1. Fill in the right hand side of the following equation by taking the derivative: (x sin x) =
|
|
- Iris Whitehead
- 5 years ago
- Views:
Transcription
1 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 both sides of the equation, I tell not to worry about the +C. They may need to be encouraged to use the Fundamental Theorem of Calculus on the left hand side. The students should, of course, get stuck on integrating x cos x, and hopefully they recognize that this is the original question. They re equations should look like: (x sin x = sin x + x cos x (x sin x = sin x + x cos x x sin x = cos x + x cos x 3. Now solve for x cos x. If we think of substitution (u substitution, section 5.3 as the inverse to the chain rule in differentiation, we may think of integration by parts as the inverse to the product rule. 1. Fill in the right hand side of the following equation by taking the derivative (using the product rule: (f(x g(x = 2. Now integrate both sides of the resulting equation. Instructor: When instructing students to integrate both sides of the equation, I tell not to worry about the +C. They may need to be encouraged to use the Fundamental Theorem of Calculus on the left hand side. They re equations should look like: (f(x g(x = f (xg(x + f(xg (x (f(x g(x = f (xg(x + f(xg (x f(x g(x = f (xg(x + f(xg (x 3. Now solve for f (xg(x. This is the formula for integration by parts!
2 L.I.P.E.T. When attempting integration by parts, the first step is to choose u and dv in the left hand side of the formula u dv = u v v du The mnemonic LIPET can help with this choice. LIPET helps us remember the order for the five kinds of functions: L: log I: inverse trig P: power E: exponential T: trig ln x, log x tan 1 x, sin 1 x x 2, 3 x, 1 x 2 x, e x, ( 1 4 x cos x, csc x When attempting integration by parts, if the left hand side of the above equation, u dv, is a product of two types of functions from the LIPET list, then choose the function of the type farthest to the left as u, and the function (along with the differential of the type farthest to the right as dv. 1. Try LIPET on the following integral: x cos x. Instructor: I especially like students to work this example after having done the first activity listed her. 2. x ln x Instructor: If there is time, I like to give the students the following list: x ln x ln x x 2 ln x x It helps them remember that 1 should be interpreted as a power function and x2 ln x that is better done with substitution. Generally, don t forget to look x for substitutions first, as they usually provide the best method for solving an integral. 3. t 2 e t dt Instructor: This is a gentle introduction into using LIPET multiple times in a single problem. 4. ln x Instructor: Here, I give them the hint to rewrite the integral as ln x = ln x 1 and to think of 1 as the power function x, then do LIPET. 5. e x cos x (a Do LIPET once, then stop. Instructor: Equation will be e x cos x = e x sin x (b Separately, do LIPET on e x sin x. e x sin x
3 (c Plug this into your original computation. Instructor: Equation should be e x cos x = e x sin x e x sin x ( = e x sin x e x cos x + e x cos x = e x sin x + e x cos x e x cos x This is a great opportunity to make sure that students are using parentheses when they do integration by parts twice. Give them time to articulate amongst themselves that they have come up against a wall, the same integral they started with. If possible, one of the students should make the leap that they can bring that integral to the other side of the equation. (d Solve for e x cos x. Instructor: If students are still stuck, hint that they can solve algebraically. You may even suggest replacing all factors of e x cos x with the variable I, then solve for I. 6. x 3 e x2 Instructor: I like to give students a chance to try LIPET on this integral, and fail. Then I ask why LIPET isn t working. If the students do not reach this conclusion themselves, I make a strong point that e x2 is NOT an exponential function (a fact that comes up again in series. (a Rewrite x 3 e x2 = x x 2 e x2 (b Perform the substitution t = x 2. (c Perform LIPET on the resulting integral. Instructor: If you are using the u-dv notation, this is a convenient place to point out that substituting with the variable u and then doing integration by parts can be confusing. Just switch variables, then do LIPET. 7. sin x (a Perform the substitution t = x. Instructor: This is a very difficult problem for students. It breaks the formal rules set up for substitutions up to this point. It can be approached as a backwards substitution (x = t 2, but I never treat it this way. That is, of course, up to you, and will avoid some of the issues that come up when having students do this activity. If you are treating the substitution t = x, students will almost certainly get stuck in the following spot: t = x dt = 1 2 x
4 and find that there is no matching dt in the original integral. I usually tell them that I would really like to do the following cheat 2 x dt = but that this mixes x s and t s. Then I ask what I can do to get around that problem. Usually with that phrasing, a student will eventually suggest replacing x with t once it has crossed over to join the dt differential, yielding 2t dt = then I have the students complete the substitution, giving sin x = 2t (b Perform LIPET on 2t (c Complete the integral by replace t with x
