluate the integral: 5 sec x tan x dx. (a) 5 sec x tan x C (b) 5 sec x C
More information9.3: Separable Equations
9.3: Separable Equations An equation is separable if one can move terms so that each side of the equation only contains 1 variable. Consider the 1st order equation = F (x, y). dx When F (x, y) = f (x)g(y),
More informationRegent College Maths Department. Core Mathematics 4 Trapezium Rule. C4 Integration Page 1
Regent College Maths Department Core Mathematics Trapezium Rule C Integration Page Integration It might appear to be a bit obvious but you must remember all of your C work on differentiation if you are
More informationNO CALCULATORS: 1. Find A) 1 B) 0 C) D) 2. Find the points of discontinuity of the function y of discontinuity.
AP CALCULUS BC NO CALCULATORS: MIDTERM REVIEW 1. Find lim 7x 6x x 7 x 9. 1 B) 0 C) D). Find the points of discontinuity of the function y of discontinuity. x 9x 0. For each discontinuity identify the type
More information90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y.
90 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions Test Form A Chapter 5 Name Class Date Section. Find the derivative: f ln. 6. Differentiate: y. ln y y y y. Find dy d if ey y. y
More informationAP Calculus Testbank (Chapter 6) (Mr. Surowski)
AP Calculus Testbank (Chapter 6) (Mr. Surowski) Part I. Multiple-Choice Questions 1. Suppose that f is an odd differentiable function. Then (A) f(1); (B) f (1) (C) f(1) f( 1) (D) 0 (E). 1 1 xf (x) =. The
More information2. Find the value of y for which the line through A and B has the given slope m: A(-2, 3), B(4, y), 2 3
. Find an equation for the line that contains the points (, -) and (6, 9).. Find the value of y for which the line through A and B has the given slope m: A(-, ), B(4, y), m.. Find an equation for the line
More information(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)
. f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula
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 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 information( 4. AP Exam Practice Questions for Chapter 7. AP Exam Practice Questions for Chapter 7 1 = = x dx. 1 3x So, the answer is A.
AP Eam Practice Questions for Chapter 7 AP Eam Practice Questions for Chapter 7. e e So, the answer is A. e e ( ) A e e d e e e e. 7 + + (, ) + (, ) (, ) 7 + + + 7 + 7 + ( ) ( ),, A d + + + + + + + d +
More informationExact Differential Equations. The general solution of the equation is f x, y C. If f has continuous second partials, then M y 2 f
APPENDIX C Additional Topics in Differential Equations APPENDIX C. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Chapter 6, ou studied applications
More information= f (x ), recalling the Chain Rule and the fact. dx = f (x )dx and. dx = x y dy dx = x ydy = xdx y dy = x dx. 2 = c
Separable Variables, differential equations, and graphs of their solutions This will be an eploration of a variety of problems that occur when stuing rates of change. Many of these problems can be modeled
More informationNO CALCULATORS: 1. Find A) 1 B) 0 C) D) 2. Find the points of discontinuity of the function y of discontinuity.
AP CALCULUS BC NO CALCULATORS: MIDTERM REVIEW. Find lim 7 7 9. B) C) D). Find the points of discontinuit of the function of discontinuit. 9. For each discontinuit identif the tpe A. Removable discontinuit
More information5.1 Separable Differential Equations
5.1 Separable Differential Equations A differential equation is an equation that has one or more derivatives in it. The order of a differential equation is the highest derivative present in the equation.
