2 ODEs Integrating Factors and Homogeneous Equations
|
|
- Harold Baldwin
- 6 years ago
- Views:
Transcription
1 2 ODEs Integrating Factors an Homogeneous Equations We begin with a slightly ifferent type of equation: 2.1 Exact Equations These are ODEs whose general solution can be obtaine by simply integrating both sies of the equation. Examples Fin the general solutions of the following ODEs. a) y = 2 sin x, b) (yx2 ) = 4x 3 a) This is completely stanar we can rewrite as the integral equation integrating both sies, we obtain y = 2 cos x + c y = 2 sin(x). Then b) Here we o the same: Integrating both sies, we obtain (yx 2 ) = 4x 3 an hence yx 2 = x 4 + c. We can now ivie through by x 2 to obtain the general solution y = x 2 + c x 2. Key Point The solution of the equation is ( ) f(x)y = g(x) f(x)y = g(x) or y = 1 f(x) g(x) Terminology: In the ifferential equation b) (yx2 ) = 4x 3, the solution has two parts, since y = x 2 + c x 2. 1
2 The part x 2 is calle the efinite part. The part c (containing the arbitrary constant of integration) is x2 calle the inefinite part. If we take the efinite part of the solution, i.e. y = x 2 then (y x 2 ) = (x2 x 2 ) = (x4 ) = 4x 3. Hence y = x 2 is a solution of the original ifferential equation (b). Now consier the inefinite part of the solution, i.e. y i = c x 2, then (y i x 2 ) = ( c x 2 x2) = (c) = 0. This works generally: Useful fact: In solving an exact equation ( ) f(x)y = g(x), the solution y has two parts: y = y (x) + y i (x) where: 1. The efinite part y (x) which is a solution of the ifferential equation. 2. The inefinite part y i (x) which satisfies a simpler version of the ifferential equation in which the right-han sie is zero. Exercise: Solve the equation (y sin x) = sinx an verify that the inefinite part of the solution satisfies the equation (y sin x) = 0. Solution: Integrate both sies of the equation, y sin x = cos x + C y = cot x + Ccosec x. Hence y (x) = cot x an y i (x) = C cosec x. Now (y i(x) sin x) = (cosec x sin x) = (C) = 0. 2
3 2.1.2 Recognising exact equations The equation b) (yx2 ) = x 2 is exact but if we expan the left-han sie of this equation using the prouct rule, we have that (yx2 ) = x 2 y + y (x2 ) = x 2 y + y(2x) an so, x 2 y + 2xy = x2. (2.1) Hence the equation (2.1) is exact but it is not in stanar form Key point The equation f(x) y + yf (x) = g(x) is exact. It can be rewritten as: so that yf(x) = g(x) an y = 1 f(x) (yf(x)) = g(x) g(x). Example. Solve the equation 2x y e + 2e2x y = x 2. Note that e2x = 2e 2x. Therefore this is just the exact equation (e2x y) = x 2. Integrating both sies gives e 2x y = x3 3 + c. Thus y = x3 3 e 2x + ce 2x is the general solution. 3
4 2.2 Integrating Factors Integrating factors can be use to transform certain ODEs which are not exact into exact ODEs. As an illustration, consier x 3 y + 4x2 y = x. As you may wish to check, this is not exact. However, if we multiply through by x, we get which can be rewritten as the exact equation x 4 y + 4x3 y = x 2 (yx4 ) = x 2. Thus we can solve this as before to get yx 4 = (yx4 ) = x 2 = x3 3 + c or y = 1 3x + c x 4. The function by which we multiply a given ODE in orer to make it exact is calle the integrating factor. In the above example x is the integrating factor. This works rather generally: Funamental Algorithm Any linear ODE of the form a(x) y + b(x)y + c(x) = 0 can be transforme into an exact equation an then (hopefully!) solve. This requires the following steps: Step I: Divie by a(x) to get an equation of the form Step II: Compute the integrating factor ( I.F. = exp y + f(x)y = g(x) ) f(x) Note that we o not nee constants of integration here the IF is any function of the form e F (x) where f(x) = F (x). Step III: Multiply through by the IF to get Step IV: The LHS of this equation is simply e F (x) y = F (x) y e + f(x)ef (x) y = e F (x) g(x). (ef (x) y) an so we can solve: (ef (x) y) = e F (x) g(x). 4
