Lecture : The Indefinite Integral MTH 124
|
|
- Teresa Harrell
- 6 years ago
- Views:
Transcription
1 Up to this point we have investigated the definite integral of a function over an interval. In particular we have done the following. Approximated integrals using left and right Riemann sums. Defined the limit of the left Riemann sum (LHS) over an interval [a, b] to be the integral of a function f(x) from a to b, denoted b a f(x) dx. Calculated definite integrals numerically using our calculators. Today we will algebraically determine the indefinite integral of an arbitrary function f(x). Antiderivative An antiderivative of a function f is a function F such that F = f. It may not be immediately clear what this definition is describing. Let s consider some examples to develop our understanding of the antiderivative. Example 1 Determine an antiderivative of 2x. This question can be reformulated as the statement below. The derivative of is 2x. So instead of being given a function and finding its derivative, we re given a derivative and we need to find a function it could have been derived from. So we need to think of a function we could differentiate and get 2x. Initially solving these problems will likely take some trial and error. Notice that 2x is a power function whose exponent is 1. When we take the derivative of a power function we lose a power so our antiderivative should have an x 2 term (Why must this be true?). We can check if x 2 works as an antiderivative by simply taking its derivative. If we get 2x then x 2 is an antiderivative of 2x. Since (x 2 ) = 2x, we see that an antiderivative of 2x is x 2. 1 Example 2 Determine an antiderivative of 8x 3. Again, it may help to rephrase this question as the statement below. The derivative of is 8x 3. Since 8x 3 is a power function we expect its antiderivative to be a power function with an exponent of 4, otherwise its derivative won t give us a power function with an exponent 3. So let s check to see if x 4 works as an antiderivative of 8x 3. 1 If we had tried something like x 5 we would have that (x 5 ) = 5x 4 so since 5x 4 2x we see x 5 is not an antiderivative of 2x. 108
2 Taking the derivative we get (x 4 ) = 4x 3. This is close but we need the coefficient to be 8 so we see by trying 2x 4 (since 2 4 = 8) that an antiderivative of 8x 3 is 2x 4 (since the derivative of 2x 4 is 8x 3 ). Antiderivatives are not unique. That is, given a function and asked to find its antiderivative, you and I might both come up with different antiderivatives. For instance in example 1 we found that an antiderivative of 2x is x 2. However the derivative of a constant is zero, so x 2 +4 is also an antiderivative of 2x since the derivative of x is 2x. Notice the issue of non-uniqueness arises because the derivative of a constant is zero. We can represent the set of all antiderivatives of 2x by x 2 +C where C is any constant. The expression x 2 +C is known as the indefinite integral of 2x. The Indefinite Integral The integral given by f(x)dx, is known as the indefinite integral. This expression represents the set of all antiderivatives. Example 3 Determine 2x dx. The indefinite integral of 2x is the set of all antiderivatives so we have 2xdx = x 2 + C, where C is some constant 2. We can check our answer by differentiating. To check the example above we check that (x 2 + C) equals 2x (it does!). Example 4 Determine 4x 3 dx. The indefinite integral of 4x 3 is the set of all antiderivatives so we have 4x 3 dx = x 4 + C, where C is some constant. The function f(x) being integrated is called the integrand, and the variable x is called the variable of integration. 2 When used in an indefinite integral we call the constant C the constant of integration. 109
3 You may have noticed that for the power functions above we can find their integral by taking the reverse derivative rule. In fact, just like derivatives, we have rules for integration. Power Rule for the Indefinite Integral ( ) 1 x n dx = x n+1 + C, for n 1. n + 1 x 1 dx = ln x + C, for n = 1. In both cases both n and C are constants. Remember x 1 = 1 x so our second rule can also be written as 1 dx = ln x + C. x Example 5 Determine x dx. Similar to differentiation, it is helpful to first rewrite the x as a power of x. This gives us xdx = x 1/2 dx Following the rule above we have that n = 1/2 so we get the following indefinite integral. ( ) x 1/2 1 dx = x 1/2+1 + C, 1/2 + 1 = 1 3/2 x3/2 + C, where C is some constant. = 2 3 x3/2 + C. 1. Determine x 42 dx. 110
