dy dx dx = 7 1 x dx dy = 7 1 x dx e u du = 1 C = 0
|
|
- Marvin Fowler
- 5 years ago
- Views:
Transcription
1 1. = 6x = 6x = 6 x = 6 x x 2 y = C = 3x2 + C General solution: y = 3x 2 + C 3. = 7 x = 7 1 x = 7 1 x General solution: y = 7 ln x + C. = e.2x = e.2x = e.2x (u =.2x, du =.2) y = e u 1.2 du = 1 e u du = eu + C = e.2x + C General solution: y = e.2x + C 7. = x2 - x; y() = = (x2 - x) y = 1 3 x3-1 2 x2 + C Given y() = : 1 3 ()3-1 2 ()2 + C = Particular solution: y = 1 3 x3 9. = -2xe-x2 ; y() = 3 = -2xe -x2 y = -2xe -x2 C = x2 Let u = -x 2, then du = -2x and -2xe -x2 = e u du = e u + c = e -x2 Thus, y = e -x2 + c Given y() = 3: 3 = e + c 3 = 1 + c c = 2 Particular solution: y = e -x c EXERCISE
2 11. = x ; y() = = x = x 1 = 2 (u = 1 + x, du = ) 1 + x y = 2 1 du = 2 ln u + C = 2 ln 1 + x + C u Given y() = : = 2 ln 1 + C = C Particular solution: y = 2 ln 1 + x Figure (B). When x = 1, = 1-1 = for any y. When x =, = - 1 = -1 for any y. When x = 2, = 2-1 = 1 for any y; and so on. These facts are consistent with the slope-field in Figure (B); they are not consistent with the slope-field in Figure (A). 1. = x = (x - 1) General solution: y = 1 2 x2 - x + C Given y() = -2: 1 2 ()2 - + C = -2 C = -2 Particular solution: y = 1 2 x2 - x y - x 19. = 2y 1 y = 2 1 y = 2 so 1 u du = 2 [u = y, du = = ] ln u = 2t + K [K an arbitrary constant] u = e 2t+K = e K e 2t u = Ce 2t [C = e K, C > ] y = Ce 2t 38 CHAPTER 6 INTEGRATION
3 21. Now, if we set y(t) = Ce 2t, C ANY constant, then y'(t) = 2Ce 2t = 2y(t), So y = Ce 2t satisfies the differential equation where C is any constant. This is the general solution. Note, the differential equation is the model for exponential growth with growth rate 2. = -.y, y() = 1 1 y = -. 1 y = -. 1 u du = -. [u = y, du = = ] ln u = -.x + K u = e -.x+k = e K e -.x y = Ce -.x, C = e K >. So, general solution: y = Ce -.x, C any constant. Given y() = 1: 1 = Ce = C; particular solution: y = 1e -.x 23. = -x 1 x = - 1 x = - 1 x = - ln x = -t + K x = e -t+k = e K e -t = Ce -t, C = e K >. General solution: x = Ce -t, C any constant. 2. = -t = -t = - t General solution: x = - t2 2 + C EXERCISE
4 27. Figure (A). When y = 1, = 1-1 = for any x. When y = 2, = 1-2 = -1 for any x; and so on. This is consistent with the slope-field in Figure (A); it is not consistent with the slope-field in Figure (B). 29. y = 1 - Ce -x = d [1 - Ce-x ] = Ce -x From the original equation, Ce -x = 1 - y Thus, we have = 1 - y and y = 1 - Ce -x is a solution of the differential equation for any number c. Given y() = : = 1 - Ce = 1 - c c = 1 Particular solution: y = 1 - e -x 31. y x y = C " x 2 = (C x 2 ) 1/2 = 1 2 (C x2 ) -1/2 "x (-2x) = = - x C " x 2 y Thus, y = C " x 2 satisfies the differential equation = - x y. Setting x = 3, y = 4, gives 4 = C " 9 16 = C 9 C = 2 The solution that passes through (3, 4) is y = 2 " x CHAPTER 6 INTEGRATION
5 37. y = Cx; C = y x = C = y x Therefore y = Cx satisfies the differential equation = y x. Setting x = -8, y = 24 gives 24 = -8C, C = -3 The solution that passes through (-8, 24) is y = -3x y = 1 + ce "t = e t e t (multiply numerator and denominator by e t ) + c = (et + c)e t " e t (e t ) ce t (e t + c) 2 = (e t + c) 2 " e t % # y(1 y) = $ # e t ' + c 1 " e t & % & $ e t ( + c = e t ' e t + c c e t + c = ce t (e t + c) 2 Thus = y(1 y) and y = 1 satisfies the differential "t 1 + ce equation. Setting t =, y = -1 gives -1 = 1, -1 c = 1, c = c The solution that passes through (, -1) is 1 y = 1 " 2e "t 41. y = 1,e.8t t 1, y 3, p = 1e -.x x 3, p N = 1(1 - e -.t ) t 1, N 1 1 1, 47. N = e!.4t t 4, N 1, EXERCISE