More informationOrdinary Differential Equations (ODEs)
c01.tex 8/10/2010 22: 55 Page 1 PART A Ordinary Differential Equations (ODEs) Chap. 1 First-Order ODEs Sec. 1.1 Basic Concepts. Modeling To get a good start into this chapter and this section, quickly
More informationSection Differential Equations: Modeling, Slope Fields, and Euler s Method
Section.. Differential Equations: Modeling, Slope Fields, and Euler s Method Preliminar Eample. Phsical Situation Modeling Differential Equation An object is taken out of an oven and placed in a room where
More informationApplications of First Order Differential Equation
Dr Mansoor Alshehri King Saud University MATH204-Differential Equations Center of Excellence in Learning and Teaching 1 / 39 Orthogonal Trajectories How to Find Orthogonal Trajectories Growth and Decay
More informationMath 2214 Solution Test 1D Spring 2015
Math 2214 Solution Test 1D Spring 2015 Problem 1: A 600 gallon open top tank initially holds 300 gallons of fresh water. At t = 0, a brine solution containing 3 lbs of salt per gallon is poured into the
More informationSeparable Differential Equations
-1- F D2f0B1I3F sknuwtqa RSAo9fjt3wdaircei TLgL4CK.w j EAMlilW mroidghttso nr3efsgefrpv0eado.q T 9MdaFdQe5 UwDibtuh8 UI0nMf6i3nZ itez scjaaljcduclguesb.k Worksheet b Kuta Software LLC Kuta Software - Infinite
More informationSlope Fields and Differential Equations
Student Stud Session Slope Fields and Differential Equations Students should be able to: Draw a slope field at a specified number of points b hand. Sketch a solution that passes through a given point on
More informationMath 2300 Calculus II University of Colorado Final exam review problems
Math 300 Calculus II University of Colorado Final exam review problems. A slope field for the differential equation y = y e x is shown. Sketch the graphs of the solutions that satisfy the given initial
More informationFirst Order Linear Ordinary Differential Equations
First Order Linear Ordinary Differential Equations The most general first order linear ODE is an equation of the form p t dy dt q t y t f t. 1 Herepqarecalledcoefficients f is referred to as the forcing
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) 94 C) ) A) 1 2
Chapter Calculus MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the average rate of change of the function over the given interval. ) = 73-5
More informationName Date. Show all work! Exact answers only unless the problem asks for an approximation.
Advanced Calculus & AP Calculus AB Summer Assignment Name Date Show all work! Eact answers only unless the problem asks for an approimation. These are important topics from previous courses that you must
More informationQuestions. x 2 e x dx. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the functions g(x) = x cost2 dt.
Questions. Evaluate the Riemann sum for f() =,, with four subintervals, taking the sample points to be right endpoints. Eplain, with the aid of a diagram, what the Riemann sum represents.. If f() = ln,
More informationDifferential Equations: Homework 2
Differential Equations: Homework Alvin Lin January 08 - May 08 Section.3 Exercise The direction field for provided x 0. dx = 4x y is shown. Verify that the straight lines y = ±x are solution curves, y
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. D) u = - x15 cos (x15) + C
AP Calculus AB Exam Review Differential Equations and Mathematical Modelling MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the general solution
More informationArkansas Council of Teachers of Mathematics 2013 State Contest Calculus Exam
0 State Contest Calculus Eam In each of the following choose the BEST answer and shade the corresponding letter on the Scantron Sheet. Answer all multiple choice questions before attempting the tie-breaker
More informationMath 132 Information for Test 2
Math 13 Information for Test Test will cover material from Sections 5.6, 5.7, 5.8, 6.1, 6., 6.3, 7.1, 7., and 7.3. The use of graphing calculators will not be allowed on the test. Some practice questions
More informationChapter 5 Review. 1. [No Calculator] Evaluate using the FTOC (the evaluation part) 2. [No Calculator] Evaluate using geometry
AP Calculus Chapter Review Name: Block:. [No Calculator] Evaluate using the FTOC (the evaluation part) a) 7 8 4 7 d b) 9 4 7 d. [No Calculator] Evaluate using geometry a) d c) 6 8 d. [No Calculator] Evaluate
More informationTOTAL NAME DATE PERIOD AP CALCULUS AB UNIT 4 ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT /6 10/8 10/9 10/10 X X X X 10/11 10/12
NAME DATE PERIOD AP CALCULUS AB UNIT ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT 0 0 0/6 0/8 0/9 0/0 X X X X 0/ 0/ 0/5 0/6 QUIZ X X X 0/7 0/8 0/9 0/ 0/ 0/ 0/5 UNIT EXAM X X X TOTAL AP Calculus
More information1 Differential Equations
Reading [Simon], Chapter 24, p. 633-657. 1 Differential Equations 1.1 Definition and Examples A differential equation is an equation involving an unknown function (say y = y(t)) and one or more of its
More informationdy x a. Sketch the slope field for the points: (1,±1), (2,±1), ( 1, ±1), and (0,±1).