5 6.2 A new way to see cylinders. 1. Graph the region R bounded by x = 2, x = 3, y =, and y = Rotate the region R about the x-axis. (a Graph this 3-d object. What geometric object is it? (b What is the volume formula for this object? Use this formula to calculate the volume of this object. Instructor: The idea here is to get students using the volume formula for a cylinder, and seeing how the area formula for the disc is used. This should help them remember to use the formula for the disc in the integral formula coming up. (c What is the formula for calculating the volume of this object using integrals? Note that the area of a disc is used in this formula, also. 3. Rotate the region R about the y-axis. (a Graph this 3-d object. What geometric object is it? Instructor: This may not be something they ve seen before, but students should articulate that the object is basically a difference of cylinders, and the class should treat it this way after that. (b What is the volume formula for this object? Use this formula to calculate the volume of this object. Instructor: Students should use this idea of a difference of cylinders to get to the following equation: V = πr 2 h πr 2 h = ( πr 2 πr 2 h We want them to interpret the volume as the area of the base times the height, and the area of the base is the area of an annulus: A = πr 2 πr 2 which will be used in the integral formula as well. (c What is the formula for calculating the volume of this object using integrals? Note that the area of an annulus is used in this formula, also. A new way to see cones. 1. Graph the region R bounded by y = x, x = 1, and y =. 2. Rotate the region R about the x-axis. (a Graph this 3-d object. What geometric object is it? (b What is the volume formula for this object? Use this formula to calculate the volume of this object. Instructor: Don t focus on the formula for the volume of a cone as much as we focused on the volume of the cylinder or the difference of cylinders. The cone volume isn t used in the integral computation, we re just allowing
6 the student to see that the integral computation matches up with other geometric formulas where they are available. Then, in a following problem, we discuss when the solid of revolution is not one of our familiar geometric, we are left only with the integral formula. (c What is the formula for calculating the volume of this object using integrals? Note that we still use the formula for the area of a disc, since slices of this object are 2-dimensional discs. 3. Rotate the region R about the y-axis. (a Graph this 3-d object. What geometric object is it? Instructor: At this point, students should collectively come to the conclusion that the object is a cylinder minus a cone. (b What is a volume formula for this object? Use this formula to calculate the volume of this object. Instructor: If there is time, you might use this opportunity to discuss how the volume of a cone is 1/3 the volume of a full cylinder, and so for this object - a cylinder minus a cone - the volume will be 2/3 the volume of a full cylinder. Again, we don t want to spend too much time on this formula as it will not be used in the integral equations. (c What is the formula for calculating the volume of this object using integrals? Note that we still use the area of an annulus, since slices of this object are 2-dimensional annuli. Seeing solids of revolution as deformations of cylinders. 1. What is the volume of the solid obtained by rotating the region R bounded by y = sin x + 2, y =, x =, and x = 2π about the x-axis. Instructor: The emphasis here is not on the actual functions being used, but that the object they graph will look like a deformation of a cylinder. This allows them to see how we are generalizing the idea of the volume of a cylinder, and also why the area for a disc still comes into play in these equations. (a Graph the region R bounded by y = sin x + 2, y =, x =, and x = 2π. (b Imagine rotating this region about the x-axis. Graph this 3-d object as best you can. Can you see it as a familiar geometric object that has been warped in some way? (c What are the slices of this solid? Are they the same as the slices of the geometric object you thought about it in part b? Note that since this object has been warped slightly, or deformed slightly, there are no geometric formulas we know to calculate its volume. But the slices are still familiar geometric objects (in this case, discs, and we re exploiting that fact to calculate the volume of this more complicated solid. (d What is the formula for calculating the volume of this object using integrals? Instructor: We re not interested in the actual volume of this object, or integrating the functions involved (in this case, a sin 2 x which involves a trig identity and may be too involved at this point. So if you like, just have the students set up the integral without trying to compute its numerical value.
