Examples from Section 7.1: Integration by Parts Page 1
|
|
- Laureen Hodges
- 5 years ago
- Views:
Transcription
1 Examples from Section 7.: Integration by Parts Page Questions Example Determine x cos x dx. Example e θ cos θ dθ Example You may wonder why we do not add a constant at the point where we integrate for v in the parts substitution. This is because any constant added there will cancel later. Redo x cos x dx and keep all constants of integration to show this. Example: / arccos x dx. Example Prove the following integral formula (# in table) using parts: [ ln w ] + c Solutions Example Determine x cos x dx. Let s do this integral using parts. The formula for parts is for u and dv based on our integral. Let s choose udv = uv vdu, so we want to choose the two substitutions u = x dv = cos x dx du = dx v = cos x dx = sin x NOTE: once you have chosen u, dv has to be everything else that s left so we can write: x cos x dx = u dv = x sin x sin x dx = x sin x ( cos x) + c (integrate) = x sin x + cos x + c (simplify) What we have done is replaced the original integral, which we could not do, with the integral do. This is the power of parts! sin x dx, which we could
2 Examples from Section 7.: Integration by Parts Page Check our answer by taking the derivative, and verifying that we get the integrand back: d [x sin x + cos x + c] = sin x + x cos x sin x = x cos x dx The problem most people have with parts is picking the u and dv correctly. Keep in mind that we want to be able to do our new integral, so generally we pick u to be a function that gets simpler when differentiated. What would happen if we had chosen differently in the above? Let s choose u = cos x dv = x dx du = sin x dx v = x dx = x x cos x dx = u dv = uv = x cos x + x sin x dx This is true, but it doesn t help us do the integral! Sometimes you have to use parts twice, and sometimes you never have to integrate explicitly to get the integral. Example e θ cos θ dθ We want to use parts, but neither e θ nor cos θ will simplify when we take a derivative. We have to do something, so let s try u = cos θ du = sin θ dθ dv = e θ dθ v = e θ dθ = e θ So the integral becomes: e θ cos θ dθ = u dv = e θ cos θ e θ ( sin θ) dθ = e θ cos θ + e θ sin θ dθ (simplify) () Our new integral is no simpler. But, it is no worse! That tells us to try parts again, on the the integral: e θ sin θ dθ
3 Examples from Section 7.: Integration by Parts Page 3 Let s choose u = sin θ du = cos θ dθ dv = e θ dθ v = e θ dθ = e θ NOTE: if we switch our choices of u and dv here, we will get the original integral back. So the integral becomes: e θ sin θ dθ = u dv = e θ sin θ e θ cos θ dθ Now, we can substitute everything back into Equation (): e θ cos θdθ = e θ cos θ + e θ sin θ dθ = e θ cos θ + e θ sin θ e θ cos θ dθ Solve for the integral algebraically: e θ cos θ dθ = [eθ cos θ + e θ sin θ] + c Notice how we simply added a constant of integration at the end, since we had a definite integral. Example You may wonder why we do not add a constant at the point where we integrate for v in the parts substitution. This is because any constant added there will cancel later. Let s redo x cos x dx to show this. Let s choose u = x dv = cos x dx du = dx v = cos x dx = sin x + c x cos x dx = u dv = x(sin x + c ) (sin x + c ) dx = x sin x + xc [( cos x) + c x] + c (integrate) = x sin x + cos x + c (simplify) Parts is also used to prove many of the integral formulas in the back of your textbook, and especially the ones that are written in terms of reduction formulas. The following example shows where #88 from the table of integrals comes from. Definite Integrals by Parts: b b u dv = uv b a. a a
4 Examples from Section 7.: Integration by Parts Page 4 Example: / arccos x dx. To begin, we use parts and choose: u = arccos x du = dx x dv = dx v = x / arccos x dx = / u dv = uv / / = x arccos x / x ( x ) dx = arccos arccos + / x x dx Substitution: t = x when x = t = dt = xdx when x = t = 3 4 = ( π ) ( π ) 3/4 dt + 3 t ( ) = π 6 = π 6 3/4 t / (/) t / dt 3/4 = π 6 ( 3/4 ) = π 6 + 3/4 Example Prove the following integral formula (# in table) using parts: [ ln w ] + c To begin, we use parts and choose: u = ln w du = w dw dv = w n dw v = w n dw = Notice that the above integral for v is only true if n. = u dv = uv
5 Examples from Section 7.: Integration by Parts Page 5 = ln w = ln w n + = ln w n + w dw w n dw ( w n+ w n+ n + = ln w + c = w n+ [ ln w ] + c ) + c (simplification) We have shown [ ln w ] + c, n. If n =, it is not too difficult to show ln w w dw = ln w + c.