Chapter 6. d x Given the differential equation: dx a. Sketch the slope field for the points: (,±), (,±), (, ±), and (0,±). b. Find the general solution for the given differential equation. c. Find the
More informationExtra Practice Recovering C
Etra Practice Recovering C 1 Given the second derivative of a function, integrate to get the first derivative, then again to find the equation of the original function. Use the given initial conditions
More information1. The following problems are not related: (a) (15 pts, 5 pts ea.) Find the following limits or show that they do not exist: arcsin(x)
APPM 5 Final Eam (5 pts) Fall. The following problems are not related: (a) (5 pts, 5 pts ea.) Find the following limits or show that they do not eist: (i) lim e (ii) lim arcsin() (b) (5 pts) Find and classify
More informationDifferential Equations & Separation of Variables
Differential Equations & Separation of Variables SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference Chapter 8. of the recommended textbook (or the equivalent
More information1998 AP Calculus AB: Section I, Part A
998 AP Calculus AB: 55 Minutes No Calculator Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number.. What is the -coordinate
More informationdy dx 1. If y 2 3xy = 18, then at the point H1, 3L is HAL 1 HBL 0 HCL 1 HDL 4 HEL 8 kx + 8 k + x The value of k is
. If = 8, then d d at the point H, L is 0 HCL HDL HEL 8. The equation of the line tangent to the curve = k + 8 k + The value of k is at = is = +. HCL HDL HEL. If f HL = and f H L =, then find f HL d 0
More informationChapters 8.1 & 8.2 Practice Problems
EXPECTED SKILLS: Chapters 8.1 & 8. Practice Problems Be able to verify that a given function is a solution to a differential equation. Given a description in words of how some quantity changes in time
More informationQuestions. x 2 e x dx. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the functions g(x) = x cost2 dt.
Questions. Evaluate the Riemann sum for f() =,, with four subintervals, taking the sample points to be right endpoints. Eplain, with the aid of a diagram, what the Riemann sum represents.. If f() = ln,
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 information3. Identify and find the general solution of each of the following first order differential equations.
Final Exam MATH 33, Sample Questions. Fall 7. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C x is the general solution of a differential
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 informationDifferentiating Functions & Expressions - Edexcel Past Exam Questions
- Edecel Past Eam Questions. (a) Differentiate with respect to (i) sin + sec, (ii) { + ln ()}. 5-0 + 9 Given that y =, ¹, ( -) 8 (b) show that = ( -). (6) June 05 Q. f() = e ln, > 0. (a) Differentiate
More informationHomework 3. (33-40) The graph of an exponential function is given. Match each graph to one of the following functions.