4 In addition to the indefinite integral of power functions we have the following rules for the other functions we have been working with. 3 Indefinite Integral of e x, b x, and x e x dx = e x + C, for If b 1 is any positive number then b x dx = bx ln(b) + C, x dx = x x 2 + C We have integral rules for sums, differences, and constant multipliers as well. Sums, Differences, and Constant Multiples [f(x) ] ± g(x) dx = f(x)dx ± g(x)dx If k is any constant, then we have the following rule. kf(x)dx = k f(x)dx Example 6 Use the rules above to determine the indefinite integral given by (10x 4 + 2x 2 3e x ) dx 3 There rules for the indefinite integral of other functions but for our purposes we will only use the rules presented here. 111
5 Example 7 By the start of 2008, Apple had sold a total of about 3.5 million iphones. From the start of 2008 through the end of 2010, sales of iphones were approximately s(t) = 4.5t t million iphones per year (0 t 3), where t is time in years since the start of Find an expression for the total sales of iphones up to time t. 2. Determine the following indefinite integrals. (a) x 5 dx (b) (1 + u) du (c) 4 x dx (d) ( ) dt t0.9 (e) ( 6.1 x0.5 + x0.5 6 ex ) dx 112
6 3. The velocity of a particle moving along a straight line is given by v(t) = 4t + 1 m/sec. given that the particle is at position s = 2 meters at time t = 1, find an expression for s in terms of t. That is, find s(t) and use the given information to determine the constant of integration. 4. Explain why the following solutions were marked incorrect and determine the correct answer. (a) 4(e x 2x) dx= (4x)(e x x 2 ) + C Wrong (b) (2 x 1) dx= 2x+1 x + 1 x + C Wrong 113
Chapter 5: Integrals
Chapter 5: Integrals Section 5.5 The Substitution Rule (u-substitution) Sec. 5.5: The Substitution Rule We know how to find the derivative of any combination of functions Sum rule Difference rule Constant
More informationMath 142 (Summer 2018) Business Calculus 6.1 Notes
Math 142 (Summer 2018) Business Calculus 6.1 Notes Antiderivatives Why? So far in the course we have studied derivatives. Differentiation is the process of going from a function f to its derivative f.
More informationChapter 13: Integral Calculus. SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M.
Chapter 13: Integral Calculus SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza Chapter 13: Integral Calculus Lecture 13.1: The Integrals Lecture
More informationSubstitutions and by Parts, Area Between Curves. Goals: The Method of Substitution Areas Integration by Parts
Week #7: Substitutions and by Parts, Area Between Curves Goals: The Method of Substitution Areas Integration by Parts 1 Week 7 The Indefinite Integral The Fundamental Theorem of Calculus, b a f(x) dx =
More informationSection 5.6. Integration By Parts. MATH 126 (Section 5.6) Integration By Parts The University of Kansas 1 / 10
Section 5.6 Integration By Parts MATH 126 (Section 5.6) Integration By Parts The University of Kansas 1 / 10 Integration By Parts Manipulating the Product Rule d dx (f (x) g(x)) = f (x) g (x) + f (x) g(x)
More informationOBJECTIVES Use the area under a graph to find total cost. Use rectangles to approximate the area under a graph.
4.1 The Area under a Graph OBJECTIVES Use the area under a graph to find total cost. Use rectangles to approximate the area under a graph. 4.1 The Area Under a Graph Riemann Sums (continued): In the following
More informationChapter 5: Integrals
Chapter 5: Integrals Section 5.3 The Fundamental Theorem of Calculus Sec. 5.3: The Fundamental Theorem of Calculus Fundamental Theorem of Calculus: Sec. 5.3: The Fundamental Theorem of Calculus Fundamental
More information4. Theory of the Integral
4. Theory of the Integral 4.1 Antidifferentiation 4.2 The Definite Integral 4.3 Riemann Sums 4.4 The Fundamental Theorem of Calculus 4.5 Fundamental Integration Rules 4.6 U-Substitutions 4.1 Antidifferentiation
More informationMath Refresher Course
Math Refresher Course Columbia University Department of Political Science Fall 2007 Day 2 Prepared by Jessamyn Blau 6 Calculus CONT D 6.9 Antiderivatives and Integration Integration is the reverse of differentiation.