Unit #11 : Integration by Parts, Average of a Function. Goals: Learning integration by parts. Computing the average value of a function.
Unit #11 : Integration by Parts, Average of a Function Goals: Learning integration by parts. Computing the average value of a function. Integration Method - By Parts - 1 Integration by Parts So far in
More informationMAT137 Calculus! Lecture 31
MAT137 Calculus! Lecture 31 Today: Next: Integration Methods: Integration Methods: Trig. Functions (v. 9.10-9.12) Rational Functions Trig. Substitution (v. 9.13-9.15) (v. 9.16-9.17) Integration by Parts
More informationPolar Coordinates. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 28 / 46
Polar Coordinates Polar Coordinates: Given any point P = (x, y) on the plane r stands for the distance from the origin (0, 0). θ stands for the angle from positive x-axis to OP. Polar coordinate: (r, θ)
More informationPolar Coordinates. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 27 / 45
: Given any point P = (x, y) on the plane r stands for the distance from the origin (0, 0). θ stands for the angle from positive x-axis to OP. Polar coordinate: (r, θ) Chapter 10: Parametric Equations
More informationIntegration by parts, sometimes multiple times in the same problem. a. Show your choices. b. Evaluate the integral. a.
Instructions Complete the homework with a pencil. Show your work. Use extra paper if necessary. Homework can be turned in as follows Turn in homework in person at KPC Scan and e-mail to UAACalculus@yahoo.com
More information7.2 Trigonometric Integrals
7. Trigonometric Integrals The three identities sin x + cos x, cos x (cos x + ) and sin x ( cos x) can be used to integrate expressions involving powers of Sine and Cosine. The basic idea is to use an
More informationParametric Surfaces. Substitution
Calculus Lia Vas Parametric Surfaces. Substitution Recall that a curve in space is given by parametric equations as a function of single parameter t x = x(t) y = y(t) z = z(t). A curve is a one-dimensional
More informationMath 113 Exam 1 Practice
Math Exam Practice January 6, 00 Exam will cover sections 6.-6.5 and 7.-7.5 This sheet has three sections. The first section will remind you about techniques and formulas that you should know. The second
More information2. Use the above table to transform the integral into a new integral problem using the integration by parts formula below:
Integration by Parts: Basic (1238178) Due: Mon Aug 27 218 9:1 AM MDT Question 1 2 3 4 5 6 7 8 9 1 11 12 13 Instructions Read today's tes and Learning Goals 1. Question Details sp16 by parts intro 1 [34458]
More informationCalculus III. Math 233 Spring In-term exam April 11th. Suggested solutions
Calculus III Math Spring 7 In-term exam April th. Suggested solutions This exam contains sixteen problems numbered through 6. Problems 5 are multiple choice problems, which each count 5% of your total
More informationMATH 19520/51 Class 15
MATH 19520/51 Class 15 Minh-Tam Trinh University of Chicago 2017-11-01 1 Change of variables in two dimensions. 2 Double integrals via change of variables. Change of Variables Slogan: An n-variable substitution
More informationScience One Math. Jan
Science One Math Jan 24 2018 Announcements Midterm : February 13 th, 2018 Details on topics and list of practice problems to appear on course webpage soon. What integration techniques have we learned so
More informationStudy Guide for Test 2
Study Guide for Test Math 6: Calculus October, 7. Overview Non-graphing calculators will be allowed. You will need to know the following:. Set Pieces 9 4.. Trigonometric Substitutions (Section 7.).. Partial
More informationFirst of all, we need to know what it means for a parameterize curve to be differentiable. FACT:
CALCULUS WITH PARAMETERIZED CURVES In calculus I we learned how to differentiate and integrate functions. In the chapter covering the applications of the integral, we learned how to find the length of
More informationMATH 261 EXAM III PRACTICE PROBLEMS
MATH 6 EXAM III PRACTICE PROBLEMS These practice problems are pulled from actual midterms in previous semesters. Exam 3 typically has 5 (not 6!) problems on it, with no more than one problem of any given
More informationAnswer: Find the volume of the solid generated by revolving the shaded region about the given axis. 2) About the x-axis. y = 9 - x π.