6 49. = ky(m - y), k, M positive constants. Set f(y) = ky(m - y) = kmy - ky2. This is a quadratic function which opens downward; it has a maximum value. Now f'(y) = km - 2ky Critical value: km - 2ky = y = M 2 f"(y) = -2k <. Thus, f has a maximum value at y = M In 1967: dq = 3.e In 1999: dq = 6e The rate of growth in 1999 is almost twice the rate of growth in da =.8A and A() = 1, is an unlimited growth model. From 4, the amount in the account after t years is: A(t) = 1e.8t.. da = ra, A() = 8, is an unlimited growth model. From 4, A(t) = 8,e rt. Since A(2) = 9,2, we solve 8,e 2r = 9,2 for r. 8e 2r = 9,2 e 2r = r = ln(92/8) ln(92 8) r =.6 2 Thus, A(t) = 8,e.6t. 7. (A) dp = rp, p() = 1 This is an UNLIMITED GROWTH MODEL. From 4, p(x) = 1e rx. Since p() = 77.88, we have = 1e r e r =.7788 r = ln(.7788) r = ln(.7788) -. Thus, p(x) = 1e -.x. (B) p(1) = 1e -.(1) = 1e -. $6.6 per unit (C) 384 CHAPTER 6 INTEGRATION
7 9. (A) dn = k(l - N); N() = This is a LIMITED GROWTH MODEL. From 4, N(t) = L(1 - e -kt ). Since N(1) =.4L, we have.4l = L(1 - e -1k ) 1 - e -1k =.4 e -1k =.6-1k = ln(.6) k = ln(.6).1!1 Thus, N(t) = L(1 - e -.1t ). (B) N() = L[1 - e -.1() ] = L[1 - e -.2 ].22L Approximately 22.% of the possible viewers will have been exposed after days. (C) Solve L(1 - e -.1t ) =.8L for t: 1 - e -.1t =.8 e -.1t =.2 -.1t = ln(.2) t = ln(.2)! It will take 32 days for 8% of the possible viewers to be exposed. (D) 61. di = -ki, I() = I This is an exponential decay model. From 4, I(x) = I e -kx with k =.942, we have I(x) = I e -.942x To find the depth at which the light is reduced to half of that at the surface, solve, I e -.942x = 1 2 I for x: e -.942x = x = ln(.) x = ln(.)! feet EXERCISE
8 63. dq = -.4Q, Q() = Q. (A) This is a model for exponential decay. From 4, Q(t) = Q e -.4t With Q = 3, we have Q(t) = 3e -.4t (B) Q(1) = 3e -.4(1) = 3e There are approximately 2.1 milliliters in the bo after 1 hours. (C) 3e -.4t = 1 (D) 3 e -.4t = t = ln(1/3) ln(1 3) t =! It will take approximately hours for Q to decrease to 1 milliliter Using the exponential decay model, we have = -ky, y() = 1, k > where y = y(t) is the amount of cesium-137 present at time t. From 4, y(t) = 1e -kt Since y(3) = 93.3, we solve 93.3 = 1e -3k for k to find the continuous compound decay rate: 93.3 = 1e -3k e -3k =.933-3k = ln(.933) k = ln(.933)!3 67. From Example 3: Q = Q e t Now, the amount of radioactive carbon-14 present is % of the original amount. Thus,.Q = Q e t or e t =.. Therefore, t = ln(.) and t 24,2 years. 69. N(k) = 18e -.11(k-1), 1 k 1 Thus, N(6) = 18e -.11(6-1) = 18e times and N(1) = 18e -.11(1-1) = 18e times. 386 CHAPTER 6 INTEGRATION
(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 informationSection 6-1 Antiderivatives and Indefinite Integrals
Name Date Class Section 6-1 Antiderivatives and Indefinite Integrals Goal: To find antiderivatives and indefinite integrals of functions using the formulas and properties Theorem 1 Antiderivatives If the
More informationSection 6-1 Antiderivatives and Indefinite Integrals
Name Date Class Section 6- Antiderivatives and Indefinite Integrals Goal: To find antiderivatives and indefinite integrals of functions using the formulas and properties Theorem Antiderivatives If the
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 informationSolving differential equations (Sect. 7.4) Review: Overview of differential equations.