More informationStudy 4.10 #465, 471, , 487, , , 515, 517, 521, 523
Goals: 1. Understand that antiderivatives are the functions from which the present derivative was found. 2. The process of finding an antiderivative or indefinite integral requires the reverse process
More informationExam 3 review for Math 1190
Exam 3 review for Math 9 Be sure to be familiar with the following : Extreme Value Theorem Optimization The antiderivative u-substitution as a method for finding antiderivatives Reimann sums (e.g. L 6
More informationIntegration Using Tables and Summary of Techniques
Integration Using Tables and Summary of Techniques Philippe B. Laval KSU Today Philippe B. Laval (KSU) Summary Today 1 / 13 Introduction We wrap up integration techniques by discussing the following topics:
More informationMath RE - Calculus II Antiderivatives and the Indefinite Integral Page 1 of 5
Math 201-203-RE - Calculus II Antiderivatives and the Indefinite Integral Page 1 of 5 What is the Antiderivative? In a derivative problem, a function f(x) is given and you find the derivative f (x) using
More informationBusiness and Life Calculus
Business and Life Calculus George Voutsadakis Mathematics and Computer Science Lake Superior State University LSSU Math 2 George Voutsadakis (LSSU) Calculus For Business and Life Sciences Fall 203 / 55
More informationAnnouncements. Topics: Homework:
Announcements Topics: - sections 7.1 (differential equations), 7.2 (antiderivatives), and 7.3 (the definite integral +area) * Read these sections and study solved examples in your textbook! Homework: -
More informationUNIT 3 INTEGRATION 3.0 INTRODUCTION 3.1 OBJECTIVES. Structure
Calculus UNIT 3 INTEGRATION Structure 3.0 Introduction 3.1 Objectives 3.2 Basic Integration Rules 3.3 Integration by Substitution 3.4 Integration of Rational Functions 3.5 Integration by Parts 3.6 Answers
More informationApplications of Differentiation
Applications of Differentiation Definitions. A function f has an absolute maximum (or global maximum) at c if for all x in the domain D of f, f(c) f(x). The number f(c) is called the maximum value of f
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 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 informationIntegration. Antiderivatives and Indefinite Integration 3/9/2015. Copyright Cengage Learning. All rights reserved.
Integration Copyright Cengage Learning. All rights reserved. Antiderivatives and Indefinite Integration Copyright Cengage Learning. All rights reserved. 1 Objectives Write the general solution of a differential
More informationMath 180 Written Homework Assignment #10 Due Tuesday, December 2nd at the beginning of your discussion class.