7 2. What is the volume of the solid obtained by rotating the region R bounded by y = sin x + 4, y = sin x + 2, x =, and x = 2π about the x-axis. Instructor: This is clearly similar to the first part of this activity, but we are now exploring the idea of a deformation of a cylinder minus a cylinder, which will still use the formula for an annulus. 3. Imagine rotating this region about the x-axis. Graph this 3-d object as best you can. Can you see it as a familiar geometric object that has been warped in some way? 4. What are the slices of this solid? Are they the same as the slices of the geometric object you thought about it in part b? Note that since this object has been warped slightly, or deformed slightly, there are no geometric formulas we know to calculate its volume. But the slices are still familiar geometric objects (in this case, annuli, and we re exploiting that fact to calculate the volume of this more complicated solid. 5. What is the formula for calculating the volume of this object using integrals? Instructor: We re not interested in the actual volume of this object, or integrating the functions involved (in this case, a sin 2 x which involves a trig identity and may be too involved at this point. So if you like, just have the students set up the integral without trying to compute its numerical value.
8 5.3 Exploring g(x = 1. g(x = x x f(t dt t dt, x (a Let f(t = t, t. Graph f(t. Make sure to use a t-axis and a y-axis. (b We re going to compute 2. g(x = x t dt, x. First, plot an x > on the t-axis. This is some positive real number, but we re not choosing its value. We re treating the variable x like a constant here, usually reserved for symbols like a, b, or especially c. (c Now graph f(x, which is just x again in this case, but on the y-axis. (d Let s use the geometric interpretation of g(x. Thus g(x is the area under the graph of f(t and above the t-axis from t = to t = x. What shape does this give us? (e Use the geometry formula for the area of a triangle to determine g(x (i.e. A = 1 2 b h. (f We see that g(x = 1 2 x2, an antiderivative of f(x. (Note that the variables have switched. g(x is an antiderivative of f(x, not f(t. g(x = x cos t dt, x 2π (a Let f(t = cos t, t 2π. Graph f(t. Make sure to use a t-axis and a y-axis. (b We re going to try to graph g(x. Let s start by filling in the following table as best as we can: x π 2 π 3π 2 2π g(x Instructor: Students should reach the following conclusions: i. g( = cos t dt = using a basic property of definite integrals. ii. g(π/2 = π/2 cos t dt: students will need more help here to see that they can not find an exact value, but the class will reach a consensus that it is some positive value α, and you want to make sure they identify α with the area of that geometric shape, the first hill of f(t.
9 iii. g(π = π cos t dt = π/2 cos t dt + π cos t dt = : students should now π/2 see that the first hill of f(t will cancel with the first valley of f(t. Point out that this means that π cos t dt = α. Make sure they π/2 identify this in the graph of f(t. iv. g(3π/2 = α: using the above logic about the graph of f(t, students should be able to see that the total integral here will be α. v. g(2π = follows similarly. The table will look like this now: x g(x π α 2 π 3π α 2 2π (c Plot the points in the chart. What curve fits these points? (d What do you guess g(x is? Instructor: It should be clear now that g(x = sin x. Point out that this is an antiderivative of f(x. Again, note the change in variables. ( d ln x x 1. First, use the property that b a f(x = Instructor: Students should get ( d ln x x c a f(x + b = d ( c + x ( c = d x c f(x ln x c + d ( ln x c 2. Second, use the property that b f(x = a a b f(x
10 Instructor: Students should get ( d c + d ( ln x x c ( = d = d x c ( x c + d + d ( ln x c ( ln x c 3. Last, use the Chain Rule with the First Part of the Fundamental Theorem of Calculus: ( d u f(t dt = d u f(t dt du a du a = f(u du Instructor: Students should get ( d x + d ( ln x c c = sin ( x d d x + sin (ln x ln x = sin ( x sin (ln x x x
11 NAME: EID: Final Exam M48S Spring 215 To receive credit, write your name on every sheet, show all work, give reasons for answers. 1. (1 points Estimate 1 centered at. x tan 1 (x 2 using the Taylor polynomial T 3 (x for x tan 1 (x 2
12 2. (1 points What is the radius of convergence for f(x = n= (2n! (n! 2 xn?
13 NAME: EID: Final Exam M48S Spring 215 To receive credit, write your name on every sheet, show all work, give reasons for answers. 3. (1 points Compute T 3 (x centered at π/3 for f(x = cos x.
14 4. (1 points The power series f(x = n=1 n n 2 + n + 1 xn has radius of convergence R = 1. What is the interval of convergence?