5 Example: Consier the example x 3 y + 4x2 y = x from before. This can be rewritten as y + 4 x y = x 2. Thus the integrating factor (I.F.) is ( ) exp 4x 1 = exp(4 log x) = exp(log x 4 ) = x 4. Therefore, we multiply through by x 4, giving the exact ODE x 4 y + 4x3 y = x 2 ; equivalently (yx4 ) = x 4 y + 4x3 y = x 2 as before. Example: Solve the ODE y + 2y = sin x. Step 1 is alreay one! Step 2. Calculate the I.F. ( I.F. = exp ) 2 = e 2x. Step 3. Multiply through by the I.F. an write the equation in exact form. 2x y e + 2e2x y = e 2x sin x. Thus ( e 2x y ) = e 2x sin x. Step 4. Integrate both sies of the equation an fin the general solution. This is a stanar integration by parts, an I will let you check that e 2x y = e 2x y = e 2x sin x = = 1 5 e2x cos x e2x sin x + C. Example: Solve the ODE Step 1. Rewrite the ifferential equation as Step 2. Calculate the I.F. ( I.F. = exp cos x y + (sin x)y = cos2 x. ) tan(x) y + tan(x)y = cos x. = exp( ln(cos x)) = exp(ln(sec x)) = sec(x). 5
6 Step 3. Multiply through by the I.F. an write the equation in exact form. sec x y + y sec x tan x = sec(x) cos x = 1 equivalently, ( ) sec(x)y = 1 Step 4. Integrate both sies of the equation an fin the general solution: ( sec(x)y ) = which has solution sec(x)y = x + c or y = (x + c) cos(x). An electrical example: Suppose we are given an electrical circuit with a resistor an an inuctance coil an a battery. R L + E - A basic fact from physics is that, if we close the circuit we get a current i flowing clockwise aroun the circuit, with voltage rop across the resistor of ir an across the coil of L i t. Thus, E = ir + L i t. [Note: These are not formulae you nee to remember for this course.] Let s put in some numbers; say E = 10, R = 8 an L = 2; with initial conitions of i = 0 at t = 0. Thus 10 = 8i + 2 i t, or i + 4i = 5. t 6
7 This can be solve either with an IF (which equals e 4t ) or by separating variables an using (1.3.1) from page 6 of the Separation of Variables notes. Either way you fin that i = 5/4 + Ce 4t. The initial conitions give 0 = 5/4 + C an so C = 5/4 an i = e 4t. This has the graph with asymptote at i = 5/4 like those on page 7 of the Separation of Variables section. Now suppose the battery goes flat an we use the mains in place, say with voltage 2 cos(t). This means that we have to solve the equation 2 i t + 8i = 2 cos(t) or i + 4i = cos(t). t This oes now nee an IF = exp( 4t) = e 4t. Thus ( e 4t i ) 4t i = e t t + 4e4t i = e 4t cos(t). Therefore, e 4t i = e 4t cos(t) = 4 17 e4t cos(t) e4t sin(t) + C, where the integral on the RHS was solve by integration by parts (twice). Solving for i we get i = 4 17 cos(t) sin(t) + Ce 4t. Finally, solving for C from i(0) = 0 gives C = 4 17 an so, finally, i = 4 17 cos(t) sin(t) 4 17 e 4t. Example: Even quite innocent-looking equations can lea to impossible integrals. For example the equation y + xy = 1 leas to the IF = exp( x) = e x2 /2 an hence to the equation e x2 /2 y = e x2 /2, which has no close form. (In fact, you can always solve this sort of equation by writing e x2 /2 as a power series in x, but that is another story an another course.) 7
8 3 Homogenous Equations These are ifferential equations of the form: y = f(x, y) where the function f(, ) satisfies f(λx, λy) = f(x, y), for any λ. Remark: Be careful about looking up homogeneous equations in the literature since this has more than one meaning. Examples of homogeneous equations: a) f(x, y) = 1 + y x ty since f(tx, ty) = 1 + tx = 1 + y x = f(x, y). b) f(x, y) = e x y since f(tx, ty) = e tx ty = e x y = f(x, y). c) f(x, y) = x y + y x tx since f(tx, ty) = ty + ty tx = x y + y x = f(x, y). 3.1 Basic Technique for Solving Homogeneous Equations. To solve homogeneous equations, we o a substitution: z = y x (or y = zx) an use the rule y = z (zx) = x + z. We illustrate how this is one with an example. Example: Solve the ODE x y = x + y. Step 1. Divie through by x to get a homogenous equation: y = 1 + y x. Step 2. Substitute y = zx to get (xz) = 1 + z. The Key Step 3. Expan the LHS of the equation using the prouct rule, x z + z z (x) = 1 + z or x + z = 1 + z. 8
9 Step 4. Hopefully (!) this can now be solve by one of the earlier techniques. In this case we can cancel the z s to get x z = 1. This can then be solve by separating variables: This has solution z = ln x + C. z = 1 x an so z = x. Step 5. Finally substitute back z = y/x; thus y = ln(x) + C or y = x ln(x) + Cx. x Lets o this with some more examples. Example. Solve the ODE x y = y + xey/x subject to y = 1 at x = 1. Solution: Divie by x to get This is homogeneous, so we set z = y x y = y x + ey/x. or y = zx to get x z + z = (zx) = z + e z. Cancel the z to get x z = ez. This can now be solve by separating variables: e z z = x 1, with solution e z = ln x + C. Substituting back z = y/x we get e y/x = ln x + C. Plugging in y = 1 at x = 1 gives e 1 = 0 + C an so C = e 1. Thus, finally, e y/x = ln x e 1. If you like you can solve for y by taking logs, but the answer is still not very elegant. 9