Final Review Study All Eams. Omit the following sections: 6.,.6,., 8. For Ch9 and, study Eam4 and Eam 4 review sheets. Find the volume of the described solid. ) The base of the solid is the disk + y 4.
More informationMath 241, Final Exam. 12/11/12.
Math, Final Exam. //. No notes, calculator, or text. There are points total. Partial credit may be given. ircle or otherwise clearly identify your final answer. Name:. (5 points): Equation of a line. Find
More information2 Second Derivatives. As we have seen, a function f (x, y) of two variables has four different partial derivatives: f xx. f yx. f x y.
2 Second Derivatives As we have seen, a function f (x, y) of two variables has four different partial derivatives: (x, y), (x, y), f yx (x, y), (x, y) It is convenient to gather all four of these into
More informationMAC2313 Final A. a. The vector r u r v lies in the tangent plane of S at a given point. b. S f(x, y, z) ds = R f(r(u, v)) r u r v du dv.
MAC2313 Final A (5 pts) 1. Let f(x, y, z) be a function continuous in R 3 and let S be a surface parameterized by r(u, v) with the domain of the parameterization given by R; how many of the following are
More informationContents 20. Trigonometric Formulas, Identities, and Equations
Contents 20. Trigonometric Formulas, Identities, and Equations 2 20.1 Basic Identities............................... 2 Using Graphs to Help Verify Identities................... 2 Example 20.1................................
More informationScience One Math. Jan
Science One Math Jan 21 2019 Today s Goal: Compute Trigonometric Integrals Apply the technique of substitution to compute trigonometric integrals of the form sin % x cos * x dx for both m > 1 and n > 1
More information302 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES. 4. Function of several variables, their domain. 6. Limit of a function of several variables
302 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES 3.8 Chapter Review 3.8.1 Concepts to Know You should have an understanding of, and be able to explain the concepts listed below. 1. Boundary and interior points
More informationMathematics 134 Calculus 2 With Fundamentals Exam 2 Answers/Solutions for Sample Questions March 2, 2018
Sample Exam Questions Mathematics 1 Calculus 2 With Fundamentals Exam 2 Answers/Solutions for Sample Questions March 2, 218 Disclaimer: The actual exam questions may be organized differently and ask questions
More informationMATH 2400: CALCULUS 3 MAY 9, 2007 FINAL EXAM
MATH 4: CALCULUS 3 MAY 9, 7 FINAL EXAM I have neither given nor received aid on this exam. Name: 1 E. Kim................ (9am) E. Angel.............(1am) 3 I. Mishev............ (11am) 4 M. Daniel...........
More informationMath 142, Exam 1 Information.
