Announcements. Topics: Homework:
|
|
- Oswald Lyons
- 5 years ago
- Views:
Transcription
1 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 lecture notes thoroughly - work on practice problems from the textbook and assignments from the coursepack as assigned on the course web page (under the SCHEDULE + HOMEWORK link)
2 Types of Integrals Indefinite Integral function of x f (x) dx = F(x) + C antiderivative of f Definite Integral b a f (x) dx = net area number
3 The Fundamental Theorem of Calculus If f is continuous on [a, b], then b a f (x) dx = F(x) a b = F(b) F(a) where F is any antiderivative of f, i.e., F'= f.
4 Evaluating Definite Integrals Example: Evaluate each definite integral using the FTC. (a) 3 0 (x ) dx (b) 2 ( 4 t + t 4 )dt 2 4 4x dx 2 (c) (d) 2 (3x ) 2 x dx
5 Evaluating Definite Integrals Example: Try to evaluate the following definite integral using the FTC. What is the problem? 4 (x 2) 2 dx
6 Differentiation and Integration as Inverse Processes If f is integrated and then differentiated, we arrive back at the original function f. d dx x a f (t) dt = f (x) FTC I If F is differentiated and then integrated, we arrive back at the original function F. b a d dx F(x) dx = F(x) b a FTC II
7 The Definite Integral - Total Change Interpretation: The definite integral represents the total amount of change during some period of time. Total change in F between times a and b: value at end F(b) F(a) = value at start b a df dt dt rate of change
8 Application Total Change Example: Suppose that the growth rate of a fish is given by the differential equation dl dt = 6.48e 0.09t where t is measured in years and L is measured in centimetres and the fish was 0.0 cm at age t=0 (time measured from fertilization).
9 Application Total Change (a) Determine the amount the fish grows between 2 and 5 years of age. (b) At approximately what age will the fish reach 45cm?
10 The Chain Rule and Integration by Recall: Substitution The chain rule for derivatives allows us to differentiate a composition of functions: derivative [ f (g(x))]' = f '(g(x))g'(x) antiderivative
11 The Chain Rule and Integration by Substitution Suppose we have an integral of the form where f (g(x))g'(x)dx F'= f. composition of functions derivative of Inside function F is an antiderivative of f Then, by reversing the chain rule for derivatives, we have f (g(x))g'(x)dx = F(g(x)) + C. integrand is the result of differentiating a composition of functions
12 Example Integrate (x 3 +) 4 3x 2 dx.
13 Integration by Substitution Algorithm:. Let u = g(x) where g(x) is the part causing problems and g'(x) cancels the remaining x terms in the integrand. 2. Substitute u = g(x) and du = g'(x)dx into the integral to obtain an equivalent (easier!) integral all in terms of u. f (g(x))g'(x)dx = f (u)du
14 Integration by Substitution Algorithm: 3. Integrate with respect to u, if possible. f (u)du = F(u) + C 4. Write final answer in terms of x again. F(u) + C = F(g(x)) + C
15 Integration by Substitution Example: Integrate each using substitution. (a) 2x + 5 x 2 + 5x 7 dx (b) xe 4 x 2 dx cos x (c) (d) x dx e 4 ln x x dx
16 The Product Rule and Integration by Parts The product rule for derivatives leads to a technique of integration that breaks a complicated integral into simpler parts. Integration by Parts Formula: udv = uv vdu given integral that we cannot solve hopefully this is a simpler Integral to evaluate
17 The Product Rule and Integration by Parts Deriving the Formula Start by writing out the Product Rule: d dx [u(x) v(x)] = du dx v(x) + u(x) dv dx Solve for u(x) dv dx : u(x) dv dx = d dx [u(x) v(x)] du dx v(x)
18 Deriving the Formula The Product Rule and Integration by Parts Integrate both sides with respect to x: u(x) dv dx dx = d dx [u(x) v(x)] dx v(x) du dx dx
19 Deriving the Formula The Product Rule and Integration by Parts Simplify: u(x) dv dx dx = d dx [u(x) v(x)] dx v(x) du dx dx u(x)dv = u(x) v(x) v(x) du
20 Integration by Parts udv = uv vdu Template: Choose: u = part which gets simpler after differentiation dv = easy to integrate part Compute: du = v =
21 Integration by Parts Example: Integrate each using integration by parts. (a) x cos4 xdx (b) x 2 e x 2 dx (c) 2 ln x dx