10 Example. Solve the ODE y (y + x)y = x(x y) 2 Solution: This is not obviously homogeneous, but we make it so by iviing the top an bottom of the fraction on RHS by x 2 : Substitute in y = zx ( y y = x + 1) y ( ) x 1 y 2 x x z + 1)z + z = (z (1 z) 2 x z = ( z 2 + z 1 z ) z 2 We can now simplify the RHS: x z = z2 z z(1 z) 2(1 z) 1 z = z2 z z + z z 1 z = 2 1 z = 2 z 1. Thus we can separate variables: ( z 1 Integrate both sies of the equation to get 2 ) z = 1 ( z x or 2 1 ) z = 1 2 x z 2 4 z 2 = ln x + c Finally we substitute back z = y x to get y 2 4x 2 y = ln x + c. 2x With a bit of an effort you can write this as y = g(x) for some function g(x) but I really o not think it is worth the bother. 10
First Order Linear Differential Equations
LECTURE 6 First Orer Linear Differential Equations A linear first orer orinary ifferential equation is a ifferential equation of the form ( a(xy + b(xy = c(x. Here y represents the unknown function, y
More informationLinear First-Order Equations
5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)
More informationReview of Differentiation and Integration for Ordinary Differential Equations
Schreyer Fall 208 Review of Differentiation an Integration for Orinary Differential Equations In this course you will be expecte to be able to ifferentiate an integrate quickly an accurately. Many stuents
More informationIntroduction to Differential Equations Math 286 X1 Fall 2009 Homework 2 Solutions
Introuction to Differential Equations Math 286 X1 Fall 2009 Homewk 2 Solutions 1. Solve each of the following ifferential equations: (a) y + 3xy = 0 (b) y + 3y = 3x (c) y t = cos(t)y () x 2 y x y = 3 Solution:
More informationcosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) = 2 2t 1 + t 2 cos x = 1 t2 We will revisit the double angle identities:
6.4 Integration using tanx/) We will revisit the ouble angle ientities: sin x = sinx/) cosx/) = tanx/) sec x/) = tanx/) + tan x/) cos x = cos x/) sin x/) tan x = = tan x/) sec x/) tanx/) tan x/). = tan
More informationdx dx [x2 + y 2 ] = y d [tan x] + tan x = 2x + 2y = y sec 2 x + tan x dy dy = tan x dy dy = [tan x 2y] dy dx = 2x y sec2 x [1 + sin y] = sin(xy)
Math 7 Activit: Implicit & Logarithmic Differentiation (Solutions) Implicit Differentiation. For each of the following equations, etermine x. a. tan x = x 2 + 2 tan x] = x x x2 + 2 ] = tan x] + tan x =
More information2.5 The Chain Rule Brian E. Veitch
2.5 The Chain Rule This is our last ifferentiation rule for this course. It s also one of the most use. The best way to memorize this (along with the other rules) is just by practicing until you can o
More informationAntiderivatives. Definition (Antiderivative) If F (x) = f (x) we call F an antiderivative of f. Alan H. SteinUniversity of Connecticut
Antierivatives Definition (Antierivative) If F (x) = f (x) we call F an antierivative of f. Antierivatives Definition (Antierivative) If F (x) = f (x) we call F an antierivative of f. Definition (Inefinite
More informationLectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs
Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent
More informationImplicit Differentiation
Implicit Differentiation Implicit Differentiation Using the Chain Rule In the previous section we focuse on the erivatives of composites an saw that THEOREM 20 (Chain Rule) Suppose that u = g(x) is ifferentiable
More informationSolutions to Practice Problems Tuesday, October 28, 2008
Solutions to Practice Problems Tuesay, October 28, 2008 1. The graph of the function f is shown below. Figure 1: The graph of f(x) What is x 1 + f(x)? What is x 1 f(x)? An oes x 1 f(x) exist? If so, what
More informationSome functions and their derivatives
Chapter Some functions an their erivatives. Derivative of x n for integer n Recall, from eqn (.6), for y = f (x), Also recall that, for integer n, Hence, if y = x n then y x = lim δx 0 (a + b) n = a n
More informationINVERSE FUNCTIONS DERIVATIVES. terms on one side and everything else on the other. (3) Factor out dy. for the following functions: 1.