Solving differential equations (Sect. 7.4 Previous class: Overview of differential equations. Exponential growth. Separable differential equations. Review: Overview of differential equations. Definition
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Basic Terminology A differential equation is an equation that contains an unknown function together with one or more of its derivatives. 1 Examples: 1. y = 2x + cos x 2. dy dt =
More informationDifferential Equations
Differential Equations A differential equation (DE) is an equation which involves an unknown function f (x) as well as some of its derivatives. To solve a differential equation means to find the unknown
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Chapter 1 Introduction and Basic Terminology Most of the phenomena studied in the sciences and engineering involve processes that change with time. For example, it is well known
More informationPRECAL REVIEW DAY 11/14/17
PRECAL REVIEW DAY 11/14/17 COPY THE FOLLOWING INTO JOURNAL 1 of 3 Transformations of logs Vertical Transformation Horizontal Transformation g x = log b x + c g x = log b x c g x = log b (x + c) g x = log
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 informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES 3.8 Exponential Growth and Decay In this section, we will: Use differentiation to solve real-life problems involving exponentially growing quantities. EXPONENTIAL
More informationSolutions to Section 1.1
Solutions to Section True-False Review: FALSE A derivative must involve some derivative of the function y f(x), not necessarily the first derivative TRUE The initial conditions accompanying a differential
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 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 informationInternet Mat117 Formulas and Concepts. d(a, B) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. ( x 1 + x 2 2., y 1 + y 2. (x h) 2 + (y k) 2 = r 2. m = y 2 y 1 x 2 x 1
Internet Mat117 Formulas and Concepts 1. The distance between the points A(x 1, y 1 ) and B(x 2, y 2 ) in the plane is d(a, B) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 2. The midpoint of the line segment from A(x
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 informationDIFFERENTIAL EQUATIONS
Mr. Isaac Akpor Adjei (MSc. Mathematics, MSc. Biostats) isaac.adjei@gmail.com April 7, 2017 ORDINARY In many physical situation, equation arise which involve differential coefficients. For example: 1 The
More informationINTERNET MAT 117. Solution for the Review Problems. (1) Let us consider the circle with equation. x 2 + 2x + y 2 + 3y = 3 4. (x + 1) 2 + (y + 3 2
INTERNET MAT 117 Solution for the Review Problems (1) Let us consider the circle with equation x 2 + y 2 + 2x + 3y + 3 4 = 0. (a) Find the standard form of the equation of the circle given above. (i) Group
More informationOrdinary Differential Equations (ODEs)
Chapter 13 Ordinary Differential Equations (ODEs) We briefly review how to solve some of the most standard ODEs. 13.1 First Order Equations 13.1.1 Separable Equations A first-order ordinary differential
More informationPractice Exam 1 Solutions
Practice Exam 1 Solutions 1a. Let S be the region bounded by y = x 3, y = 1, and x. Find the area of S. What is the volume of the solid obtained by rotating S about the line y = 1? Area A = Volume 1 1
More informationFirst Order Differential Equations Chapter 1
First Order Differential Equations Chapter 1 Doreen De Leon Department of Mathematics, California State University, Fresno 1 Differential Equations and Mathematical Models Section 1.1 Definitions: An equation