22 Strategy for Integration Method Basic antiderivative Applies when the integrand is recognized as the reversal of a differentiation formula, such as Guess-and-check the integrand differs from a basic antiderivative in that x is replaced by ax+b, for example Substitution both a function and its derivative (up to a constant) appear in the integrand, such as Integration by parts the integrand is the product of a power of x and one of sin x, cos x, and e x, such as the integrand contains a single function whose derivative we know, such as
23 Strategy for Integration What if the integrand does not have a formula for its antiderivative? Example: impossible to integrate 0 e x 2 dx
24 Approximating Functions with Polynomials Recall: The quadratic approximation to f (x) = e x 2 around the base point x=0 is T 2 (x) = x 2. base point 0.5 f (x) = e x T 2 (x) = x 2
25 Integration Using Taylor Polynomials We approximate the function with an appropriate Taylor polynomial and then integrate this Taylor polynomial instead! Example: impossible to integrate easy to integrate e x 2 dx ( x 2 ) dx 0 0 for x-values near 0
26 Integration Using Taylor Polynomials We can obtain a better approximation by using a higher degree Taylor polynomial to represent the integrand Example: 0 e x2 dx 0 ( x x 4 6 x6 ) dx x 3 x3 + 0 x5 42 x
27 The Definite Integral Area Between Curves The area between the curves y = f (x) and between and is y = g(x) x = a x = b and A = b a f (x) g(x) dx Recall: f (x) g(x) = f (x) g(x) when f (x) g(x) g(x) f (x) when f (x) g(x)
28 The Definite Integral Area Between Curves Examples: Sketch the region enclosed by the given curves and then find the area of the region. (a) y = x 2 2x, y = x + 4 (b) y = x, y = x, x = 2, x = 2
29 The Definite Integral - Average Value f (x) area = b a f (x)dx area = base height = (b a) f f b a f (x) dx = (b a) f
30 The Definite Integral - Average Value The average value of a function f on the interval from a to b is f (x) f = b a b a f (x) dx f For a positive function, average height = area width
31 Application Example: Several very skinny 2.0-m-long snakes are collected in the Amazon. Each snake has a density of ρ(x) = x 2 (300 x) where ρ is measured in grams per centimeter and is measured in centimeters from the tip of the tail. Find the average density of the snake. x
32 Application ρ(x) ρ(x) x x200 x
33 Application (a) Find the total mass of each snake. (b) Find the average density of each snake.
Announcements. 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 informationAnnouncements. Topics: Homework:
Announcements Topics: - sections 7.5 (additional techniques of integration), 7.6 (applications of integration), * Read these sections and study solved examples in your textbook! Homework: - review lecture
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 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 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 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 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 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 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 informationAnnouncements. Topics: Homework: - sections , 6.1 (extreme values) * Read these sections and study solved examples in your textbook!
Announcements Topics: - sections 5.2 5.7, 6.1 (extreme values) * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems
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 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 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 informationIntegration by Parts. MAT 126, Week 2, Thursday class. Xuntao Hu
MAT 126, Week 2, Thursday class Xuntao Hu Recall that the substitution rule is a combination of the FTC and the chain rule. We can also combine the FTC and the product rule: d dx [f (x)g(x)] = f (x)g (x)
More informationThe Relation between the Integral and the Derivative Graphs. Unit #10 : Graphs of Antiderivatives, Substitution Integrals
Graphs of Antiderivatives - Unit #0 : 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.
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 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 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 informationAnnouncements. Topics: Homework: - sections 4.5 and * Read these sections and study solved examples in your textbook!