INVERSE FUNCTIONS DERIVATIVES Recall the steps for computing y implicitly: (1) Take of both sies, treating y like a function. (2) Expan, a, subtract to get the y terms on one sie an everything else on
More informationIntegration by Parts
Integration by Parts 6-3-207 If u an v are functions of, the Prouct Rule says that (uv) = uv +vu Integrate both sies: (uv) = uv = uv + u v + uv = uv vu, vu v u, I ve written u an v as shorthan for u an
More informationExam 3 Review. Lesson 19: Concavity, Inflection Points, and the Second Derivative Test. Lesson 20: Absolute Extrema on an Interval
Exam 3 Review Lessons 17-18: Relative Extrema, Critical Numbers, an First Derivative Test (from exam 2 review neee for curve sketching) Critical Numbers: where the erivative of a function is zero or unefine.
More informationFebruary 21 Math 1190 sec. 63 Spring 2017
February 21 Math 1190 sec. 63 Spring 2017 Chapter 2: Derivatives Let s recall the efinitions an erivative rules we have so far: Let s assume that y = f (x) is a function with c in it s omain. The erivative
More informationMath 115 Section 018 Course Note
Course Note 1 General Functions Definition 1.1. A function is a rule that takes certain numbers as inputs an assigns to each a efinite output number. The set of all input numbers is calle the omain of
More information3 Algebraic Methods. we can differentiate both sides implicitly to obtain a differential equation involving x and y:
3 Algebraic Methods b The first appearance of the equation E Mc 2 in Einstein s handwritten notes. So far, the only general class of differential equations that we know how to solve are directly integrable
More informationThe derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)
Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)
More informationf(x) f(a) Limit definition of the at a point in slope notation.
Lesson 9: Orinary Derivatives Review Hanout Reference: Brigg s Calculus: Early Transcenentals, Secon Eition Topics: Chapter 3: Derivatives, p. 126-235 Definition. Limit Definition of Derivatives at a point
More informationAP Calculus AB One Last Mega Review Packet of Stuff. Take the derivative of the following. 1.) 3.) 5.) 7.) Determine the limit of the following.
AP Calculus AB One Last Mega Review Packet of Stuff Name: Date: Block: Take the erivative of the following. 1.) x (sin (5x)).) x (etan(x) ) 3.) x (sin 1 ( x3 )) 4.) x (x3 5x) 4 5.) x ( ex sin(x) ) 6.)
More informationSYDE 112, LECTURE 1: Review & Antidifferentiation
SYDE 112, LECTURE 1: Review & Antiifferentiation 1 Course Information For a etaile breakown of the course content an available resources, see the Course Outline. Other relevant information for this section
More informationMath 2163, Practice Exam II, Solution
Math 63, Practice Exam II, Solution. (a) f =< f s, f t >=< s e t, s e t >, an v v = , so D v f(, ) =< ()e, e > =< 4, 4 > = 4. (b) f =< xy 3, 3x y 4y 3 > an v =< cos π, sin π >=, so
More informationMA 2232 Lecture 08 - Review of Log and Exponential Functions and Exponential Growth
MA 2232 Lecture 08 - Review of Log an Exponential Functions an Exponential Growth Friay, February 2, 2018. Objectives: Review log an exponential functions, their erivative an integration formulas. Exponential
More information( 3x +1) 2 does not fit the requirement of the power rule that the base be x
Section 3 4A: The Chain Rule Introuction The Power Rule is state as an x raise to a real number If y = x n where n is a real number then y = n x n-1 What if we wante to fin the erivative of a variable
More informationUsing the definition of the derivative of a function is quite tedious. f (x + h) f (x)
Derivative Rules Using te efinition of te erivative of a function is quite teious. Let s prove some sortcuts tat we can use. Recall tat te efinition of erivative is: Given any number x for wic te limit
More information016A Homework 10 Solution
016A Homework 10 Solution Jae-young Park November 2, 2008 4.1 #14 Write each expression in the form of 2 kx or 3 kx, for a suitable constant k; (3 x 3 x/5 ) 5, (16 1/4 16 3/4 ) 3x Solution (3 x 3 x/5 )
More information1 Lecture 18: The chain rule
1 Lecture 18: The chain rule 1.1 Outline Comparing the graphs of sin(x) an sin(2x). The chain rule. The erivative of a x. Some examples. 1.2 Comparing the graphs of sin(x) an sin(2x) We graph f(x) = sin(x)
More informationExam 2 Review Solutions
Exam Review Solutions 1. True or False, an explain: (a) There exists a function f with continuous secon partial erivatives such that f x (x, y) = x + y f y = x y False. If the function has continuous secon
More informationMath 210 Midterm #1 Review
Math 20 Miterm # Review This ocument is intene to be a rough outline of what you are expecte to have learne an retaine from this course to be prepare for the first miterm. : Functions Definition: A function
More informationSolutions to Math 41 Second Exam November 4, 2010
Solutions to Math 41 Secon Exam November 4, 2010 1. (13 points) Differentiate, using the metho of your choice. (a) p(t) = ln(sec t + tan t) + log 2 (2 + t) (4 points) Using the rule for the erivative of
More informationChapter 7. Integrals and Transcendental Functions
7. The Logarithm Define as an Integral Chapter 7. Integrals an Transcenental Functions 7.. The Logarithm Define as an Integral Note. In this section, we introuce the natural logarithm function using efinite
More informationFinal Exam Study Guide and Practice Problems Solutions
Final Exam Stuy Guie an Practice Problems Solutions Note: These problems are just some of the types of problems that might appear on the exam. However, to fully prepare for the exam, in aition to making
More informationQF101: Quantitative Finance September 5, Week 3: Derivatives. Facilitator: Christopher Ting AY 2017/2018. f ( x + ) f(x) f(x) = lim
QF101: Quantitative Finance September 5, 2017 Week 3: Derivatives Facilitator: Christopher Ting AY 2017/2018 I recoil with ismay an horror at this lamentable plague of functions which o not have erivatives.