More informationA population is modeled by the differential equation
Math 2, Winter 2016 Weekly Homework #8 Solutions 9.1.9. A population is modeled by the differential equation dt = 1.2 P 1 P ). 4200 a) For what values of P is the population increasing? P is increasing
More informationThe Fundamental Theorem of Calculus: Suppose f continuous on [a, b]. 1.) If G(x) = x. f(t)dt = F (b) F (a) where F is any antiderivative
1 Calulus pre-requisites you must know. Derivative = slope of tangent line = rate. Integral = area between curve and x-axis (where area can be negative). The Fundamental Theorem of Calculus: Suppose f
More information7.1. Calculus of inverse functions. Text Section 7.1 Exercise:
Contents 7. Inverse functions 1 7.1. Calculus of inverse functions 2 7.2. Derivatives of exponential function 4 7.3. Logarithmic function 6 7.4. Derivatives of logarithmic functions 7 7.5. Exponential
More informationLast quiz Comments. ! F '(t) dt = F(b) " F(a) #1: State the fundamental theorem of calculus version I or II. Version I : Version II :
Last quiz Comments #1: State the fundamental theorem of calculus version I or II. Version I : b! F '(t) dt = F(b) " F(a) a Version II : x F( x) =! f ( t) dt F '( x) = f ( x) a Comments of last quiz #1:
More informationMATH 236 ELAC FALL 2017 CA 9 NAME: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
MATH 236 ELAC FALL 207 CA 9 NAME: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. ) 27 p 3 27 p 3 ) 2) If 9 t 3 4t 9-2t = 3, find t. 2) Solve the equation.
More informationECONOMICS 207 SPRING 2006 LABORATORY EXERCISE 5 KEY. 8 = 10(5x 2) = 9(3x + 8), x 50x 20 = 27x x = 92 x = 4. 8x 2 22x + 15 = 0 (2x 3)(4x 5) = 0
ECONOMICS 07 SPRING 006 LABORATORY EXERCISE 5 KEY Problem. Solve the following equations for x. a 5x 3x + 8 = 9 0 5x 3x + 8 9 8 = 0(5x ) = 9(3x + 8), x 0 3 50x 0 = 7x + 7 3x = 9 x = 4 b 8x x + 5 = 0 8x
More informationSection 2.2 Solutions to Separable Equations
Section. Solutions to Separable Equations Key Terms: Separable DE Eponential Equation General Solution Half-life Newton s Law of Cooling Implicit Solution (The epression has independent and dependent variables
More informationExponential and logarithmic functions (pp ) () Supplement October 14, / 1. a and b positive real numbers and x and y real numbers.
MA123, Supplement Exponential and logaritmic functions pp. 315-319) Capter s Goal: Review te properties of exponential and logaritmic functions. Learn ow to differentiate exponential and logaritmic functions.
More informationInternet Mat117 Formulas and Concepts. d(a, B) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2., y 1 + y 2. ( x 1 + x 2 2
Internet Mat117 Formulas and Concepts 1. The distance between the points A(x 1, y 1 ) and B(x 2, y 2 ) in the plane is d(a, B) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 2. The midpoint of the line segment from A(x
More informationChapter 6 Differential Equations and Mathematical Modeling. 6.1 Antiderivatives and Slope Fields
Chapter 6 Differential Equations and Mathematical Modeling 6. Antiderivatives and Slope Fields Def: An equation of the form: = y ln x which contains a derivative is called a Differential Equation. In this
More informationIntroductory Differential Equations
Introductory Differential Equations Lecture Notes June 3, 208 Contents Introduction Terminology and Examples 2 Classification of Differential Equations 4 2 First Order ODEs 5 2 Separable ODEs 5 22 First
More informationy0 = F (t0)+c implies C = y0 F (t0) Integral = area between curve and x-axis (where I.e., f(t)dt = F (b) F (a) wheref is any antiderivative 2.