Homework Section 4. (-40) The graph of an exponential function is given. Match each graph to one of the following functions. (a)y = x (b)y = x (c)y = x (d)y = x (e)y = x (f)y = x (g)y = x (h)y = x (46,
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 informationC3 A Booster Course. Workbook. 1. a) Sketch, on the same set of axis the graphs of y = x and y = 2x 3. (3) b) Hence, or otherwise, solve the equation
C3 A Booster Course Workbook 1. a) Sketch, on the same set of axis the graphs of y = x and y = 2x 3. b) Hence, or otherwise, solve the equation x = 2x 3 (3) (4) BlueStar Mathematics Workshops (2011) 1
More informationMath 231 Final Exam Review
Math Final Eam Review Find the equation of the line tangent to the curve 4y y at the point (, ) Find the slope of the normal line to y ) ( e at the point (,) dy Find d if cos( y) y 4 y 4 Find the eact
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 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 informationCHAPTER 6 Differential Equations
CHAPTER 6 Differential Equations Section 6. Slope Fields and Euler s Method.............. 55 Section 6. Differential Equations: Growth and Deca........ 557 Section 6. Separation of Variables and the Logistic
More information2413 Exam 3 Review. 14t 2 Ë. dt. t 6 1 dt. 3z 2 12z 9 z 4 8 Ë. n 7 4,4. Short Answer. 1. Find the indefinite integral 9t 2 ˆ
3 Eam 3 Review Short Answer. Find the indefinite integral 9t ˆ t dt.. Find the indefinite integral of the following function and check the result by differentiation. 6t 5 t 6 dt 3. Find the indefinite
More informationcos 5x dx e dt dx 20. CALCULUS AB WORKSHEET ON SECOND FUNDAMENTAL THEOREM AND REVIEW Work the following on notebook paper. No calculator.
WORKSHEET ON SECOND FUNDAMENTAL THEOREM AND REVIEW Work the following on notebook paper. No calculator. Find the derivative. Do not leave negative eponents or comple fractions in our answers. 4. 8 4 f
More informationAP Calculus (BC) Summer Assignment (169 points)
AP Calculus (BC) Summer Assignment (69 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion
More informationSlope Fields and Differential Equations
Student Stud Session Slope Fields and Differential Equations Students should be able to: Draw a slope field at a specified number of points b hand. Sketch a solution that passes through a given point on
More informationdx dx x sec tan d 1 4 tan 2 2 csc d 2 ln 2 x 2 5x 6 C 2 ln 2 ln x ln x 3 x 2 C Now, suppose you had observed that x 3
CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 8. Partial Fractions Understand the concept of a partial fraction decomposition. Use partial fraction decomposition with
More information1 Exam 1 Spring 2007.
Exam Spring 2007.. An object is moving along a line. At each time t, its velocity v(t is given by v(t = t 2 2 t 3. Find the total distance traveled by the object from time t = to time t = 5. 2. Use the
More informationSolutions Of Homework 4
Solutions Of Homework 1. Two parallel sides of a rectangle are being lengthened at the rate of 3 in/sec, while the other two sides are shortened in such a way that the figure remains a rectangle with constant
More informationUnit 5 Applications of Antidifferentiation
Warmup 1) If f ( ) cos(ln ) for > 0, then f () (a) sin(ln ) (b) sin(ln ) (c) sin(ln ) (d) sin(ln ) (e) ln sin 2) If f ( ) 2, then f () (a) 2 ( ln 2) (b) 2 (1 ln 2) (c) 2 ln 2 (d) 2 (1 ln 2) (e) 2 (1 ln
More informationAP Calculus Multiple Choice Questions - Chapter 7
Find the general solution to the eact differential equation dy/d = t cos(t ) a y = cos(6t) + b y = sin(6t) + c y = cos(t ) + d y = sin(t ) + Find the general solution to the eact differential equation
More informationFirst Order Differential Equations
Chapter 2 First Order Differential Equations Introduction Any first order differential equation can be written as F (x, y, y )=0 by moving all nonzero terms to the left hand side of the equation. Of course,
More informationModeling with Differential Equations
Modeling with Differential Equations 1. Exponential Growth and Decay models. Definition. A quantity y(t) is said to have an exponential growth model if it increases at a rate proportional to the amount
More information1998 AP Calculus AB: Section I, Part A