Math 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 informationMath 2260 Exam #1 Practice Problem Solutions
Math 6 Exam # Practice Problem Solutions. What is the area bounded by the curves y x and y x + 7? Answer: As we can see in the figure, the line y x + 7 lies above the parabola y x in the region we care
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 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 informationIn this chapter, we will investigate what have become the standard applications of the integral:
Chapter 8 Overview: Applications of Integrals Calculus, like most mathematical fields, began with trying to solve everyday problems. The theory and operations were formalized later. As early as 70 BC,
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 informationSection 7.2 Volume: The Disk Method
Section 7. Volume: The Disk Method White Board Challenge Find the volume of the following cylinder: No Calculator 6 ft 1 ft V 3 1 108 339.9 ft 3 White Board Challenge Calculate the volume V of the solid
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 informationIntegration. Edexcel GCE. Core Mathematics C4
Edexcel GCE Core Mathematics C Integration Materials required for examination Mathematical Formulae (Green) Items included with question papers Nil Advice to Candidates You must ensure that your answers
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 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 informationMATH 104 Sample problems for first exam - Fall MATH 104 First Midterm Exam - Fall (d) 256 3
MATH 14 Sample problems for first exam - Fall 1 MATH 14 First Midterm Exam - Fall 1. Find the area between the graphs of y = 9 x and y = x + 1. (a) 4 (b) (c) (d) 5 (e) 4 (f) 81. A solid has as its base
More informationMath 126 Winter CHECK that your exam contains 8 problems.
Math 126 Winter 2016 Your Name Your Signature Student ID # Quiz Section Professor s Name TA s Name CHECK that your exam contains 8 problems. This exam is closed book. You may use one 8 1 11 sheet of hand-written
More informationThe Fundamental Theorem of Calculus Using the Rule of Three
The Fundamental Theorem of Calculus Using the Rule of Three A. Approimations with Riemann sums. The area under a curve can be approimated through the use of Riemann (or rectangular) sums: n Area f ( k
More informationMATH 104 First Midterm Exam - Fall (d) A solid has as its base the region in the xy-plane the region between the curve y = 1 x2
MATH 14 First Midterm Exam - Fall 214 1. Find the area between the graphs of y = x 2 + x + 5 and y = 2x 2 x. 1. Find the area between the graphs of y = x 2 + 4x + 6 and y = 2x 2 x. 1. Find the area between
More informationCHAPTER 6: APPLICATIONS OF INTEGRALS
(Exercises for Section 6.1: Area) E.6.1 CHAPTER 6: APPLICATIONS OF INTEGRALS SECTION 6.1: AREA 1) For parts a) and b) below, in the usual xy-plane i) Sketch the region R bounded by the graphs of the given
More informationMA 114 Worksheet #17: Average value of a function
Spring 2019 MA 114 Worksheet 17 Thursday, 7 March 2019 MA 114 Worksheet #17: Average value of a function 1. Write down the equation for the average value of an integrable function f(x) on [a, b]. 2. Find
More informationNotice that the height of each rectangle is and the width of each rectangle is.