Calulus pre-requisites you must know. Derivative = slope of tangent line = rate. Integral = area between curve and x-axis (where area can be negative). The Fundamental Theorem of Calculus: Suppose f continuous
More informationScience One Math. October 23, 2018
Science One Math October 23, 2018 Today A general discussion about mathematical modelling A simple growth model Mathematical Modelling A mathematical model is an attempt to describe a natural phenomenon
More informationSolutions: Homework 8
Solutions: Homework 8 1 Chapter 7.1 Problem 3 This is one of those problems where we substitute in solutions and just check to see if they work or not. (a) Substituting y = e rx into the differential equation
More informationAPPENDIX : PARTIAL FRACTIONS
APPENDIX : PARTIAL FRACTIONS Appendix : Partial Fractions Given the expression x 2 and asked to find its integral, x + you can use work from Section. to give x 2 =ln( x 2) ln( x + )+c x + = ln k x 2 x+
More informationMath 315: Differential Equations Lecture Notes Patrick Torres
Introduction What is a Differential Equation? A differential equation (DE) is an equation that relates a function (usually unknown) to its own derivatives. Example 1: The equation + y3 unknown function,
More informationMathsGeeks. Everything You Need to Know A Level Edexcel C4. March 2014 MathsGeeks Copyright 2014 Elite Learning Limited
Everything You Need to Know A Level Edexcel C4 March 4 Copyright 4 Elite Learning Limited Page of 4 Further Binomial Expansion: Make sure it starts with a e.g. for ( x) ( x ) then use ( + x) n + nx + n(n
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS 1. Basic Terminology A differential equation is an equation that contains an unknown function together with one or more of its derivatives. 1 Examples: 1. y = 2x + cos x 2. dy dt
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 information3.8 Exponential Growth and Decay
October 15, 2010 Population growth Population growth If y = f (t) is the number of individuals in a population of animals or humans at time t, then it seems reasonable to expect that the rate of growth
More informationChapter1. Ordinary Differential Equations
Chapter1. Ordinary Differential Equations In the sciences and engineering, mathematical models are developed to aid in the understanding of physical phenomena. These models often yield an equation that
More informationMath Applied Differential Equations
Math 256 - Applied Differential Equations Notes Existence and Uniqueness The following theorem gives sufficient conditions for the existence and uniqueness of a solution to the IVP for first order nonlinear
More informationIntroduction to Differential Equations
Chapter 1 Introduction to Differential Equations 1.1 Basic Terminology Most of the phenomena studied in the sciences and engineering involve processes that change with time. For example, it is well known
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS 1. Basic Terminology A differential equation is an equation that contains an unknown function together with one or more of its derivatives. 1 Examples: 1. y = 2x + cos x 2. dy dt
More informationINTERNET MAT 117 Review Problems. (1) Let us consider the circle with equation. (b) Find the center and the radius of the circle given above.
INTERNET MAT 117 Review Problems (1) Let us consider the circle with equation x 2 + y 2 + 2x + 3y + 3 4 = 0. (a) Find the standard form of the equation of the circle given above. (b) Find the center and
More informationNotes for exponential functions The week of March 6. Math 140
Notes for exponential functions The week of March 6 Math 140 Exponential functions: formulas An exponential function has the formula f (t) = ab t, where b is a positive constant; a is the initial value
More informationName Class. 5. Find the particular solution to given the general solution y C cos x and the. x 2 y
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
More informationSolutions to the Review Questions
Solutions to the Review Questions Short Answer/True or False. True or False, and explain: (a) If y = y + 2t, then 0 = y + 2t is an equilibrium solution. False: This is an isocline associated with a slope
More informationRadioactivity. Radioactivity
The Law of Radioactive Decay. 72 The law of radioactive decay. It turns out that the probability per unit time for any radioactive nucleus to decay is a constant, called the decay constant, lambda, ".
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 information3.9 Derivatives of Exponential and Logarithmic Functions
322 Chapter 3 Derivatives 3.9 Derivatives of Exponential and Logarithmic Functions Learning Objectives 3.9.1 Find the derivative of exponential functions. 3.9.2 Find the derivative of logarithmic functions.
More information1.1. Bacteria Reproduce like Rabbits. (a) A differential equation is an equation. a function, and both the function and its
G. NAGY ODE August 28, 2018 1 1.1. Bacteria Reproduce like Rabbits Section Objective(s): Overview of Differential Equations. The Discrete Equation. The Continuum Equation. Summary and Consistency. 1.1.1.