Math 18 Written Homework Assignment #1 Due Tuesday, December 2nd at the beginning of your discussion class. Directions. You are welcome to work on the following problems with other MATH 18 students, but
More informationQMI Lesson 19: Integration by Substitution, Definite Integral, and Area Under Curve
QMI Lesson 19: Integration by Substitution, Definite Integral, and Area Under Curve C C Moxley Samford University Brock School of Business Substitution Rule The following rules arise from the chain rule
More information1 Lesson 13: Methods of Integration
Lesson 3: Methods of Integration Chapter 6 Material: pages 273-294 in the textbook: Lesson 3 reviews integration by parts and presents integration via partial fraction decomposition as the third of the
More information1 Introduction; Integration by Parts
1 Introduction; Integration by Parts September 11-1 Traditionally Calculus I covers Differential Calculus and Calculus II covers Integral Calculus. You have already seen the Riemann integral and certain
More informationAnnouncements. Topics: Homework:
Announcements Topics: - sections 7.3 (the definite integral +area), 7.4 (FTC), 7.5 (additional techniques of integration) * Read these sections and study solved examples in your textbook! Homework: - review
More informationSection 4.4. Using the Fundamental Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 4.4 Using the Fundamental Theorem As we saw in Section 4.3, using the Fundamental Theorem of Integral Calculus reduces the problem of evaluating a
More informationMATH 250 TOPIC 13 INTEGRATION. 13B. Constant, Sum, and Difference Rules
Math 5 Integration Topic 3 Page MATH 5 TOPIC 3 INTEGRATION 3A. Integration of Common Functions Practice Problems 3B. Constant, Sum, and Difference Rules Practice Problems 3C. Substitution Practice Problems
More informationA quadratic expression is a mathematical expression that can be written in the form 2
118 CHAPTER Algebra.6 FACTORING AND THE QUADRATIC EQUATION Textbook Reference Section 5. CLAST OBJECTIVES Factor a quadratic expression Find the roots of a quadratic equation A quadratic expression is
More informationMethods of Integration
Methods of Integration Professor D. Olles January 8, 04 Substitution The derivative of a composition of functions can be found using the chain rule form d dx [f (g(x))] f (g(x)) g (x) Rewriting the derivative
More informationf(x) g(x) = [f (x)g(x) dx + f(x)g (x)dx
Chapter 7 is concerned with all the integrals that can t be evaluated with simple antidifferentiation. Chart of Integrals on Page 463 7.1 Integration by Parts Like with the Chain Rule substitutions with
More informationMath 122 Fall Unit Test 1 Review Problems Set A
Math Fall 8 Unit Test Review Problems Set A We have chosen these problems because we think that they are representative of many of the mathematical concepts that we have studied. There is no guarantee
More informationdy = f( x) dx = F ( x)+c = f ( x) dy = f( x) dx
Antiderivatives and The Integral Antiderivatives Objective: Use indefinite integral notation for antiderivatives. Use basic integration rules to find antiderivatives. Another important question in calculus
More informationChapter 6: The Definite Integral
Name: Date: Period: AP Calc AB Mr. Mellina Chapter 6: The Definite Integral v v Sections: v 6.1 Estimating with Finite Sums v 6.5 Trapezoidal Rule v 6.2 Definite Integrals 6.3 Definite Integrals and Antiderivatives
More informationintegration integration
13 Contents integration integration 1. Basic concepts of integration 2. Definite integrals 3. The area bounded by a curve 4. Integration by parts 5. Integration by substitution and using partial fractions
More informationExample. Evaluate. 3x 2 4 x dx.
3x 2 4 x 3 + 4 dx. Solution: We need a new technique to integrate this function. Notice that if we let u x 3 + 4, and we compute the differential du of u, we get: du 3x 2 dx Going back to our integral,
More informationIntegration by Substitution
Integration by Substitution Dr. Philippe B. Laval Kennesaw State University Abstract This handout contains material on a very important integration method called integration by substitution. Substitution
More informationCalculus II (Fall 2015) Practice Problems for Exam 1
Calculus II (Fall 15) Practice Problems for Exam 1 Note: Section divisions and instructions below are the same as they will be on the exam, so you will have a better idea of what to expect, though I will
More informationMATH 1271 Monday, 21 November 2018
MATH 1271 Monday, 21 November 218 Today: Section 5.4 - Indefinite Integrals and the Theorem Homework: 5-17 odd, 21-45 odd, 51-63 odd, 67, 71 1/13 Def Total displacement is the integral of the velocity
More information8.3 Partial Fraction Decomposition
8.3 partial fraction decomposition 575 8.3 Partial Fraction Decomposition Rational functions (polynomials divided by polynomials) and their integrals play important roles in mathematics and applications,