Math 1410 Worksheet #40: Section 6.3 Name: In some cases, computing the volume of a solid of revolution with cross-sections can be difficult or even impossible. Is there another way to compute volumes
More informationSolving Trigonometric Equations
OpenStax-CNX module: m49398 1 Solving Trigonometric Equations OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section, you
More informationAB Student Notes: Area and Volume
AB Student Notes: Area and Volume An area and volume problem has appeared on every one of the free response sections of the AP Calculus exam AB since year 1. They are straightforward and only occasionally
More informationCurves: We always parameterize a curve with a single variable, for example r(t) =
Final Exam Topics hapters 16 and 17 In a very broad sense, the two major topics of this exam will be line and surface integrals. Both of these have versions for scalar functions and vector fields, and
More informationVolumes of Rotation with Solids of Known Cross Sections
Volumes of Rotation with Solids of Known Cross Sections In this lesson we are going to learn how to find the volume of a solid which is swept out by a curve revolving about an ais. There are three main
More informationAP Calculus AB. Table of Contents. Slide 1 / 180. Slide 2 / 180. Slide 3 / 180. Review Unit
Slide 1 / 180 Slide 2 / 180 P alculus Review Unit 2015-10-20 www.njctl.org Table of ontents lick on the topic to go to that section Slide 3 / 180 Slopes Equations of Lines Functions Graphing Functions
More informationInternational Conference Las Vegas, NV, USA March 7-9, 2014
International Conference Las Vegas, NV, USA March 7-9, 2014 Overview About ETS (engineering school) Why Nspire CAS? Why Computer Algebra? Examples in pre-calculus Examples in single variable calculus Examples
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 informationMEI Desmos Tasks for AS Pure
Task 1: Coordinate Geometry Intersection of a line and a curve 1. Add a quadratic curve, e.g. y = x² 4x + 1 2. Add a line, e.g. y = x 3 3. Select the points of intersection of the line and the curve. What
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 informationAP * Calculus Review. Area and Volume
AP * Calculus Review Area and Volume Student Packet Advanced Placement and AP are registered trademark of the College Entrance Examination Board. The College Board was not involved in the production of,
More informationMultivariate Calculus Review Problems for Examination Two
Multivariate Calculus Review Problems for Examination Two Note: Exam Two is on Thursday, February 28, class time. The coverage is multivariate differential calculus and double integration: sections 13.3,
More informationHSC Mathematics - Extension 1. Workshop E2
HSC Mathematics - Extension Workshop E Presented by Richard D. Kenderdine BSc, GradDipAppSc(IndMaths), SurvCert, MAppStat, GStat School of Mathematics and Applied Statistics University of Wollongong Moss
More informationDefinition 3.1 The partial derivatives of a function f(x, y) defined on an open domain containing (x, y) are denoted by f
Chapter 3 Draft October 3, 009 3. Partial Derivatives Overview: Partial derivatives are defined by differentiation in one variable, viewing all others as constant (frozen at some value). The reduction
More informationLinear algebra deals with matrixes: two-dimensional arrays of values. Here s a matrix: [ x + 5y + 7z 9x + 3y + 11z
Basic Linear Algebra Linear algebra deals with matrixes: two-dimensional arrays of values. Here s a matrix: [ 1 5 ] 7 9 3 11 Often matrices are used to describe in a simpler way a series of linear equations.
More informationMultivariate Calculus: Review Problems for Examination Two
Multivariate Calculus: Review Problems for Examination Two Note: Exam Two is on Tuesday, August 16. The coverage is multivariate differential calculus and double integration. You should review the double
More information5/27/12. Objectives 7.1. Area of a Region Between Two Curves. Find the area of a region between two curves using integration.
Objectives 7.1 Find the area of a region between two curves using integration. Find the area of a region between intersecting curves using integration. Describe integration as an accumulation process.
More informationMath 32, August 20: Review & Parametric Equations
Math 3, August 0: Review & Parametric Equations Section 1: Review This course will continue the development of the Calculus tools started in Math 30 and Math 31. The primary difference between this course
More informationUPCAT Reviewer Booklet
UPCAT Reviewer Booklet I. Linear Equations y = y-value at a certain point in the graph x = x-value at a certain point in the graph b = a constant m = the slope of the line Section 1 Mathematics Linear
More informationObjectives. Materials
Activity 13 Objectives Understand what a slope field represents in terms of Create a slope field for a given differential equation Materials TI-84 Plus / TI-83 Plus Graph paper Introduction One of the
More informationVolume by Slicing (Disks & Washers)
Volume by Slicing (Disks & Washers) SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference Chapter 6.2 of the recommended textbook (or the equivalent chapter
More informationEuler s Method for Approximating Solution Curves
Euler s Method for Approximating Solution Curves As you may have begun to suspect at this point, time constraints will allow us to learn only a few of the many known methods for solving differential equations.