More informationHyperbolic Functions. Notice: this material must not be used as a substitute for attending. the lectures
Hyperbolic Functions Notice: this material must not be use as a substitute for attening the lectures 0. Hyperbolic functions sinh an cosh The hyperbolic functions sinh (pronounce shine ) an cosh are efine
More informationFall 2016: Calculus I Final
Answer the questions in the spaces provie on the question sheets. If you run out of room for an answer, continue on the back of the page. NO calculators or other electronic evices, books or notes are allowe
More informationDifferentiation Rules Derivatives of Polynomials and Exponential Functions
Derivatives of Polynomials an Exponential Functions Differentiation Rules Derivatives of Polynomials an Exponential Functions Let s start with the simplest of all functions, the constant function f(x)
More informationImplicit Differentiation
Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,
More informationd dx [xn ] = nx n 1. (1) dy dx = 4x4 1 = 4x 3. Theorem 1.3 (Derivative of a constant function). If f(x) = k and k is a constant, then f (x) = 0.
Calculus refresher Disclaimer: I claim no original content on this ocument, which is mostly a summary-rewrite of what any stanar college calculus book offers. (Here I ve use Calculus by Dennis Zill.) I
More informationDifferentiation ( , 9.5)
Chapter 2 Differentiation (8.1 8.3, 9.5) 2.1 Rate of Change (8.2.1 5) Recall that the equation of a straight line can be written as y = mx + c, where m is the slope or graient of the line, an c is the
More informationCalculus I Announcements
Slie 1 Calculus I Announcements Office Hours: Amos Eaton 309, Monays 12:50-2:50 Exam 2 is Thursay, October 22n. The stuy guie is now on the course web page. Start stuying now, an make a plan to succee.
More informationFinal Exam: Sat 12 Dec 2009, 09:00-12:00
MATH 1013 SECTIONS A: Professor Szeptycki APPLIED CALCULUS I, FALL 009 B: Professor Toms C: Professor Szeto NAME: STUDENT #: SECTION: No ai (e.g. calculator, written notes) is allowe. Final Exam: Sat 1
More informationFind the indicated derivative. 1) Find y(4) if y = 3 sin x. A) y(4) = 3 cos x B) y(4) = 3 sin x C) y(4) = - 3 cos x D) y(4) = - 3 sin x
Assignment 5 Name Find the indicated derivative. ) Find y(4) if y = sin x. ) A) y(4) = cos x B) y(4) = sin x y(4) = - cos x y(4) = - sin x ) y = (csc x + cot x)(csc x - cot x) ) A) y = 0 B) y = y = - csc
More informationCore Mathematics 3 Differentiation
http://kumarmaths.weebly.com/ Core Mathematics Differentiation C differentiation Page Differentiation C Specifications. By the end of this unit you should be able to : Use chain rule to find the derivative
More informationYou should also review L Hôpital s Rule, section 3.6; follow the homework link above for exercises.
BEFORE You Begin Calculus II If it has been awhile since you ha Calculus, I strongly suggest that you refresh both your ifferentiation an integration skills. I woul also like to remin you that in Calculus,
More informationOptimization Notes. Note: Any material in red you will need to have memorized verbatim (more or less) for tests, quizzes, and the final exam.