More informationC.3 First-Order Linear Differential Equations
A34 APPENDIX C Differential Equations C.3 First-Order Linear Differential Equations Solve first-order linear differential equations. Use first-order linear differential equations to model and solve real-life
More informationSection 11.1 What is a Differential Equation?
1 Section 11.1 What is a Differential Equation? Example 1 Suppose a ball is dropped from the top of a building of height 50 meters. Let h(t) denote the height of the ball after t seconds, then it is known
More informationMath 3313: Differential Equations First-order ordinary differential equations
Math 3313: Differential Equations First-order ordinary differential equations Thomas W. Carr Department of Mathematics Southern Methodist University Dallas, TX Outline Math 2343: Introduction Separable
More informationFundamentals of Mathematics (MATH 1510)
Fundamentals of Mathematics () Instructor: Email: shenlili@yorku.ca Department of Mathematics and Statistics York University February 3-5, 2016 Outline 1 growth (doubling time) Suppose a single bacterium
More informationSolution: APPM 1350 Final Exam Spring 2014
APPM 135 Final Exam Spring 214 1. (a) (5 pts. each) Find the following derivatives, f (x), for the f given: (a) f(x) = x 2 sin 1 (x 2 ) (b) f(x) = 1 1 + x 2 (c) f(x) = x ln x (d) f(x) = x x d (b) (15 pts)
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Basic Terminology A differential equation is an equation that contains an unknown function together with one or more of its derivatives. 1 Examples: 1. y = 2x + cos x 2. dy dt =
More information1.5. Applications. Theorem The solution of the exponential decay equation with N(0) = N 0 is N(t) = N 0 e kt.
6 Section Objective(s): The Radioactive Decay Equation Newton s Cooling Law Salt in a Water Tanks 151 Exponential Decay 15 Applications Definition 151 The exponential decay equation for N is N = k N, k
More informationExponential Functions Concept Summary See pages Vocabulary and Concept Check.
Vocabulary and Concept Check Change of Base Formula (p. 548) common logarithm (p. 547) exponential decay (p. 524) exponential equation (p. 526) exponential function (p. 524) exponential growth (p. 524)
More informationMath 225 Differential Equations Notes Chapter 1
Math 225 Differential Equations Notes Chapter 1 Michael Muscedere September 9, 2004 1 Introduction 1.1 Background In science and engineering models are used to describe physical phenomena. Often these
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 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 information2r 2 e rx 5re rx +3e rx = 0. That is,
Math 4, Exam 1, Solution, Spring 013 Write everything on the blank paper provided. You should KEEP this piece of paper. If possible: turn the problems in order (use as much paper as necessary), use only
More information33. The gas law for an ideal gas at absolute temperature T (in. 34. In a fish farm, a population of fish is introduced into a pond
SECTION 3.8 EXPONENTIAL GROWTH AND DECAY 2 3 3 29. The cost, in dollars, of producing x yards of a certain fabric is Cx 1200 12x 0.1x 2 0.0005x 3 (a) Find the marginal cost function. (b) Find C200 and
More information2. (12 points) Find an equation for the line tangent to the graph of f(x) =
November 23, 2010 Name The total number of points available is 153 Throughout this test, show your work Throughout this test, you are expected to use calculus to solve problems Graphing calculator solutions
More informationHonors Advanced Algebra Chapter 8 Exponential and Logarithmic Functions and Relations Target Goals
Honors Advanced Algebra Chapter 8 Exponential and Logarithmic Functions and Relations Target Goals By the end of this chapter, you should be able to Graph exponential growth functions. (8.1) Graph exponential
More informationMath 110 Final Exam General Review. Edward Yu
Math 110 Final Exam General Review Edward Yu Da Game Plan Solving Limits Regular limits Indeterminate Form Approach Infinities One sided limits/discontinuity Derivatives Power Rule Product/Quotient Rule
More information2.1 Exponential Growth
2.1 Exponential Growth A mathematical model is a description of a real-world system using mathematical language and ideas. Differential equations are fundamental to modern science and engineering. Many
More informationToday: 5.4 General log and exp functions (continued) Warm up:
Today: 5.4 General log and exp functions (continued) Warm up: log a (x) =ln(x)/ ln(a) d dx log a(x) = 1 ln(a)x 1. Evaluate the following functions. log 5 (25) log 7 p 7 log4 8 log 4 2 2. Di erentiate the
More informationEx. 1. Find the general solution for each of the following differential equations:
MATH 261.007 Instr. K. Ciesielski Spring 2010 NAME (print): SAMPLE TEST # 2 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1.