More informationLinear Transformations
Linear Transformations The two basic vector operations are addition and scaling From this perspective, the nicest functions are those which preserve these operations: Def: A linear transformation is a
More informationNAME: Section # SSN: X X X X
Math 155 FINAL EXAM A May 5, 2003 NAME: Section # SSN: X X X X Question Grade 1 5 (out of 25) 6 10 (out of 25) 11 (out of 20) 12 (out of 20) 13 (out of 10) 14 (out of 10) 15 (out of 16) 16 (out of 24)
More informationSecond Edition. Concept Builders. Jana Kohout
Second Edition Concept Builders Jana Kohout First published in Australia as an online resource in 016. Edited and printed in 017. Jana Kohout 017 Reproduction and Communication for educational purposes
More information8.6 Other Trigonometric Functions
8.6 Other Trigonometric Functions I have already discussed all the trigonometric functions and their relationship to the sine and cosine functions and the x and y coordinates on the unit circle, but let
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surface of revolution is formed when a curve is rotated about a line. Such a surface is the lateral boundar of a solid of revolution of the tpe discussed
More informationFinal Exam May 2, 2017
Math 07 Calculus II Name: Final Exam May, 07 Circle the name of your instructor and, in the appropriate column, the name of your recitation leader. The second row is the time of your lecture. Radu Ledder
More informationAppendix D Trigonometry
Math 151 c Lynch 1 of 8 Appendix D Trigonometry Definition. Angles can be measure in either degree or radians with one complete revolution 360 or 2 rad. Then Example 1. rad = 180 (a) Convert 3 4 into degrees.
More informationColumbus State Community College Mathematics Department Public Syllabus. Course and Number: MATH 1172 Engineering Mathematics A
Columbus State Community College Mathematics Department Public Syllabus Course and Number: MATH 1172 Engineering Mathematics A CREDITS: 5 CLASS HOURS PER WEEK: 5 PREREQUISITES: MATH 1151 with a C or higher
More informationExploring Fractals through Geometry and Algebra. Kelly Deckelman Ben Eggleston Laura Mckenzie Patricia Parker-Davis Deanna Voss
Exploring Fractals through Geometry and Algebra Kelly Deckelman Ben Eggleston Laura Mckenzie Patricia Parker-Davis Deanna Voss Learning Objective and skills practiced Students will: Learn the three criteria
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 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 informationMEI GeoGebra Tasks for A2 Core
Task 1: Functions The Modulus Function 1. Plot the graph of y = x : use y = x or y = abs(x) 2. Plot the graph of y = ax+b : use y = ax + b or y = abs(ax+b) If prompted click Create Sliders. What combination
More informationCalculators ARE NOT Permitted On This Portion Of The Exam 28 Questions - 55 Minutes
1 of 11 1) Give f(g(1)), given that Calculators ARE NOT Permitted On This Portion Of The Exam 28 Questions - 55 Minutes 2) Find the slope of the tangent line to the graph of f at x = 4, given that 3) Determine
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 informationwe wish to minimize this function; to make life easier, we may minimize
Optimization and Lagrange Multipliers We studied single variable optimization problems in Calculus 1; given a function f(x), we found the extremes of f relative to some constraint. Our ability to find
More informationPolar Coordinates. 2, π and ( )
Polar Coordinates Up to this point we ve dealt exclusively with the Cartesian (or Rectangular, or x-y) coordinate system. However, as we will see, this is not always the easiest coordinate system to work
More information11.4. Imagine that you are, right now, facing a clock and reading the time on that. Spin to Win. Volume of Cones and Pyramids
Spin to Win Volume of Cones and Pyramids.4 Learning Goals In this lesson, you will: Rotate two-dimensional plane figures to generate three-dimensional figures. Give an informal argument for the volume
More informationMEI GeoGebra Tasks for AS Pure
Task 1: Coordinate Geometry Intersection of a line and a curve 1. Add a quadratic curve, e.g. y = x 2 4x + 1 2. Add a line, e.g. y = x 3 3. Use the Intersect tool to find the points of intersection of
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 informationMath Lab 6: Powerful Fun with Power Series Representations of Functions Due noon Thu. Jan. 11 in class *note new due time, location for winter quarter
Matter & Motion Winter 2017 18 Name: Math Lab 6: Powerful Fun with Power Series Representations of Functions Due noon Thu. Jan. 11 in class *note new due time, location for winter quarter Goals: 1. Practice
More informationSummer Packet 7 th into 8 th grade. Name. Integer Operations = 2. (-7)(6)(-4) = = = = 6.