MATH 2250 Calculus I Date: October 5, 2017 Eric Perkerson Optimization Notes 1 Chapter 4 Note: Any material in re you will nee to have memorize verbatim (more or less) for tests, quizzes, an the final
More information1 Lecture 20: Implicit differentiation
Lecture 20: Implicit ifferentiation. Outline The technique of implicit ifferentiation Tangent lines to a circle Derivatives of inverse functions by implicit ifferentiation Examples.2 Implicit ifferentiation
More informationSection The Chain Rule and Implicit Differentiation with Application on Derivative of Logarithm Functions
Section 3.4-3.6 The Chain Rule an Implicit Differentiation with Application on Derivative of Logarithm Functions Ruipeng Shen September 3r, 5th Ruipeng Shen MATH 1ZA3 September 3r, 5th 1 / 3 The Chain
More informationThe Chain Rule. Composition Review. Intuition. = 2(1.5) = 3 times faster than (X)avier.
The Chain Rule In the previous section we ha to use a trig ientity to etermine the erivative of. h(x) = sin(2x). We can view h(x) as the composition of two functions. Let g(x) = 2x an f (x) = sin x. Then
More informationMath 1B, lecture 8: Integration by parts
Math B, lecture 8: Integration by parts Nathan Pflueger 23 September 2 Introuction Integration by parts, similarly to integration by substitution, reverses a well-known technique of ifferentiation an explores
More informationSingle Variable Calculus Warnings
Single Variable Calculus Warnings These notes highlight number of common, but serious, first year calculus errors. Warning. The formula g(x) = g(x) is vali only uner the hypothesis g(x). Discussion. In
More informationMath Chapter 2 Essentials of Calculus by James Stewart Prepared by Jason Gaddis
Math 231 - Chapter 2 Essentials of Calculus by James Stewart Prepare by Jason Gais Chapter 2 - Derivatives 21 - Derivatives an Rates of Change Definition A tangent to a curve is a line that intersects
More information1.4 Techniques of Integration
.4 Techniques of Integration Recall the following strategy for evaluating definite integrals, which arose from the Fundamental Theorem of Calculus (see Section.3). To calculate b a f(x) dx. Find a function
More informationDifferentiability, Computing Derivatives, Trig Review. Goals:
Secants vs. Derivatives - Unit #3 : Goals: Differentiability, Computing Derivatives, Trig Review Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an
More informationUnit #6 - Families of Functions, Taylor Polynomials, l Hopital s Rule
Unit # - Families of Functions, Taylor Polynomials, l Hopital s Rule Some problems an solutions selecte or aapte from Hughes-Hallett Calculus. Critical Points. Consier the function f) = 54 +. b) a) Fin
More information2.3 Linear Equations 69
2.3 Linear Equations 69 2.3 Linear Equations An equation y = fx,y) is called first-order linear or a linear equation provided it can be rewritten in the special form 1) y + px)y = rx) for some functions
More informationPartial Derivatives for Math 229. Our puropose here is to explain how one computes partial derivatives. We will not attempt
Partial Derivatives for Math 229 Our puropose here is to explain how one computes partial derivatives. We will not attempt to explain how they arise or why one would use them; that is left to other courses
More informationDifferentiability, Computing Derivatives, Trig Review
Unit #3 : Differentiability, Computing Derivatives, Trig Review Goals: Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an original function Compute
More informationTrigonometric Functions
4 Trigonometric Functions So far we have use only algebraic functions as examples when fining erivatives, that is, functions that can be built up by the usual algebraic operations of aition, subtraction,
More informationTable of Contents Derivatives of Logarithms
Derivatives of Logarithms- Table of Contents Derivatives of Logarithms Arithmetic Properties of Logarithms Derivatives of Logarithms Example Example 2 Example 3 Example 4 Logarithmic Differentiation Example
More informationDefine each term or concept.
Chapter Differentiation Course Number Section.1 The Derivative an the Tangent Line Problem Objective: In this lesson you learne how to fin the erivative of a function using the limit efinition an unerstan
More informationMathematics 104 Fall Term 2006 Solutions to Final Exam. sin(ln t) dt = e x sin(x) dx.
Mathematics 14 Fall Term 26 Solutions to Final Exam 1. Evaluate sin(ln t) dt. Solution. We first make the substitution t = e x, for which dt = e x. This gives sin(ln t) dt = e x sin(x). To evaluate the
More information0.1 The Chain Rule. db dt = db
0. The Chain Rule A basic illustration of the chain rules comes in thinking about runners in a race. Suppose two brothers, Mark an Brian, hol an annual race to see who is the fastest. Last year Mark won
More informationLINEAR DIFFERENTIAL EQUATIONS OF ORDER 1. where a(x) and b(x) are functions. Observe that this class of equations includes equations of the form
LINEAR DIFFERENTIAL EQUATIONS OF ORDER 1 We consier ifferential equations of the form y + a()y = b(), (1) y( 0 ) = y 0, where a() an b() are functions. Observe that this class of equations inclues equations
More informationDifferential Equations DIRECT INTEGRATION. Graham S McDonald
Differential Equations DIRECT INTEGRATION Graham S McDonald A Tutorial Module introducing ordinary differential equations and the method of direct integration Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk
More informationa x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x).