Mth 14, Exm 1 Informtion. 9/14/10, LC 41, 9:30-10:45. Exm 1 will be bsed on: Sections 7.1-7.5. The corresponding ssigned homework problems (see http://www.mth.sc.edu/ boyln/sccourses/14f10/14.html) At
More informationMath 2374 Spring 2007 Midterm 3 Solutions - Page 1 of 6 April 25, 2007
Math 374 Spring 7 Midterm 3 Solutions - Page of 6 April 5, 7. (3 points) Consider the surface parametrized by (x, y, z) Φ(x, y) (x, y,4 (x +y )) between the planes z and z 3. (i) (5 points) Set up the
More information10.1 Curves Defined by Parametric Equations
10.1 Curves Defined by Parametric Equations Ex: Consider the unit circle from Trigonometry. What is the equation of that circle? There are 2 ways to describe it: x 2 + y 2 = 1 and x = cos θ y = sin θ When
More information1. Fill in the right hand side of the following equation by taking the derivative: (x sin x) =
7.1 What is x cos x? 1. Fill in the right hand side of the following equation by taking the derivative: (x sin x = 2. Integrate both sides of the equation. Instructor: When instructing students to integrate
More informationMath 136 Exam 1 Practice Problems
Math Exam Practice Problems. Find the surface area of the surface of revolution generated by revolving the curve given by around the x-axis? To solve this we use the equation: In this case this translates
More informationTrigonometric Integrals
Most trigonometric integrals can be solved by using trigonometric identities or by following a strategy based on the form of the integrand. There are some that are not so easy! Basic Trig Identities and
More informationR f da (where da denotes the differential of area dxdy (or dydx)
Math 28H Topics for the second exam (Technically, everything covered on the first exam, plus) Constrained Optimization: Lagrange Multipliers Most optimization problems that arise naturally are not unconstrained;
More informationContents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Homework 1 - Solutions 3. 2 Homework 2 - Solutions 13
MATH 32B-2 (8) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Homework - Solutions 3 2 Homework 2 - Solutions 3 3 Homework 3 - Solutions 9 MATH 32B-2 (8) (L) G. Liu / (TA) A. Zhou Calculus
More informationHomework 8. Due: Tuesday, March 31st, 2009
MATH 55 Applied Honors Calculus III Winter 9 Homework 8 Due: Tuesday, March 3st, 9 Section 6.5, pg. 54: 7, 3. Section 6.6, pg. 58:, 3. Section 6.7, pg. 66: 3, 5, 47. Section 6.8, pg. 73: 33, 38. Section
More information(c) 0 (d) (a) 27 (b) (e) x 2 3x2
1. Sarah the architect is designing a modern building. The base of the building is the region in the xy-plane bounded by x =, y =, and y = 3 x. The building itself has a height bounded between z = and
More informationMath 407 Solutions to Exam 3 April 24, Let f = u + iv, and let Γ be parmetrized by Γ = (x (t),y(t)) = x (t) +iy (t) for a t b.
Math 7 Solutions to Exam 3 April, 3 1. (1 Define the following: (a f (z dz, where Γ is a smooth curve with finite length. Γ Let f u + iv, and let Γ be parmetrized by Γ (x (t,y(t x (t +iy (t for a t b.
More informationChanging Variables in Multiple Integrals
Changing Variables in Multiple Integrals 3. Examples and comments; putting in limits. If we write the change of variable formula as (x, y) (8) f(x, y)dx dy = g(u, v) dudv, (u, v) where (x, y) x u x v (9)
More informationChapter 5 Partial Differentiation
Chapter 5 Partial Differentiation For functions of one variable, y = f (x), the rate of change of the dependent variable can dy be found unambiguously by differentiation: f x. In this chapter we explore
More informationChapter 4 Using Fundamental Identities Section USING FUNDAMENTAL IDENTITIES. Fundamental Trigonometric Identities. Reciprocal Identities
Chapter 4 Using Fundamental Identities Section 4.1 4.1 USING FUNDAMENTAL IDENTITIES Fundamental Trigonometric Identities Reciprocal Identities csc x sec x cot x Quotient Identities tan x cot x Pythagorean
More informationMath Triple Integrals in Cylindrical Coordinates
Math 213 - Triple Integrals in Cylindrical Coordinates Peter A. Perry University of Kentucky November 2, 218 Homework Re-read section 15.7 Work on section 15.7, problems 1-13 (odd), 17-21 (odd) from Stewart
More information1. Suppose that the equation F (x, y, z) = 0 implicitly defines each of the three variables x, y, and z as functions of the other two:
Final Solutions. Suppose that the equation F (x, y, z) implicitly defines each of the three variables x, y, and z as functions of the other two: z f(x, y), y g(x, z), x h(y, z). If F is differentiable
More informationPolar Coordinates. Calculus 2 Lia Vas. If P = (x, y) is a point in the xy-plane and O denotes the origin, let
Calculus Lia Vas Polar Coordinates If P = (x, y) is a point in the xy-plane and O denotes the origin, let r denote the distance from the origin O to the point P = (x, y). Thus, x + y = r ; θ be the angle
More informationMath 241, Exam 3 Information.