55 Minutes No Calculator Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number.. What is the -coordinate of the point
More informationSMA 208: Ordinary differential equations I
SMA 208: Ordinary differential equations I Modeling with First Order differential equations Lecturer: Dr. Philip Ngare (Contacts: pngare@uonbi.ac.ke, Tue 12-2 PM) School of Mathematics, University of Nairobi
More information(x! 4) (x! 4)10 + C + C. 2 e2x dx = 1 2 (1 + e 2x ) 3 2e 2x dx. # 8 '(4)(1 + e 2x ) 3 e 2x (2) = e 2x (1 + e 2x ) 3 & dx = 1
33. x(x - 4) 9 Let u = x - 4, then du = and x = u + 4. x(x - 4) 9 = (u + 4)u 9 du = (u 0 + 4u 9 )du = u + 4u0 0 = (x! 4) + 2 5 (x! 4)0 (x " 4) + 2 5 (x " 4)0 ( '( = ()(x - 4)0 () + 2 5 (0)(x - 4)9 () =
More informationfor any C, including C = 0, because y = 0 is also a solution: dy
Math 3200-001 Fall 2014 Practice exam 1 solutions 2/16/2014 Each problem is worth 0 to 4 points: 4=correct, 3=small error, 2=good progress, 1=some progress 0=nothing relevant. If the result is correct,
More informationC3 Revision and Exam Answers: Simpson s Rule
C3 Revision and Exam Answers: Simpson s Rule Simpson s Rule is a way of accurately finding the area under a curve It is more accurate than the Trapezium Rule which we have seen before You start it the
More information2018 Pre-Cal Spring Semester Review Name: Per:
08 Pre-Cal Spring Semester Review Name: Per: For # 4, find the domain of each function. USE INTERVAL NOTATION!!. 4 f ( ) 5. f ( ) 6 5. f( ) 5 4. f( ) 4 For #5-6, find the domain and range of each graph.
More informationENGI 3424 Mid Term Test Solutions Page 1 of 9
ENGI 344 Mid Term Test 07 0 Solutions Page of 9. The displacement st of the free end of a mass spring sstem beond the equilibrium position is governed b the ordinar differential equation d s ds 3t 6 5s
More informationAP Calculus (BC) Summer Assignment (104 points)
AP Calculus (BC) Summer Assignment (0 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion
More informationA MATH 1225 Practice Test 4 NAME: SOLUTIONS CRN:
A MATH 5 Practice Test 4 NAME: SOLUTIONS CRN: Multiple Choice No partial credit will be given. Clearly circle one answer. No calculator!. Which of the following must be true (you may select more than one
More informationKey- Math 231 Final Exam Review
Key- Math Final Eam Review Find the equation of the line tangent to the curve y y at the point (, ) y-=(-/)(-) Find the slope of the normal line to y ) ( e at the point (,) dy Find d if cos( y) y y (ysiny+y)/(-siny-y^-^)
More informationName: MA 160 Dr. Katiraie (100 points) Test #3 Spring 2013
Name: MA 160 Dr. Katiraie (100 points) Test #3 Spring 2013 Show all of your work on the test paper. All of the problems must be solved symbolically using Calculus. You may use your calculator to confirm
More informationDifferential Equations
Differential Equations Big Ideas Slope fields draw a slope field, sketch a particular solution Separation of variables separable differential equations General solution Particular solution Growth decay
More informationUNIVERSITI TEKNOLOGI MALAYSIA SSE 1893 ENGINEERING MATHEMATICS TUTORIAL Determine the domain and range for each of the following functions.
UNIVERSITI TEKNOLOGI MALAYSIA SSE 1893 ENGINEERING MATHEMATICS TUTORIAL 1 1 Determine the domain and range for each of the following functions a = + b = 1 c = d = ln( ) + e = e /( 1) Sketch the level curves
More informationPartial Fractions. dx dx x sec tan d 1 4 tan 2. 2 csc d. csc cot C. 2x 5. 2 ln. 2 x 2 5x 6 C. 2 ln. 2 ln x
460_080.qd //04 :08 PM Page CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 8. Partial Fractions Understand the concept of a partial fraction decomposition. Use partial
More informationMath Spring 2014 Homework 2 solution
Math 3-00 Spring 04 Homework solution.3/5 A tank initially contains 0 lb of salt in gal of weater. A salt solution flows into the tank at 3 gal/min and well-stirred out at the same rate. Inflow salt concentration
More informationLog1 Contest Round 2 Theta Logarithms & Exponents. 4 points each
5 Log Contest Round Theta Logarithms & Eponents Name: points each Simplify: log log65 log6 log6log9 log5 Evaluate: log Find the sum:... A square has a diagonal whose length is feet, enclosed by the square.