You should prepare the following topics for our final exam. () Pre-calculus. (2) Inverses. (3) Algebra of Limits. (4) Derivative Formulas and Rules. (5) Graphing Techniques. (6) Optimization (Maxima and
More informationChapter 3. Modeling with First-Order Differential Equations
Chapter 3 Moeling with First-Orer Differential Equations i GROWTH AND DECAY: The initial-value problem x = kx, x(t 0 ) = x 0, (1) where k is a constant of proportionality, serves as a moel for iverse phenomena
More information2.5 SOME APPLICATIONS OF THE CHAIN RULE
2.5 SOME APPLICATIONS OF THE CHAIN RULE The Chain Rule will help us etermine the erivatives of logarithms an exponential functions a x. We will also use it to answer some applie questions an to fin slopes
More informationMath 106 Exam 2 Topics. du dx
The Chain Rule Math 106 Exam 2 Topics Composition (g f)(x 0 ) = g(f(x 0 )) ; (note: we on t know what g(x 0 ) is.) (g f) ought to have something to o with g (x) an f (x) in particular, (g f) (x 0 ) shoul
More informationOutline. MS121: IT Mathematics. Differentiation Rules for Differentiation: Part 1. Outline. Dublin City University 4 The Quotient Rule
MS2: IT Mathematics Differentiation Rules for Differentiation: Part John Carroll School of Mathematical Sciences Dublin City University Pattern Observe You may have notice the following pattern when we
More informationThe above statement is the false product rule! The correct product rule gives g (x) = 3x 4 cos x+ 12x 3 sin x. for all angles θ.
Math 7A Practice Midterm III Solutions Ch. 6-8 (Ebersole,.7-.4 (Stewart DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual exam. You
More informationMath 251 Notes. Part I.
Math 251 Notes. Part I. F. Patricia Meina May 6, 2013 Growth Moel.Consumer price inex. [Problem 20, page 172] The U.S. consumer price inex (CPI) measures the cost of living base on a value of 100 in the
More informationChapter 2. Exponential and Log functions. Contents
Chapter. Exponential an Log functions This material is in Chapter 6 of Anton Calculus. The basic iea here is mainly to a to the list of functions we know about (for calculus) an the ones we will stu all
More information2.12: Derivatives of Exp/Log (cont d) and 2.15: Antiderivatives and Initial Value Problems
2.12: Derivatives of Exp/Log (cont d) and 2.15: Antiderivatives and Initial Value Problems Mathematics 3 Lecture 14 Dartmouth College February 03, 2010 Derivatives of the Exponential and Logarithmic Functions
More informationJUST THE MATHS UNIT NUMBER ORDINARY DIFFERENTIAL EQUATIONS 3 (First order equations (C)) A.J.Hobson
JUST THE MATHS UNIT NUMBER 15.3 ORDINARY DIFFERENTIAL EQUATIONS 3 (First order equations (C)) by A.J.Hobson 15.3.1 Linear equations 15.3.2 Bernouilli s equation 15.3.3 Exercises 15.3.4 Answers to exercises
More information1 Definition of the derivative
Math 20A - Calculus by Jon Rogawski Chapter 3 - Differentiation Prepare by Jason Gais Definition of the erivative Remark.. Recall our iscussion of tangent lines from way back. We now rephrase this in terms
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 informationThe Exact Form and General Integrating Factors
7 The Exact Form an General Integrating Factors In the previous chapters, we ve seen how separable an linear ifferential equations can be solve using methos for converting them to forms that can be easily
More informationAntiderivatives Introduction
Antierivatives 40. Introuction So far much of the term has been spent fining erivatives or rates of change. But in some circumstances we alreay know the rate of change an we wish to etermine the original
More informationThe Explicit Form of a Function
Section 3 5 Implicit Differentiation The Eplicit Form of a Function The normal way we see function notation has f () on one sie of an equation an an epression in terms of on the other sie. We know the
More informationFINAL EXAM 1 SOLUTIONS Below is the graph of a function f(x). From the graph, read off the value (if any) of the following limits: x 1 +
FINAL EXAM 1 SOLUTIONS 2011 1. Below is the graph of a function f(x). From the graph, rea off the value (if any) of the following its: x 1 = 0 f(x) x 1 + = 1 f(x) x 0 = x 0 + = 0 x 1 = 1 1 2 FINAL EXAM
More informationChapter 2. First-Order Differential Equations
Chapter 2 First-Order Differential Equations i Let M(x, y) + N(x, y) = 0 Some equations can be written in the form A(x) + B(y) = 0 DEFINITION 2.2. (Separable Equation) A first-order differential equation
More informationYour signature: (1) (Pre-calculus Review Set Problems 80 and 124.)