Math 241, xam 3 Information. 11/28/12, LC 310, 11:15-12:05. xam 3 will be based on: Sections 15.2-15.4, 15.6-15.8. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationMEI STRUCTURED MATHEMATICS METHODS FOR ADVANCED MATHEMATICS, C3. Practice Paper C3-B
MEI Mathematics in Education and Industry MEI STRUCTURED MATHEMATICS METHODS FOR ADVANCED MATHEMATICS, C3 Practice Paper C3-B Additional materials: Answer booklet/paper Graph paper List of formulae (MF)
More informationMAC2313 Test 3 A E g(x, y, z) dy dx dz
MAC2313 Test 3 A (5 pts) 1. If the function g(x, y, z) is integrated over the cylindrical solid bounded by x 2 + y 2 = 3, z = 1, and z = 7, the correct integral in Cartesian coordinates is given by: A.
More informationMATH 122 FINAL EXAM WINTER March 15, 2011
MATH 1 FINAL EXAM WINTER 010-011 March 15, 011 NAME: SECTION: ONLY THE CORRECT ANSWER AND ALL WORK USED TO REACH IT WILL EARN FULL CREDIT. Simplify all answers as much as possible unless explicitly stated
More informationPartial Derivatives (Online)
7in x 10in Felder c04_online.tex V3 - January 21, 2015 9:44 A.M. Page 1 CHAPTER 4 Partial Derivatives (Online) 4.7 Tangent Plane Approximations and Power Series It is often helpful to use a linear approximation
More informationCALCULUS II. Parametric Equations and Polar Coordinates. Paul Dawkins
CALCULUS II Parametric Equations and Polar Coordinates Paul Dawkins Table of Contents Preface... ii Parametric Equations and Polar Coordinates... 3 Introduction... 3 Parametric Equations and Curves...
More informationExam 3 SCORE. MA 114 Exam 3 Spring Section and/or TA:
MA 114 Exam 3 Spring 217 Exam 3 Name: Section and/or TA: Last Four Digits of Student ID: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test.
More informationMath 142 Fall 2000 Rotation of Axes. In section 11.4, we found that every equation of the form. (1) Ax 2 + Cy 2 + Dx + Ey + F =0,
Math 14 Fall 000 Rotation of Axes In section 11.4, we found that every equation of the form (1) Ax + Cy + Dx + Ey + F =0, with A and C not both 0, can be transformed by completing the square into a standard
More informationMath Exam III Review
Math 213 - Exam III Review Peter A. Perry University of Kentucky April 10, 2019 Homework Exam III is tonight at 5 PM Exam III will cover 15.1 15.3, 15.6 15.9, 16.1 16.2, and identifying conservative vector
More informationMath 265 Exam 3 Solutions
C Roettger, Fall 16 Math 265 Exam 3 Solutions Problem 1 Let D be the region inside the circle r 5 sin θ but outside the cardioid r 2 + sin θ. Find the area of D. Note that r and θ denote polar coordinates.