More informationApplied Calculus I. Lecture 29
Applied Calculus I Lecture 29 Integrals of trigonometric functions We shall continue learning substitutions by considering integrals involving trigonometric functions. Integrals of trigonometric functions
More informationChapter 6: Messy Integrals
Chapter 6: Messy Integrals Review: Solve the following integrals x 4 sec x tan x 0 0 Find the average value of 3 1 x 3 3 Evaluate 4 3 3 ( x 1), then find the area of ( x 1) 4 Section 6.1: Slope Fields
More informationIntegration by Substitution
November 22, 2013 Introduction 7x 2 cos(3x 3 )dx =? 2xe x2 +5 dx =? Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation
More informationName: Partners: PreCalculus. Review 5 Version A
Name: Partners: PreCalculus Date: Review 5 Version A [A] Circle whether each statement is true or false. 1. 3 log 3 5x = 5x 2. log 2 16 x+3 = 4x + 3 3. ln x 6 + ln x 5 = ln x 30 4. If ln x = 4, then e
More informationEvaluating Integrals (Section 5.3) and the Fundamental Theorem of Calculus (Section 1 / 15 (5.4
Evaluating Integrals (Section 5.3) and the Fundamental Theorem of Calculus (Section (5.4) Evaluating Integrals (Section 5.3) and the Fundamental Theorem of Calculus (Section 1 / 15 (5.4 Intro to 5.3 Today
More informationAnnouncements. Topics: Homework:
Announcements Topics: - sections 7.4 (FTC), 7.5 (additional techniques of integration), 7.6 (applications of integration) * Read these sections and study solved examples in your textbook! Homework: - review
More informationThe Definite Integral. Day 5 The Fundamental Theorem of Calculus (Evaluative Part)
The Definite Integral Day 5 The Fundamental Theorem of Calculus (Evaluative Part) Practice with Properties of Integrals 5 Given f d 5 f d 3. 0 5 5. 0 5 5 3. 0 0. 5 f d 0 f d f d f d - 0 8 5 F 3 t dt
More information4 Integration. Copyright Cengage Learning. All rights reserved.
4 Integration Copyright Cengage Learning. All rights reserved. 4.1 Antiderivatives and Indefinite Integration Copyright Cengage Learning. All rights reserved. Objectives! Write the general solution of
More information5.5. The Substitution Rule
INTEGRALS 5 INTEGRALS 5.5 The Substitution Rule In this section, we will learn: To substitute a new variable in place of an existing expression in a function, making integration easier. INTRODUCTION Due
More informationIntegrals. D. DeTurck. January 1, University of Pennsylvania. D. DeTurck Math A: Integrals 1 / 61
Integrals D. DeTurck University of Pennsylvania January 1, 2018 D. DeTurck Math 104 002 2018A: Integrals 1 / 61 Integrals Start with dx this means a little bit of x or a little change in x If we add up
More informationChapter 6 Section Antiderivatives and Indefinite Integrals
Chapter 6 Section 6.1 - Antiderivatives and Indefinite Integrals Objectives: The student will be able to formulate problems involving antiderivatives. The student will be able to use the formulas and properties
More informationPrelim 1 Solutions V2 Math 1120
Feb., Prelim Solutions V Math Please show your reasoning and all your work. This is a 9 minute exam. Calculators are not needed or permitted. Good luck! Problem ) ( Points) Calculate the following: x a)
More information3. On the grid below, sketch and label graphs of the following functions: y = sin x, y = cos x, and y = sin(x π/2). π/2 π 3π/2 2π 5π/2
AP Physics C Calculus C.1 Name Trigonometric Functions 1. Consider the right triangle to the right. In terms of a, b, and c, write the expressions for the following: c a sin θ = cos θ = tan θ =. Using
More informationINTEGRATION: THE FUNDAMENTAL THEOREM OF CALCULUS MR. VELAZQUEZ AP CALCULUS
INTEGRATION: THE FUNDAMENTAL THEOREM OF CALCULUS MR. VELAZQUEZ AP CALCULUS RECALL: ANTIDERIVATIVES When we last spoke of integration, we examined a physics problem where we saw that the area under the
More informationCalculus I Announcements
Slide 1 Calculus I Announcements Read sections 4.2,4.3,4.4,4.1 and 5.3 Do the homework from sections 4.2,4.3,4.4,4.1 and 5.3 Exam 3 is Thursday, November 12th See inside for a possible exam question. Slide
More informationChapter 5 Integrals. 5.1 Areas and Distances
Chapter 5 Integrals 5.1 Areas and Distances We start with a problem how can we calculate the area under a given function ie, the area between the function and the x-axis? If the curve happens to be something
More information1. The accumulated net change function or area-so-far function
Name: Section: Names of collaborators: Main Points: 1. The accumulated net change function ( area-so-far function) 2. Connection to antiderivative functions: the Fundamental Theorem of Calculus 3. Evaluating
More informationExploring Substitution
I. Introduction Exploring Substitution Math Fall 08 Lab We use the Fundamental Theorem of Calculus, Part to evaluate a definite integral. If f is continuous on [a, b] b and F is any antiderivative of f
More informationPlease read for extra test points: Thanks for reviewing the notes you are indeed a true scholar!