Integer Operations Name Adding Integers If the signs are the same, add the numbers and keep the sign. 7 + 9 = 16 - + -6 = -8 If the signs are different, find the difference between the numbers and keep
More informationTIME 2014 Technology in Mathematics Education July 1 st -5 th 2014, Krems, Austria
TIME 2014 Technology in Mathematics Education July 1 st -5 th 2014, Krems, Austria Overview Introduction Using a 2D Plot Window in a CAS Perspective Plotting a circle and implicit differentiation Helping
More informationTable of Laplace Transforms
Table of Laplace Transforms 1 1 2 3 4, p > -1 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Heaviside Function 27 28. Dirac Delta Function 29 30. 31 32. 1 33 34. 35 36. 37 Laplace Transforms
More informationTuesday 22 January 2008 Afternoon Time: 1 hour 30 minutes
Paper Reference(s) 6666/0 Edexcel GCE Core Mathematics C4 Advanced Level Tuesday 22 January 2008 Afternoon Time: hour 30 minutes Materials required for examination Mathematical Formulae (Green) Items included
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 informationy= sin( x) y= cos( x)
. The graphs of sin(x) and cos(x). Now I am going to define the two basic trig functions: sin(x) and cos(x). Study the diagram at the right. The circle has radius. The arm OP starts at the positive horizontal
More informationGCSE Maths: Formulae you ll need to know not
GCSE Maths: Formulae you ll need to know As provided by AQA, these are the formulae required for the new GCSE These will not be given in the exam, so you will need to recall as well as use these formulae.
More information2.9 Linear Approximations and Differentials
2.9 Linear Approximations and Differentials 2.9.1 Linear Approximation Consider the following graph, Recall that this is the tangent line at x = a. We had the following definition, f (a) = lim x a f(x)
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 informationWe imagine the egg being the three dimensional solid defined by rotating this ellipse around the x-axis:
CHAPTER 6. INTEGRAL APPLICATIONS 7 Example. Imagine we want to find the volume of hard boiled egg. We could put the egg in a measuring cup and measure how much water it displaces. But we suppose we want
More informationAlgebra II Trigonometric Functions
Slide 1 / 162 Slide 2 / 162 Algebra II Trigonometric Functions 2015-12-17 www.njctl.org Slide 3 / 162 Trig Functions click on the topic to go to that section Radians & Degrees & Co-terminal angles Arc
More informationand F is an antiderivative of f
THE EVALUATION OF DEFINITE INTEGRALS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions Comments to ingrid.stewart@csn.edu. Thank you! We have finally reached a point,
More information5 Applications of Definite Integrals
5 Applications of Definite Integrals The previous chapter introduced the concepts of a definite integral as an area and as a limit of Riemann sums, demonstrated some of the properties of integrals, introduced
More informationApplications of Integration. Copyright Cengage Learning. All rights reserved.
Applications of Integration Copyright Cengage Learning. All rights reserved. Volume: The Disk Method Copyright Cengage Learning. All rights reserved. Objectives Find the volume of a solid of revolution
More informationAP Calculus. Slide 1 / 95. Slide 2 / 95. Slide 3 / 95. Applications of Definite Integrals
Slide 1 / 95 Slide 2 / 95 AP Calculus Applications of Definite Integrals 2015-11-23 www.njctl.org Table of Contents Slide 3 / 95 Particle Movement Area Between Curves Volume: Known Cross Sections Volume:
More informationInverse Kinematics (part 1) CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2018
Inverse Kinematics (part 1) CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2018 Welman, 1993 Inverse Kinematics and Geometric Constraints for Articulated Figure Manipulation, Chris
More informationLECTURE 3-1 AREA OF A REGION BOUNDED BY CURVES
7 CALCULUS II DR. YOU 98 LECTURE 3- AREA OF A REGION BOUNDED BY CURVES If y = f(x) and y = g(x) are continuous on an interval [a, b] and f(x) g(x) for all x in [a, b], then the area of the region between
More informationChapter 5 Accumulating Change: Limits of Sums and the Definite Integral
Chapter 5 Accumulating Change: Limits of Sums and the Definite Integral 5.1 Results of Change and Area Approximations So far, we have used Excel to investigate rates of change. In this chapter we consider
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 informationCalculus I Review Handout 1.3 Introduction to Calculus - Limits. by Kevin M. Chevalier
Calculus I Review Handout 1.3 Introduction to Calculus - Limits by Kevin M. Chevalier We are now going to dive into Calculus I as we take a look at the it process. While precalculus covered more static
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 informationPrerequisites for Math 130
Prerequisites for Math 0 The material below represents only some of the basic material with which you should be familiar We will not be reviewing this material You may wish to consult Appendix A in your
More informationAPPM/MATH Problem Set 4 Solutions
APPM/MATH 465 Problem Set 4 Solutions This assignment is due by 4pm on Wednesday, October 16th. You may either turn it in to me in class on Monday or in the box outside my office door (ECOT 35). Minimal
More informationEdexcel Core Mathematics 4 Integration
Edecel Core Mathematics 4 Integration Edited by: K V Kumaran kumarmaths.weebly.com Integration It might appear to be a bit obvious but you must remember all of your C work on differentiation if you are
More informationAP Calculus BC Course Description
AP Calculus BC Course Description COURSE OUTLINE: The following topics define the AP Calculus BC course as it is taught over three trimesters, each consisting of twelve week grading periods. Limits and
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 informationCoordinate Transformations in Advanced Calculus
Coordinate Transformations in Advanced Calculus by Sacha Nandlall T.A. for MATH 264, McGill University Email: sacha.nandlall@mail.mcgill.ca Website: http://www.resanova.com/teaching/calculus/ Fall 2006,
More informationMath 104, Spring 2010 Course Log
Math 104, Spring 2010 Course Log Date: 1/11 Sections: 1.3, 1.4 Log: Lines in the plane. The point-slope and slope-intercept formulas. Functions. Domain and range. Compositions of functions. Inverse functions.