More informationTopic 6: Calculus Integration Volume of Revolution Paper 2
Topic 6: Calculus Integration Standard Level 6.1 Volume of Revolution Paper 1. Let f(x) = x ln(4 x ), for < x
More informationBackground for Surface Integration
Background for urface Integration 1 urface Integrals We have seen in previous work how to define and compute line integrals in R 2. You should remember the basic surface integrals that we will need to
More informationWriting Equations of Parallel and Perpendicular Lines
Writing Equations of Parallel and Perpendicular Lines The coordinate plane provides a connection between algebra and geometry. Postulates 17 and 18 establish a simple way to find lines that are parallel
More informationMath 205 Test 3 Grading Guidelines Problem 1 Part a: 1 point for figuring out r, 2 points for setting up the equation P = ln 2 P and 1 point for the initial condition. Part b: All or nothing. This is really
More informationSection Parametrized Surfaces and Surface Integrals. (I) Parametrizing Surfaces (II) Surface Area (III) Scalar Surface Integrals
Section 16.4 Parametrized Surfaces and Surface Integrals (I) Parametrizing Surfaces (II) Surface Area (III) Scalar Surface Integrals MATH 127 (Section 16.4) Parametrized Surfaces and Surface Integrals
More informationTest 1 - Answer Key Version A
MATH 8 Test - Answer Key Sring 6 Sections 6. - 6.5, 7. - 7.3 Student s Printed Name: Instructor: CUID: Section: Instructions: You are not ermitted to use a calculator on any ortion of this test. You are
More informationGrad operator, triple and line integrals. Notice: this material must not be used as a substitute for attending the lectures
Grad operator, triple and line integrals Notice: this material must not be used as a substitute for attending the lectures 1 .1 The grad operator Let f(x 1, x,..., x n ) be a function of the n variables
More informationVersion 001 Polar Curve Review rittenhouse (RittBCblock2) correct
Version 1 Polar Curve Review rittenhouse (RittBCblock 1 This print-out should have 9 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. BC 1993
More information2 Unit Bridging Course Day 10
1 / 31 Unit Bridging Course Day 10 Circular Functions III The cosine function, identities and derivatives Clinton Boys / 31 The cosine function The cosine function, abbreviated to cos, is very similar
More informationMath 32B Discussion Session Week 2 Notes January 17 and 24, 2017
Math 3B Discussion Session Week Notes January 7 and 4, 7 This week we ll finish discussing the double integral for non-rectangular regions (see the last few pages of the week notes) and then we ll touch
More informationChapter 15 Notes, Stewart 7e
Contents 15.2 Iterated Integrals..................................... 2 15.3 Double Integrals over General Regions......................... 5 15.4 Double Integrals in Polar Coordinates..........................
More information27. Tangent Planes & Approximations
27. Tangent Planes & Approximations If z = f(x, y) is a differentiable surface in R 3 and (x 0, y 0, z 0 ) is a point on this surface, then it is possible to construct a plane passing through this point,
More informationFocusing properties of spherical and parabolic mirrors
Physics 5B Winter 008 Focusing properties of spherical and parabolic mirrors 1 General considerations Consider a curved mirror surface that is constructed as follows Start with a curve, denoted by y()
More informationMIDTERM. Section: Signature:
MIDTERM Math 32B 8/8/2 Name: Section: Signature: Read all of the following information before starting the exam: Check your exam to make sure all pages are present. NO CALCULATORS! Show all work, clearly
More information(i) h(7,8,24) (ii) h(6,5,6) (iii) h( 7,8,9) (iv) h(10,9, 16) (iv) g 3,32 8
M252 Practice Eam for 12.1-12.9 1. Find and simplify the function values. f ( y, ) = 5 10y (i) f(0,0) (ii) f(0,1) (iii) f(3,9) (iv) f(1,y) (v) f(,0) (vi) f(t,1) 2. Find and simplify the function values.
More informationSolutions to the Exercises of Chapter 13 = 2( ) + ( ) + ( ) + ( ) = =67.
Solutions to the Exercises of Chapter A. Riemann Sums. a,b5,m. The partition is < < < < 5. So x, x, x, x 5, and p x,p x,p x, and p p p p 5 p x 5. So fp i x i fp +fp +fp +fp i 5 + 5 + 5 + 5 5 +6+9+67..
More informationCalculus II (Math 122) Final Exam, 11 December 2013
Name ID number Sections B Calculus II (Math 122) Final Exam, 11 December 2013 This is a closed book exam. Notes and calculators are not allowed. A table of trigonometric identities is attached. To receive
More informationPolar Coordinates
Polar Coordinates 7-7-2 Polar coordinates are an alternative to rectangular coordinates for referring to points in the plane. A point in the plane has polar coordinates r,θ). r is roughly) the distance
More informationTriple Integrals in Rectangular Coordinates
Triple Integrals in Rectangular Coordinates P. Sam Johnson April 10, 2017 P. Sam Johnson (NIT Karnataka) Triple Integrals in Rectangular Coordinates April 10, 2017 1 / 28 Overview We use triple integrals
More informationMATH 31A HOMEWORK 9 (DUE 12/6) PARTS (A) AND (B) SECTION 5.4. f(x) = x + 1 x 2 + 9, F (7) = 0
FROM ROGAWSKI S CALCULUS (2ND ED.) SECTION 5.4 18.) Express the antiderivative F (x) of f(x) satisfying the given initial condition as an integral. f(x) = x + 1 x 2 + 9, F (7) = 28.) Find G (1), where
More informationPre Calculus Worksheet: Fundamental Identities Day 1
Pre Calculus Worksheet: Fundamental Identities Day 1 Use the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy AND those strategies before. Strategy
More informationUse the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy AND those strategies before.