Please read for extra test points: Thanks for reviewing the notes you are indeed a true scholar! See me any time B4 school tomorrow and mention to me that you have reviewed your integration notes and you
More informationMath 111 lecture for Friday, Week 10
Math lecture for Friday, Week Finding antiderivatives mean reversing the operation of taking derivatives. Today we ll consider reversing the chain rule and the product rule. Substitution technique. Recall
More informationSubstitution and change of variables Integration by parts
Substitution and change of variables Integration by parts Math 1A October 11, 216 Announcements I have been back since Friday night but will be leaving for another short trip on Thursday. James will preside
More informationAntiderivatives and Indefinite Integrals
Antiderivatives and Indefinite Integrals MATH 151 Calculus for Management J. Robert Buchanan Department of Mathematics Fall 2018 Objectives After completing this lesson we will be able to use the definition
More informationMath 1310 Lab 10. (Sections )
Math 131 Lab 1. (Sections 5.1-5.3) Name/Unid: Lab section: 1. (Properties of the integral) Use the properties of the integral in section 5.2 for answering the following question. (a) Knowing that 2 2f(x)
More informationMAT137 - Term 2, Week 4
MAT137 - Term 2, Week 4 Reminders: Your Problem Set 6 is due tomorrow at 3pm. Test 3 is next Friday, February 3, at 4pm. See the course website for details. Today we will: Talk more about substitution.
More informationSection 5.6 Integration by Parts
.. 98 Section.6 Integration by Parts Integration by parts is another technique that we can use to integrate problems. Typically, we save integration by parts as a last resort when substitution will not
More informationSchool of the Art Institute of Chicago. Calculus. Frank Timmes. flash.uchicago.edu/~fxt/class_pages/class_calc.
School of the Art Institute of Chicago Calculus Frank Timmes ftimmes@artic.edu flash.uchicago.edu/~fxt/class_pages/class_calc.shtml Syllabus 1 Aug 29 Pre-calculus 2 Sept 05 Rates and areas 3 Sept 12 Trapezoids
More informationChapter 4 Integration
Chapter 4 Integration SECTION 4.1 Antiderivatives and Indefinite Integration Calculus: Chapter 4 Section 4.1 Antiderivative A function F is an antiderivative of f on an interval I if F '( x) f ( x) for
More informationQuestions from Larson Chapter 4 Topics. 5. Evaluate
Math. Questions from Larson Chapter 4 Topics I. Antiderivatives. Evaluate the following integrals. (a) x dx (4x 7) dx (x )(x + x ) dx x. A projectile is launched vertically with an initial velocity of
More informationTest 2 - Answer Key Version A
MATH 8 Student s Printed Name: Instructor: CUID: Section: Fall 27 8., 8.2,. -.4 Instructions: You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook,
More informationChapter 6. Techniques of Integration. 6.1 Differential notation
Chapter 6 Techniques of Integration In this chapter, we expand our repertoire for antiderivatives beyond the elementary functions discussed so far. A review of the table of elementary antiderivatives (found
More informationSections 4.9: Antiderivatives
Sections 4.9: Antiderivatives For the last few sections, we have been developing the notion of the derivative and determining rules which allow us to differentiate all different types of functions. In
More informationUnit #10 : Graphs of Antiderivatives; Substitution Integrals
Unit #10 : Graphs of Antiderivatives; Substitution Integrals Goals: Relationship between the graph of f(x) and its anti-derivative F(x) The guess-and-check method for anti-differentiation. The substitution
More informationMATH 1271 Wednesday, 5 December 2018
MATH 27 Wednesday, 5 December 208 Today: Review for Exam 3 Exam 3: Thursday, December 6; Sections 4.8-6. /6 Information on Exam 3 Six numbered problems First problem is multiple choice (five parts) See
More informationCalculus Lecture 7. Oktay Ölmez, Murat Şahin and Serhan Varma. Oktay Ölmez, Murat Şahin and Serhan Varma Calculus Lecture 7 1 / 10
Calculus Lecture 7 Oktay Ölmez, Murat Şahin and Serhan Varma Oktay Ölmez, Murat Şahin and Serhan Varma Calculus Lecture 7 1 / 10 Integration Definition Antiderivative A function F is an antiderivative
More informationNO CALCULATORS. NO BOOKS. NO NOTES. TURN OFF YOUR CELL PHONES AND PUT THEM AWAY.