(1) (Pre-calculus Review Set Problems 80 an 14.) (a) Determine if each of the following statements is True or False. If it is true, explain why. If it is false, give a counterexample. (i) If a an b are
More informationMath 1 Lecture 20. Dartmouth College. Wednesday
Math 1 Lecture 20 Dartmouth College Wenesay 10-26-16 Contents Reminers/Announcements Last Time Derivatives of Trigonometric Functions Reminers/Announcements WebWork ue Friay x-hour problem session rop
More informationSec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes
Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Infinite
More informationSec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes
Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Infinite
More informationBy writing (1) as y (x 5 1). (x 5 1), we can find the derivative using the Product Rule: y (x 5 1) 2. we know this from (2)
3.5 Chain Rule 149 3.5 Chain Rule Introuction As iscusse in Section 3.2, the Power Rule is vali for all real number exponents n. In this section we see that a similar rule hols for the erivative of a power
More informationInverse Trig Functions
6.6i Inverse Trigonometric Functions Inverse Sine Function Does g(x) = sin(x) have an inverse? What restriction would we need to make so that at least a piece of this function has an inverse? Given f (x)
More informationA Brief Review of Elementary Ordinary Differential Equations
A A Brief Review of Elementary Ordinary Differential Equations At various points in the material we will be covering, we will need to recall and use material normally covered in an elementary course on
More information4 Differential Equations
Advanced Calculus Chapter 4 Differential Equations 65 4 Differential Equations 4.1 Terminology Let U R n, and let y : U R. A differential equation in y is an equation involving y and its (partial) derivatives.
More informationMath Differential Equations Material Covering Lab 2
Math 366 - Differential Equations Material Covering Lab 2 Separable Equations A separable equation is a first-orer ODE that can be written in the form y'= g x $h y This is the general metho to solve such
More information(x 3)(x + 5) = (x 3)(x 1) = x + 5. sin 2 x e ax bx 1 = 1 2. lim
SMT Calculus Test Solutions February, x + x 5 Compute x x x + Answer: Solution: Note that x + x 5 x x + x )x + 5) = x )x ) = x + 5 x x + 5 Then x x = + 5 = Compute all real values of b such that, for fx)
More informationSection 7.1: Integration by Parts
Section 7.1: Integration by Parts 1. Introuction to Integration Techniques Unlike ifferentiation where there are a large number of rules which allow you (in principle) to ifferentiate any function, the
More information2.1 Derivatives and Rates of Change
1a 1b 2.1 Derivatives an Rates of Change Tangent Lines Example. Consier y f x x 2 0 2 x-, 0 4 y-, f(x) axes, curve C Consier a smooth curve C. A line tangent to C at a point P both intersects C at P an
More informationUNIVERSITY OF SOUTHAMPTON. A foreign language dictionary (paper version) is permitted provided it contains no notes, additions or annotations.
UNIVERSITY OF SOUTHAMPTON MATH055W SEMESTER EXAMINATION 03/4 MATHEMATICS FOR ELECTRONIC & ELECTRICAL ENGINEERING Duration: 0 min Solutions Only University approved calculators may be used. A foreign language
More informationDerivative Methods: (csc(x)) = csc(x) cot(x)
EXAM 2 IS TUESDAY IN QUIZ SECTION Allowe:. A Ti-30x IIS Calculator 2. An 8.5 by inch sheet of hanwritten notes (front/back) 3. A pencil or black/blue pen Covers: 3.-3.6, 0.2, 3.9, 3.0, 4. Quick Review
More informationTable of Common Derivatives By David Abraham
Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec
More informationx = c of N if the limit of f (x) = L and the right-handed limit lim f ( x)
Limit We say the limit of f () as approaches c equals L an write, lim L. One-Sie Limits (Left an Right-Hane Limits) Suppose a function f is efine near but not necessarily at We say that f has a left-hane
More informationChapter 6: Integration: partial fractions and improper integrals
Chapter 6: Integration: partial fractions an improper integrals Course S3, 006 07 April 5, 007 These are just summaries of the lecture notes, an few etails are inclue. Most of what we inclue here is to
More informationMath 131 Final Exam Spring 2016
Math 3 Final Exam Spring 06 Name: ID: multiple choice questions worth 5 points each. Exam is only out of 00 (so there is the possibility of getting more than 00%) Exam covers sections. through 5.4 No graphing
More information