More informationFind the volume of the solid generated by revolving the shaded region about the given axis. Use the disc/washer method 1) About the x-axis
Final eam practice for Math 6 Disclaimer: The actual eam is different Find the volume of the solid generated b revolving the shaded region about the given ais. Use the disc/washer method ) About the -ais
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Sketch the region bounded between the given curves and then find the area of the region. ) y =, y = )
More information(A) when x = 0 (B) where the tangent line is horizontal (C) when f '(x) = 0 (D) when there is a sharp corner on the graph (E) None of the above
AP Physics C - Problem Drill 10: Differentiability and Rules of Differentiation Question No. 1 of 10 Question 1. A derivative does not eist Question #01 (A) when 0 (B) where the tangent line is horizontal
More informationFP1 PAST EXAM QUESTIONS ON NUMERICAL METHODS: NEWTON-RAPHSON ONLY
FP PAST EXAM QUESTIONS ON NUMERICAL METHODS: NEWTON-RAPHSON ONLY A number of questions demand that you know derivatives of functions now not included in FP. Just look up the derivatives in the mark scheme,
More informationName Date Period. Worksheet 5.5 Partial Fractions & Logistic Growth Show all work. No calculator unless stated. Multiple Choice
Name Date Period Worksheet 5.5 Partial Fractions & Logistic Growth Show all work. No calculator unless stated. Multiple Choice 1. The spread of a disease through a community can be modeled with the logistic
More information6.1 Antiderivatives and Slope Fields Calculus
6. Antiderivatives and Slope Fields Calculus 6. ANTIDERIVATIVES AND SLOPE FIELDS Indefinite Integrals In the previous chapter we dealt with definite integrals. Definite integrals had limits of integration.
More informationFind the rectangular coordinates for each of the following polar coordinates:
WORKSHEET 13.1 1. Plot the following: 7 3 A. 6, B. 3, 6 4 5 8 D. 6, 3 C., 11 2 E. 5, F. 4, 6 3 Find the rectangular coordinates for each of the following polar coordinates: 5 2 2. 4, 3. 8, 6 3 Given the
More informationIntegration Techniques for the AB exam
For the AB eam, students need to: determine antiderivatives of the basic functions calculate antiderivatives of functions using u-substitution use algebraic manipulation to rewrite the integrand prior
More informationTrigonometric Identities Exam Questions
Trigonometric Identities Exam Questions Name: ANSWERS January 01 January 017 Multiple Choice 1. Simplify the following expression: cos x 1 cot x a. sin x b. cos x c. cot x d. sec x. Identify a non-permissible
More informationPaper Reference. Core Mathematics C3 Advanced. Thursday 11 June 2009 Morning Time: 1 hour 30 minutes. Mathematical Formulae (Orange or Green)
Centre No. Candidate No. Paper Reference(s) 6665/01 Edecel GCE Core Mathematics C3 Advanced Thursday 11 June 009 Morning Time: 1 hour 30 minutes Materials required for eamination Mathematical Formulae
More information( ) 9 b) y = x x c) y = (sin x) 7 x d) y = ( x ) cos x
NYC College of Technology, CUNY Mathematics Department Spring 05 MAT 75 Final Eam Review Problems Revised by Professor Africk Spring 05, Prof. Kostadinov, Fall 0, Fall 0, Fall 0, Fall 0, Fall 00 # Evaluate
More information