FINAL EXAM-MATH 3 FALL TERM, R. Blute & A. Novruzi Name(Print LEGIBLY) I.D. Number Instructions- This final examination consists of multiple choice questions worth 3 points each. Your answers to the multiple
More informationChapter 6. Techniques of Integration. 6.1 Differential notation
Chapter 6 Techniques of Integration In this chapter, we expand our repertoire for antiderivatives beyond the elementary functions discussed so far. A review of the table of elementary antiderivatives (found
More informationMain topics for the First Midterm
Main topics for the First Midterm Midterm 2 will cover Sections 7.7-7.9, 8.1-8.5, 9.1-9.2, 11.1-11.2. This is roughly the material from the first five homeworks and and three quizzes. In particular, I
More informationLecture : The Definite Integral & Fundamental Theorem of Calculus MTH 124. We begin with a theorem which is of fundamental importance.
We begin with a theorem which is of fundamental importance. The Fundamental Theorem of Calculus (FTC) If F (t) is continuous for a t b, then b a F (t) dt = F (b) F (a). Moreover the antiderivative F is
More information6.1 Matrices. Definition: A Matrix A is a rectangular array of the form. A 11 A 12 A 1n A 21. A 2n. A m1 A m2 A mn A 22.
61 Matrices Definition: A Matrix A is a rectangular array of the form A 11 A 12 A 1n A 21 A 22 A 2n A m1 A m2 A mn The size of A is m n, where m is the number of rows and n is the number of columns The
More informationChange of Variables: Indefinite Integrals
Change of Variables: Indefinite Integrals Mathematics 11: Lecture 39 Dan Sloughter Furman University November 29, 2007 Dan Sloughter (Furman University) Change of Variables: Indefinite Integrals November
More informationp324 Section 5.2: The Natural Logarithmic Function: Integration
p324 Section 5.2: The Natural Logarithmic Function: Integration Theorem 5.5: Log Rule for Integration Let u be a differentiable function of x 1. 2. Example 1: Using the Log Rule for Integration ** Note:
More informationChapter 4. Section Derivatives of Exponential and Logarithmic Functions
Chapter 4 Section 4.2 - Derivatives of Exponential and Logarithmic Functions Objectives: The student will be able to calculate the derivative of e x and of lnx. The student will be able to compute the
More informationAssignment 16 Solution. Please do not copy and paste my answer. You will get similar questions but with different numbers!
Assignment 6 Solution Please do not copy and paste my answer. You will get similar questions but with different numbers! Suppose f is a continuous, positive, decreasing function on [, ) and let a n = f
More information4.9 Anti-derivatives. Definition. An anti-derivative of a function f is a function F such that F (x) = f (x) for all x.