More informationLesson 6 MA Nick Egbert
Overview In this lesson we start our stu of differential equations. We start by considering only exponential growth and decay, and in the next lesson we will extend this idea to the general method of separation
More informationCHAPTER 2 POLYNOMIALS KEY POINTS
CHAPTER POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, and 3 are called linear, quadratic and cubic polynomials respectively.. A quadratic polynomial in x with real coefficient is of the form a x
More information1 What is a differential equation
Math 10B - Calculus by Hughes-Hallett, et al. Chapter 11 - Differential Equations Prepared by Jason Gaddis 1 What is a differential equation Remark 1.1. We have seen basic differential equations already
More informationEXAM. Exam #1. Math 3350 Summer II, July 21, 2000 ANSWERS
EXAM Exam #1 Math 3350 Summer II, 2000 July 21, 2000 ANSWERS i 100 pts. Problem 1. 1. In each part, find the general solution of the differential equation. dx = x2 e y We use the following sequence of
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 informationEven-Numbered Homework Solutions
Even-Numbered Homework Solutions Chapter 1 1.1 8. Using the decay-rate parameter you computed in 1.1.7, determine the time since death if: (a) 88% of the original C-14 is still in the material The decay-rate
More information1.1. Bacteria Reproduce like Rabbits. (a) A differential equation is an equation. a function, and both the function and its
G NAGY ODE January 7, 2018 1 11 Bacteria Reproduce like Rabbits Section Objective(s): Overview of Differential Equations The Discrete Equation The Continuum Equation Summary and Consistency 111 Overview
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 31S. Rumbos Fall Solutions to Exam 1
Math 31S. Rumbos Fall 2011 1 Solutions to Exam 1 1. When people smoke, carbon monoxide is released into the air. Suppose that in a room of volume 60 m 3, air containing 5% carbon monoxide is introduced
More informationSection , #5. Let Q be the amount of salt in oz in the tank. The scenario can be modeled by a differential equation.
Section.3.5.3, #5. Let Q be the amount of salt in oz in the tank. The scenario can be modeled by a differential equation dq = 1 4 (1 + sin(t) ) + Q, Q(0) = 50. (1) 100 (a) The differential equation given
More informationIntegral Curve (generic name for a member of the collection of known as the general solution)
Section 1.2 Solutions of Some Differential Equations Key Terms/Ideas: Phase Line (This topic is not in this section of the book.) Classification of Equilibrium Solutions: Source, Sink, Node SPECIAL CASE:
More informationSection 5.6 Integration by Parts
.. 98 Section.6 Integration by Parts Integration by parts is another technique that we can use to integrate problems. Typically, we save integration by parts as a last resort when substitution will not
More informationMethod of Frobenius. General Considerations. L. Nielsen, Ph.D. Dierential Equations, Fall Department of Mathematics, Creighton University
Method of Frobenius General Considerations L. Nielsen, Ph.D. Department of Mathematics, Creighton University Dierential Equations, Fall 2008 Outline 1 The Dierential Equation and Assumptions 2 3 Main Theorem
More information) = nlog b ( m) ( m) log b ( ) ( ) = log a b ( ) Algebra 2 (1) Semester 2. Exponents and Logarithmic Functions
Exponents and Logarithmic Functions Algebra 2 (1) Semester 2! a. Graph exponential growth functions!!!!!! [7.1]!! - y = ab x for b > 0!! - y = ab x h + k for b > 0!! - exponential growth models:! y = a(
More informationSolutions: Section 2.5
Solutions: Section 2.5. Problem : Given dy = ay + by2 = y(a + by) with a, b > 0. For the more general case, we will let y 0 be any real number. Always look for the equilibria first! In this case, y(a +