Announcements Topics: - sections 4.5 and 5.1-5.5 * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems from the textbook
More informationIntegration by Parts
Calculus 2 Lia Vas Integration by Parts Using integration by parts one transforms an integral of a product of two functions into a simpler integral. Divide the initial function into two parts called u
More informationGraphs 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 112 Section 10 Lecture notes, 1/7/04
Math 11 Section 10 Lecture notes, 1/7/04 Section 7. Integration by parts To integrate the product of two functions, integration by parts is used when simpler methods such as substitution or simplifying
More informationLecture 4: Integrals and applications
Lecture 4: Integrals and applications Lejla Batina Institute for Computing and Information Sciences Digital Security Version: autumn 2013 Lejla Batina Version: autumn 2013 Calculus en Kansrekenen 1 / 18
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 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 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 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 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 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 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 informationMath 3B: Lecture 11. Noah White. October 25, 2017
Math 3B: Lecture 11 Noah White October 25, 2017 Introduction Midterm 1 Introduction Midterm 1 Average is 73%. This is higher than I expected which is good. Introduction Midterm 1 Average is 73%. This is
More informationFundamental Theorem of Calculus
Fundamental Theorem of Calculus MATH 6 Calculus I J. Robert Buchanan Department of Mathematics Summer 208 Remarks The Fundamental Theorem of Calculus (FTC) will make the evaluation of definite integrals
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 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 informationLecture 22: Integration by parts and u-substitution
Lecture 22: Integration by parts and u-substitution Victoria LEBED, lebed@maths.tcd.ie MA1S11A: Calculus with Applications for Scientists December 1, 2017 1 Integration vs differentiation From our first
More informationSection 5.5 More Integration Formula (The Substitution Method) 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 5.5 More Integration Formula (The Substitution Method) 2 Lectures College of Science MATHS : Calculus I (University of Bahrain) Integrals / 7 The Substitution Method Idea: To replace a relatively
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 informationLecture : The Indefinite Integral MTH 124
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
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 informationDerivative and Integral Rules These are on the inside of the back cover of your text.
Derivative and Integral Rules These are on the inside of the back cover of your text. General Derivative Rule General Integral Rule d dx u(x) r = r u(x) r - 1 u(x) u(x)r u(x) dx = u(x) r1 r1 + C r U -1
More informationM152: Calculus II Midterm Exam Review
M52: Calculus II Midterm Exam Review Chapter 4. 4.2 : Mean Value Theorem. - Know the statement and idea of Mean Value Theorem. - Know how to find values of c making the theorem true. - Realize the importance
More information6.1 Antiderivatives and Slope Fields Calculus
6. Antiderivatives and Slope Fields Calculus 6. ANTIDERIVATIVES AND SLOPE FIELDS Indefinite Integrals In the previous chapter we dealt with definite integrals. Definite integrals had limits of integration.
More information1 Lecture 39: The substitution rule.
Lecture 39: The substitution rule. Recall the chain rule and restate as the substitution rule. u-substitution, bookkeeping for integrals. Definite integrals, changing limits. Symmetry-integrating even
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 informationIntegrated Calculus II Exam 1 Solutions 2/6/4
Integrated Calculus II Exam Solutions /6/ Question Determine the following integrals: te t dt. We integrate by parts: u = t, du = dt, dv = e t dt, v = dv = e t dt = e t, te t dt = udv = uv vdu = te t (
More informationa x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x).
You should prepare the following topics for our final exam. () Pre-calculus. (2) Inverses. (3) Algebra of Limits. (4) Derivative Formulas and Rules. (5) Graphing Techniques. (6) Optimization (Maxima and
More informationCalculus II Practice Test Problems for Chapter 7 Page 1 of 6
Calculus II Practice Test Problems for Chapter 7 Page of 6 This is a set of practice test problems for Chapter 7. This is in no way an inclusive set of problems there can be other types of problems on
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 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 informationThe Fundamental Theorem of Calculus Part 3
The Fundamental Theorem of Calculus Part FTC Part Worksheet 5: Basic Rules, Initial Value Problems, Rewriting Integrands A. It s time to find anti-derivatives algebraically. Instead of saying the anti-derivative
More informationMAT 271 Recitation. MAT 271 Recitation. Sections 7.1 and 7.2. Lindsey K. Gamard, ASU SoMSS. 30 August 2013
MAT 271 Recitation Sections 7.1 and 7.2 Lindsey K. Gamard, ASU SoMSS 30 August 2013 Agenda Today s agenda: 1. Review 2. Review Section 7.2 (Trigonometric Integrals) 3. (If time) Start homework in pairs
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 informationScience One Integral Calculus. January 9, 2019
Science One Integral Calculus January 9, 2019 Recap: What have we learned so far? The definite integral is defined as a limit of Riemann sums Riemann sums can be constructed using any point in a subinterval
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 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 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 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 informationGrade: The remainder of this page has been left blank for your workings. VERSION D. Midterm D: Page 1 of 12
First Name: Student-No: Last Name: Section: Grade: The remainder of this page has been left blank for your workings. Midterm D: Page of 2 Indefinite Integrals. 9 marks Each part is worth marks. Please
More information7.5 Partial Fractions and Integration
650 CHPTER 7. DVNCED INTEGRTION TECHNIQUES 7.5 Partial Fractions and Integration In this section we are interested in techniques for computing integrals of the form P(x) dx, (7.49) Q(x) where P(x) and
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 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 informationStudy 5.5, # 1 5, 9, 13 27, 35, 39, 49 59, 63, 69, 71, 81. Class Notes: Prof. G. Battaly, Westchester Community College, NY Homework.