4.9 Anti-derivatives Anti-differentiation is exactly what it sounds like: the opposite of differentiation. That is, given a function f, can we find a function F whose derivative is f. Definition. An anti-derivative
More informationTrigonometric integrals by basic methods
Roberto s Notes on Integral Calculus Chapter : Integration methods Section 7 Trigonometric integrals by basic methods What you need to know already: Integrals of basic trigonometric functions. Basic trigonometric
More informationSpring 2017 Midterm 1 04/26/2017
Math 2B Spring 2017 Midterm 1 04/26/2017 Time Limit: 50 Minutes Name (Print): Student ID This exam contains 10 pages (including this cover page) and 5 problems. Check to see if any pages are missing. Enter
More informationIntegration by Substitution
Integration by Substitution MATH 151 Calculus for Management J. Robert Buchanan Department of Mathematics Fall 2018 Objectives After this lesson we will be able to use the method of integration by substitution
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Fall 2017, WEEK 14 JoungDong Kim Week 14 Section 5.4, 5.5, 6.1, Indefinite Integrals and the Net Change Theorem, The Substitution Rule, Areas Between Curves. Section
More informationMath 152 Take Home Test 1
Math 5 Take Home Test Due Monday 5 th October (5 points) The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I
More informationMathematics Notes for Class 12 chapter 7. Integrals
1 P a g e Mathematics Notes for Class 12 chapter 7. Integrals Let f(x) be a function. Then, the collection of all its primitives is called the indefinite integral of f(x) and is denoted by f(x)dx. Integration
More informationChapter 8 Indeterminate Forms and Improper Integrals Math Class Notes
Chapter 8 Indeterminate Forms and Improper Integrals Math 1220-004 Class Notes Section 8.1: Indeterminate Forms of Type 0 0 Fact: The it of quotient is equal to the quotient of the its. (book page 68)
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 informationWorksheet Week 1 Review of Chapter 5, from Definition of integral to Substitution method
Worksheet Week Review of Chapter 5, from Definition of integral to Substitution method This worksheet is for improvement of your mathematical writing skill. Writing using correct mathematical expressions
More informationDefinition of differential equations and their classification. Methods of solution of first-order differential equations
Introduction to differential equations: overview Definition of differential equations and their classification Solutions of differential equations Initial value problems Existence and uniqueness Mathematical
More informationThe Definite Integral. Day 6 Motion Problems Strategies for Finding Total Area
The Definite Integral Day 6 Motion Problems Strategies for Finding Total Area ARRIVAL---HW Questions Working in PODS Additional Practice Packet p. 13 and 14 Make good use of your time! Practice makes perfect!
More information6x 2 8x + 5 ) = 12x 8. f (x) ) = d (12x 8) = 12
AMS/ECON 11A Class Notes 11/6/17 UCSC *) Higher order derivatives Example. If f = x 3 x + 5x + 1, then f = 6x 8x + 5 Observation: f is also a differentiable function... d f ) = d 6x 8x + 5 ) = 1x 8 dx
More informationGoal: Approximate the area under a curve using the Rectangular Approximation Method (RAM) RECTANGULAR APPROXIMATION METHODS
AP Calculus 5. Areas and Distances Goal: Approximate the area under a curve using the Rectangular Approximation Method (RAM) Exercise : Calculate the area between the x-axis and the graph of y = 3 2x.
More informationMath 181, Exam 2, Study Guide 2 Problem 1 Solution. 1 + dx. 1 + (cos x)2 dx. 1 + cos2 xdx. = π ( 1 + cos π 2
Math 8, Exam, Study Guide Problem Solution. Use the trapezoid rule with n to estimate the arc-length of the curve y sin x between x and x π. Solution: The arclength is: L b a π π + ( ) dy + (cos x) + cos
More informationChapter 3 The Integral Business Calculus 197
Chapter The Integral Business Calculus 97 Chapter Exercises. Let A(x) represent the area bounded by the graph and the horizontal axis and vertical lines at t=0 and t=x for the graph in Fig.. Evaluate A(x)
More informationSection 6-1 Antiderivatives and Indefinite Integrals
Name Date Class Section 6-1 Antiderivatives and Indefinite Integrals Goal: To find antiderivatives and indefinite integrals of functions using the formulas and properties Theorem 1 Antiderivatives If the
More informationLecture 5: Integrals and Applications
Lecture 5: Integrals and Applications Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2012 Lejla Batina Version: spring 2012 Wiskunde 1 1 / 21 Outline The
More information