Pre Calculus Worksheet: Fundamental Identities Day 1 Use the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy AND those strategies before. Strategy
More information18.01 Single Variable Calculus Fall 2006
MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lecture 6: Trigonometric
More informationFirst we consider how to parameterize a surface (similar to a parameterized curve for line integrals). Surfaces will need two parameters.
Math 55 - Vector Calculus II Notes 14.6 urface Integrals Let s develop some surface integrals. First we consider how to parameterize a surface (similar to a parameterized curve for line integrals). urfaces
More informationMath 210, Exam 2, Spring 2010 Problem 1 Solution
Math, Exam, Spring Problem Solution. Find and classify the critical points of the function f(x,y) x 3 +3xy y 3. Solution: By definition, an interior point (a,b) in the domain of f is a critical point of
More informationTHS Step By Step Calculus Chapter 3
Name: Class Period: Throughout this packet there will be blanks you are expected to fill in prior to coming to class. This packet follows your Larson Textbook. Do NOT throw away! Keep in 3 ring-binder
More informationReview 1. Richard Koch. April 23, 2005
Review Richard Koch April 3, 5 Curves From the chapter on curves, you should know. the formula for arc length in section.;. the definition of T (s), κ(s), N(s), B(s) in section.4. 3. the fact that κ =
More informationThe diagram above shows a sketch of the curve C with parametric equations
1. The diagram above shows a sketch of the curve C with parametric equations x = 5t 4, y = t(9 t ) The curve C cuts the x-axis at the points A and B. (a) Find the x-coordinate at the point A and the x-coordinate
More informationWorksheet 3.4: Triple Integrals in Cylindrical Coordinates. Warm-Up: Cylindrical Volume Element d V
Boise State Math 275 (Ultman) Worksheet 3.4: Triple Integrals in Cylindrical Coordinates From the Toolbox (what you need from previous classes) Know what the volume element dv represents. Be able to find
More informationMath 209, Fall 2009 Homework 3
Math 209, Fall 2009 Homework 3 () Find equations of the tangent plane and the normal line to the given surface at the specified point: x 2 + 2y 2 3z 2 = 3, P (2,, ). Solution Using implicit differentiation
More information38. Triple Integration over Rectangular Regions
8. Triple Integration over Rectangular Regions A rectangular solid region S in R can be defined by three compound inequalities, a 1 x a, b 1 y b, c 1 z c, where a 1, a, b 1, b, c 1 and c are constants.
More informationName: Signature: Section and TA:
Name: Signature: Section and TA: Math 7. Lecture 00 (V. Reiner) Midterm Exam I Thursday, February 8, 00 This is a 50 minute exam. No books, notes, calculators, cell phones or other electronic devices are
More informationMATH. 2153, Spring 16, MWF 12:40 p.m. QUIZ 1 January 25, 2016 PRINT NAME A. Derdzinski Show all work. No calculators. The problem is worth 10 points.
MATH. 2153, Spring 16, MWF 12:40 p.m. QUIZ 1 January 25, 2016 PRINT NAME A. Derdzinski Show all work. No calculators. The problem is worth 10 points. 1. Evaluate the area A of the triangle with the vertices
More information2.2 Limit of a Function and Limit Laws
Limit of a Function and Limit Laws Section Notes Page Let s look at the graph y What is y()? That s right, its undefined, but what if we wanted to find the y value the graph is approaching as we get close
More informationTopic 5.1: Line Elements and Scalar Line Integrals. Textbook: Section 16.2
Topic 5.1: Line Elements and Scalar Line Integrals Textbook: Section 16.2 Warm-Up: Derivatives of Vector Functions Suppose r(t) = x(t) î + y(t) ĵ + z(t) ˆk parameterizes a curve C. The vector: is: r (t)
More informationLecture 23. Surface integrals, Stokes theorem, and the divergence theorem. Dan Nichols
Lecture 23 urface integrals, tokes theorem, and the divergence theorem an Nichols nichols@math.umass.edu MATH 233, pring 218 University of Massachusetts April 26, 218 (2) Last time: Green s theorem Theorem
More informationMATH 2400, Analytic Geometry and Calculus 3
MATH 2400, Analytic Geometry and Calculus 3 List of important Definitions and Theorems 1 Foundations Definition 1. By a function f one understands a mathematical object consisting of (i) a set X, called
More informationYou can take the arccos of both sides to get θ by itself.