Goals: 1. Recognize an integrand that is the derivative of a composite function. 2. Generalize the Basic Integration Rules to include composite functions. 3. Use substitution to simplify the process of
More informationGrade: The remainder of this page has been left blank for your workings. VERSION E. Midterm E: Page 1 of 12
First Name: Student-No: Last Name: Section: Grade: The remainder of this page has been left blank for your workings. Midterm E: Page of Indefinite Integrals. 9 marks Each part is worth 3 marks. Please
More informationWed. Sept 28th: 1.3 New Functions from Old Functions: o vertical and horizontal shifts o vertical and horizontal stretching and reflecting o
Homework: Appendix A: 1, 2, 3, 5, 6, 7, 8, 11, 13-33(odd), 34, 37, 38, 44, 45, 49, 51, 56. Appendix B: 3, 6, 7, 9, 11, 14, 16-21, 24, 29, 33, 36, 37, 42. Appendix D: 1, 2, 4, 9, 11-20, 23, 26, 28, 29,
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 106: Review for Exam II - SOLUTIONS
Math 6: Review for Exam II - SOLUTIONS INTEGRATION TIPS Substitution: usually let u a function that s inside another function, especially if du (possibly off by a multiplying constant) is also present
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 informationScience One Math. January
Science One Math January 10 2018 (last time) The Fundamental Theorem of Calculus (FTC) Let f be continuous on an interval I containing a. 1. Define F(x) = f t dt with F (x) = f(x). on I. Then F is differentiable
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 informationSOLUTIONS FOR PRACTICE FINAL EXAM
SOLUTIONS FOR PRACTICE FINAL EXAM ANDREW J. BLUMBERG. Solutions () Short answer questions: (a) State the mean value theorem. Proof. The mean value theorem says that if f is continuous on (a, b) and differentiable
More informationSolutions to Math 41 Final Exam December 10, 2012
Solutions to Math 4 Final Exam December,. ( points) Find each of the following limits, with justification. If there is an infinite limit, then explain whether it is or. x ln(t + ) dt (a) lim x x (5 points)
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 informationCalculus II Lecture Notes
Calculus II Lecture Notes David M. McClendon Department of Mathematics Ferris State University 206 edition Contents Contents 2 Review of Calculus I 5. Limits..................................... 7.2 Derivatives...................................3
More informationChapter 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 informationReview for the Final Exam
Math 171 Review for the Final Exam 1 Find the limits (4 points each) (a) lim 4x 2 3; x x (b) lim ( x 2 x x 1 )x ; (c) lim( 1 1 ); x 1 ln x x 1 sin (x 2) (d) lim x 2 x 2 4 Solutions (a) The limit lim 4x
More informationPositive Series: Integral Test & p-series
Positive Series: Integral Test & p-series Calculus II Josh Engwer TTU 3 March 204 Josh Engwer (TTU) Positive Series: Integral Test & p-series 3 March 204 / 8 Bad News about Summing a (Convergent) Series...
More informationReform Calculus: Part II. Marcel B. Finan Arkansas Tech University c All Rights Reserved
Reform Calculus: Part II Marcel B. Finan Arkansas Tech University c All Rights Reserved PREFACE This supplement consists of my lectures of a sophomore-level mathematics class offered at Arkansas Tech University.
More informationDRAFT - Math 102 Lecture Note - Dr. Said Algarni
Math02 - Term72 - Guides and Exercises - DRAFT 7 Techniques of Integration A summery for the most important integrals that we have learned so far: 7. Integration by Parts The Product Rule states that if
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 informationChapter 7 Notes, Stewart 7e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m xcos n (x)dx...