More informationMEI Casio Tasks for A2 Core
Task 1: Functions The Modulus Function The modulus function, abs(x), is found using OPTN > NUMERIC > Abs 2. Add the graph y = x, Y1=Abs(x): iyqfl 3. Add the graph y = ax+b, Y2=Abs(Ax+B): iyqaff+agl 4.
More informationDifferentiation and Integration
Edexcel GCE Core Mathematics C Advanced Subsidiary Differentiation and Integration Materials required for examination Mathematical Formulae (Pink or Green) Items included with question papers Nil Advice
More information4.7 Approximate Integration
4.7 Approximate Integration Some anti-derivatives are difficult to impossible to find. For example, 1 0 e x2 dx or 1 1 1 + x3 dx We came across this situation back in calculus I when we introduced the
More informationMath 2250 Lab #3: Landing on Target
Math 2250 Lab #3: Landing on Target 1. INTRODUCTION TO THE LAB PROGRAM. Here are some general notes and ideas which will help you with the lab. The purpose of the lab program is to expose you to problems
More informationApplications of Integration
Week 12. Applications of Integration 12.1.Areas Between Curves Example 12.1. Determine the area of the region enclosed by y = x 2 and y = x. Solution. First you need to find the points where the two functions
More informationPRACTICE FINAL - MATH 1210, Spring 2012 CHAPTER 1
PRACTICE FINAL - MATH 2, Spring 22 The Final will have more material from Chapter 4 than other chapters. To study for chapters -3 you should review the old practice eams IN ADDITION TO what appears here.
More information1. Let be a point on the terminal side of θ. Find the 6 trig functions of θ. (Answers need not be rationalized). b. P 1,3. ( ) c. P 10, 6.
Q. Right Angle Trigonometry Trigonometry is an integral part of AP calculus. Students must know the basic trig function definitions in terms of opposite, adjacent and hypotenuse as well as the definitions
More information. The differential of y f (x)
Calculus I - Prof D Yuen Exam Review version 11/14/01 Please report any typos Derivative Rules Of course you have to remember all your derivative rules Implicit Differentiation Differentiate both sides
More information1 Double Integrals over Rectangular Regions
Contents ouble Integrals over Rectangular Regions ouble Integrals Over General Regions 7. Introduction.................................... 7. Areas of General Regions............................. 9.3 Region
More information(Section 6.2: Volumes of Solids of Revolution: Disk / Washer Methods)
(Section 6.: Volumes of Solids of Revolution: Disk / Washer Methods) 6.. PART E: DISK METHOD vs. WASHER METHOD When using the Disk or Washer Method, we need to use toothpicks that are perpendicular to
More informationMAT01B1: Surface Area of Solids of Revolution
MAT01B1: Surface Area of Solids of Revolution Dr Craig 02 October 2018 My details: acraig@uj.ac.za Consulting hours: Monday 14h40 15h25 Thursday 11h20 12h55 Friday 11h20 12h55 Office C-Ring 508 https://andrewcraigmaths.wordpress.com/
More informationProblem #3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page Mark Sparks 2012
Problem # Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 490 Mark Sparks 01 Finding Anti-derivatives of Polynomial-Type Functions If you had to explain to someone how to find
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 information