More informationCalculus (Math 1A) Lecture 4
Calculus (Math 1A) Lecture 4 Vivek Shende August 31, 2017 Hello and welcome to class! Last time We discussed shifting, stretching, and composition. Today We finish discussing composition, then discuss
More informationEXPONENTIAL, LOGARITHMIC, AND TRIGONOMETRIC FUNCTIONS
Calculus for the Life Sciences nd Edition Greenwell SOLUTIONS MANUAL Full download at: https://testbankreal.com/download/calculus-for-the-life-sciences-nd-editiongreenwell-solutions-manual-/ Calculus for
More informationSolutions to Exercises, Section 2.5
Instructor s Solutions Manual, Section 2.5 Exercise 1 Solutions to Exercises, Section 2.5 For Exercises 1 4, write the domain of the given function r as a union of intervals. 1. r(x) 5x3 12x 2 + 13 x 2
More information0.1 Problems to solve
0.1 Problems to solve Homework Set No. NEEP 547 Due September 0, 013 DLH Nonlinear Eqs. reducible to first order: 1. 5pts) Find the general solution to the differential equation: y = [ 1 + y ) ] 3/. 5pts)
More informationC. HECKMAN TEST 1A SOLUTIONS 170
C. HECKMAN TEST 1A SOLUTIONS 170 1) Thornley s Bank of Atlanta offers savings accounts which earn 4.5% per year. You have $00, which you want to invest. a) [10 points] If the bank compounds the interest
More informationAlgebra I. Course Outline
Algebra I Course Outline I. The Language of Algebra A. Variables and Expressions B. Order of Operations C. Open Sentences D. Identity and Equality Properties E. The Distributive Property F. Commutative
More informationToday. Doubling time, half life, characteristic time. Exponential behaviour as solution to DE. Nonlinear DE (e.g. y = y (1-y) )
Today Bacterial growth example Doubling time, half life, characteristic time Exponential behaviour as solution to DE Linear DE ( y = ky ) Nonlinear DE (e.g. y = y (1-y) ) Qualitative analysis (phase line)
More informationPartial Fractions. (Do you see how to work it out? Substitute u = ax + b, so du = a dx.) For example, 1 dx = ln x 7 + C, x x (x 3)(x + 1) = a
Partial Fractions 7-9-005 Partial fractions is the opposite of adding fractions over a common denominator. It applies to integrals of the form P(x) dx, wherep(x) and Q(x) are polynomials. Q(x) The idea
More informationHigher-order ordinary differential equations
Higher-order ordinary differential equations 1 A linear ODE of general order n has the form a n (x) dn y dx n +a n 1(x) dn 1 y dx n 1 + +a 1(x) dy dx +a 0(x)y = f(x). If f(x) = 0 then the equation is called
More informationSection 6.8 Exponential Models; Newton's Law of Cooling; Logistic Models
Section 6.8 Exponential Models; Newton's Law of Cooling; Logistic Models 197 Objective #1: Find Equations of Populations that Obey the Law of Uninhibited Growth. In the last section, we saw that when interest
More informationCalculus (Math 1A) Lecture 4
Calculus (Math 1A) Lecture 4 Vivek Shende August 30, 2017 Hello and welcome to class! Hello and welcome to class! Last time Hello and welcome to class! Last time We discussed shifting, stretching, and
More informationAn idea how to solve some of the problems. diverges the same must hold for the original series. T 1 p T 1 p + 1 p 1 = 1. dt = lim
An idea how to solve some of the problems 5.2-2. (a) Does not converge: By multiplying across we get Hence 2k 2k 2 /2 k 2k2 k 2 /2 k 2 /2 2k 2k 2 /2 k. As the series diverges the same must hold for the
More informationSecond-Order Linear ODEs
Chap. 2 Second-Order Linear ODEs Sec. 2.1 Homogeneous Linear ODEs of Second Order On pp. 45-46 we extend concepts defined in Chap. 1, notably solution and homogeneous and nonhomogeneous, to second-order
More information