Contents 7.1 Integration by Parts........................................ 2 7.2 Trigonometric Integrals...................................... 8 7.2.1 Evaluating sin m xcos n (x)dx..............................
More information1 Antiderivatives graphically and numerically
Math B - Calculus by Hughes-Hallett, et al. Chapter 6 - Constructing antiderivatives Prepared by Jason Gaddis Antiderivatives graphically and numerically Definition.. The antiderivative of a function f
More informationChapter 6 Differential Equations and Mathematical Modeling. 6.1 Antiderivatives and Slope Fields
Chapter 6 Differential Equations and Mathematical Modeling 6. Antiderivatives and Slope Fields Def: An equation of the form: = y ln x which contains a derivative is called a Differential Equation. In this
More informationCalculus II Study Guide Fall 2015 Instructor: Barry McQuarrie Page 1 of 8
Calculus II Study Guide Fall 205 Instructor: Barry McQuarrie Page of 8 You should be expanding this study guide as you see fit with details and worked examples. With this extra layer of detail you will
More informationCalculus II Practice Test 1 Problems: , 6.5, Page 1 of 10
Calculus II Practice Test Problems: 6.-6.3, 6.5, 7.-7.3 Page of This is in no way an inclusive set of problems there can be other types of problems on the actual test. To prepare for the test: review homework,
More informationAnnouncements. Topics: Homework: - sections 1.4, 2.2, and 2.3 * Read these sections and study solved examples in your textbook!
Announcements Topics: - sections 1.4, 2.2, and 2.3 * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems from the textbook
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 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 informationWe can regard an indefinite integral as representing an entire family of functions (one antiderivative for each value of the constant C).
4.4 Indefinite Integrals and the Net Change Theorem Because of the relation given by the Fundamental Theorem of Calculus between antiderivatives and integrals, the notation f(x) dx is traditionally used
More informationSec 4.1 Limits, Informally. When we calculated f (x), we first started with the difference quotient. f(x + h) f(x) h
1 Sec 4.1 Limits, Informally When we calculated f (x), we first started with the difference quotient f(x + h) f(x) h and made h small. In other words, f (x) is the number f(x+h) f(x) approaches as h gets
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 informationFinal Examination Solutions
Math. 5, Sections 5 53 (Fulling) 7 December Final Examination Solutions Test Forms A and B were the same except for the order of the multiple-choice responses. This key is based on Form A. Name: Section:
More informationSummary: Primer on Integral Calculus:
Physics 2460 Electricity and Magnetism I, Fall 2006, Primer on Integration: Part I 1 Summary: Primer on Integral Calculus: Part I 1. Integrating over a single variable: Area under a curve Properties of
More informationINTEGRALS5 INTEGRALS
INTEGRALS5 INTEGRALS INTEGRALS 5.3 The Fundamental Theorem of Calculus In this section, we will learn about: The Fundamental Theorem of Calculus and its significance. FUNDAMENTAL THEOREM OF CALCULUS The
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 information10550 PRACTICE FINAL EXAM SOLUTIONS. x 2 4. x 2 x 2 5x +6 = lim x +2. x 2 x 3 = 4 1 = 4.
55 PRACTICE FINAL EXAM SOLUTIONS. First notice that x 2 4 x 2x + 2 x 2 5x +6 x 2x. This function is undefined at x 2. Since, in the it as x 2, we only care about what happens near x 2 an for x less than
More informationMath. 151, WebCalc Sections December Final Examination Solutions
Math. 5, WebCalc Sections 507 508 December 00 Final Examination Solutions Name: Section: Part I: Multiple Choice ( points each) There is no partial credit. You may not use a calculator.. Another word for
More informationStudy 5.3 # , 157; 5.2 # 111, 113
Goals: 1. Recognize and understand the Fundamental Theorem of Calculus. 2. Use the Fundamental Theorum of Calculus to evaluate Definite Integrals. 3. Recognize and understand the Mean Value Theorem for
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 informationCalculus Review Problems for Math 341 (Probability)
Calculus Review Problems for Math 34 (Probability) Steven J. Miller: sjm@williams.edu June 22, 29 2 Contents 3 CONTENTS CONTENTS 4 Chapter Calculus Review Problems Calculus is an essential tool in probability
More information