.7 SOLVING TRIG EQUATIONS Example on p. 8 How do you solve cos ½ for? You can tae the arccos of both sides to get by itself. cos - (cos ) cos - ( ½) / However, arccos only gives us an answer between 0
More informationSOME PROPERTIES OF TRIGONOMETRIC FUNCTIONS. 5! x7 7! + = 6! + = 4! x6
SOME PROPERTIES OF TRIGONOMETRIC FUNCTIONS PO-LAM YUNG We defined earlier the sine cosine by the following series: sin x = x x3 3! + x5 5! x7 7! + = k=0 cos x = 1 x! + x4 4! x6 6! + = k=0 ( 1) k x k+1
More information" dx v(x) $ % You may also have seen this written in shorthand form as. & ' v(x) + u(x) '# % ! d
Calculus II MAT 146 Mthods of Intgration: Intgration by Parts Just as th mthod of substitution is an intgration tchniqu that rvrss th drivativ procss calld th chain rul, Intgration by parts is a mthod
More informationOrdinary differential equations solving methods
Radim Hošek, A07237 radhost@students.zcu.cz Ordinary differential equations solving methods Problem: y = y2 (1) y = x y (2) y = sin ( + y 2 ) (3) Where it is possible we try to solve the equations analytically,
More information2/3 Unit Math Homework for Year 12
Yimin Math Centre 2/3 Unit Math Homework for Year 12 Student Name: Grade: Date: Score: Table of contents 12 Trigonometry 2 1 12.1 The Derivative of Trigonometric Functions....................... 1 12.2
More informationPURE MATHEMATICS 212 Multivariable Calculus CONTENTS. Page. 1. Assignment Summary... i 2. Summary Assignments...2
PURE MATHEMATICS 212 Multivariable Calculus CONTENTS Page 1. Assignment Summary... i 2. Summary...1 3. Assignments...2 i PMTH212, Multivariable Calculus Assignment Summary 2010 Assignment Date to be Posted
More informationDr. Allen Back. Nov. 21, 2014
Dr. Allen Back of Nov. 21, 2014 The most important thing you should know (e.g. for exams and homework) is how to setup (and perhaps compute if not too hard) surface integrals, triple integrals, etc. But
More informationSolution of final examination
of final examination Math 20, pring 201 December 9, 201 Problem 1 Let v(t) (2t e t ) i j + π cos(πt) k be the velocity of a particle with initial position r(0) ( 1, 0, 2). Find the accelaration at the
More informationMath 241 Spring 2015 Final Exam Solutions
Math 4 Spring 5 Final Exam Solutions. Find the equation of the plane containing the line x y z+ and the point (,,). Write [ pts] your final answer in the form ax+by +cz d. Solution: A vector parallel to
More informationMathematics 205 HWK 21 Solutions Section 16.5 p766
Mathematics 5 HK 1 Solutions Section 16.5 p766 Problem 5, 16.5, p766. For the region shown (a rectangular slab of dimensions 1 5; see the text), choose coordinates and set up a triple integral, including
More informationMath 113 Calculus III Final Exam Practice Problems Spring 2003
Math 113 Calculus III Final Exam Practice Problems Spring 23 1. Let g(x, y, z) = 2x 2 + y 2 + 4z 2. (a) Describe the shapes of the level surfaces of g. (b) In three different graphs, sketch the three cross
More informationPreliminary Mathematics of Geometric Modeling (3)
Preliminary Mathematics of Geometric Modeling (3) Hongxin Zhang and Jieqing Feng 2006-11-27 State Key Lab of CAD&CG, Zhejiang University Differential Geometry of Surfaces Tangent plane and surface normal
More information