Antiderivatives! Outline. James K. Peterson. January 28, Antiderivatives. Simple Fractional Power Antiderivatives
|
|
- Julian Hampton
- 5 years ago
- Views:
Transcription
1 Antiderivatives! James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 28, 2014 Outline Antiderivatives Simple Fractional Power Antiderivatives
2 Abstract This lecture is going to talk about a thing called antiderivatives which is like the inverse of taking a derivative. We ll work you through all of the details and you ll find it is not so bad. It trains your mind to do stuff in your head which is a good thing! The idea of an Antiderivative or Primitive is very simple. We just guess! We say F is the antiderivative of f if F = f. Since we know a group of simple derivatives, we can guess a group of simple antiderivatives! From this definition, we can see immediately that antiderivatives are not unique. The derivative of any constant is zero, so adding a constant to an antiderivative just gives a new antiderivative!
3 As we said, we can guess many antiderivatives. The symbol we previously introduced as the symbol for a Riemann integral ( be patient, we will be getting to that soon!) is also used to denote the antiderivative. This common symbol for the antiderivative of f has thus evolved to be f because of the close connection between the antiderivative of f and the Riemann integral of f which is given in the Cauchy Fundamental Theorem of Calculus which we will get to in a bit. The usual Riemann integral, b a f (t) dt of f on [a, b] computes a definite value hence, the symbol b a f (t) dt to contrast it with the family of functions represented by the antiderivative f. We will discuss this thing called a Riemann integral shortly. We discussed the idea of it earlier and showed it was a kind of limit. Since the antiderivatives are arbitrary up to a constant, most of us refer to the antiderivative as the indefinite integral of f Also, we hardly ever say let s find the antiderivative of f instead, we just say, let s integrate f. We will begin using this shorthand now!
4 Let s begin with antiderivative we can guess for the function t. We know the derivative of t 2 is 2t so it follows the derivative of 1/2 t 2 must be t. In fact, adding a constant doesn t change the result. In general, we have for any constant C that ( ) d 1/2t 2 + C = t dt We say that 1/2t 2 + C is the family of antiderivatives of t or more simply, just 1/2t 2 + C is the antiderivative of t. Note we could also say these are the primitives of t too. The usual symbol for the antiderivative is the Riemann integral symbol without the a and b. So we would say t dt represents the antiderivative of t and t dt = 1/2t 2 + C. Next, let s look at the function t 2. We know the derivative of t 3 is 3t 2 so it follows the derivative of 1/3 t 3 must be t 2. In fact, adding a constant doesn t change the result. In general, we have for any constant C that ( ) d 1/3t 3 + C = t 2 dt We say that 1/3t 3 + C is the family of antiderivatives of t 2 or more simply, just 1/3t 3 + C is the antiderivative of t 2. Note we could also say these are the primitives of t 2 too. Also, we would say for any constant C that t 2 dt = 1/3t 3 + C.
5 We can do a similar analysis for other powers. You should be able to convince yourself that for these positive powers, we have 1 dt = t + C. t dt = 1/2 t 2 + C. t 2 dt = 1/3 t 3 + C. t 3 dt = 1/4 t 4 + C. Further, we can guess for negative powers also. We can do a similar analysis for negative powers. You should be able to convince yourself that t 2 dt = t 1 + C. t 3 dt = 1/2t 2 + C. t 4 dt = 1/3 t 3 + C. t 5 dt = 1/4 t 4 + C. We can then glue together these antiderivatives to handle polynomials!
6 Find (2t + 3) dt. (2t + 3) dt = (2t) dt + 3 dt = 2 t dt dt = t 2 + 3t + C where the C indicates that we can add any constant we want and still get an antiderivative. C is often called the integration constant. Find (5t 2 + 8t 2) dt. ( ) ( ) ( ) (5t 2 + 8t 2) dt = 5 t 3 /3 + 8 t 2 /2 2 t + C.
7 Find (5t 3 + 8t 2 2t) dt. ( ) (5t 3 + 8t 2 2t) dt = 5 t 2 /( 2) + ( ) ( ) 8 t 1 /( 1) 2 t 2 /2 + C Find (5t 5 + 4t ) dt. ( ) ( ) (5t 5 + 4t ) dt = 5 t 6 /6 + 4 t 3 /3 + 20t + C.
8 Homework Find (15t 4 + 4t 3 + 9t + 7) dt Find (6t 3 4t 2 12) dt Find (50t t 2/(t 2 )) dt Find (12 + 8t 7 2t 12 ) dt Find (4 + 7t) dt Find (6 + 3t 4 ) dt Find (1 + 2t + 3t 2 ) dt Find ( t 4t 5 ) dt. Now we haven t yet discussed derivative of fractional powers of x. It is not that hard but it is easy to get blown away by a listing of too many mathy things, boom, one after the other. Here is a simple example to show you how to do it. Consider f (x) = x 2/3. For convenience, let y = f (x). Then we have y = x 2/3. Cube both sides to get y 3 = x 2. Now use the chain rule on the left hand side and a regular derivative on the right hand side to get 3 y 2 y = 2 x. Now we just manipulate 3 y 2 y = 2 x = y = 2 x 3 y 2. But, we can plug in for y 2 = x 4/3 to get y = 2 x 3 x 4/3 = 2 3 x 1 4/3 = 2 3 x 1/3.
9 This mix of chain rule and regular differentiation is an easy trick. You can see we can do this for any fraction p/q. We get another theorem! Theorem The Simple Power Rule: Fractions! If f is the function x p/q for any integer p and q except q = 0, of course, then the derivative of f with respect to x satisfies (x p/q) = p q x p/q 1 Proof We did the example for the power 2/3 but the reasoning is the same for any fraction! So we can also find antiderivatives of fractional powers. You should be able to convince yourself that t 1 2 dt = 2/3 t 3/2 + C. t 4 5 dt = 5/9 t 9/5 + C. t 5 8 dt = 8/13 t 13/8 + C. t 2 3 dt = 3 t 1/3 + C.
10 It is thus easy to guess the antiderivative of a power of t as we have already mentioned. Here is the Theorem. Theorem Antiderivatives Of Simple Fractional Powers If u is any fractional power other than 1, then the antiderivative of f (t) = t u is F (t) = t u+1 /(u + 1) + C. This is also expressed as t u dt = t u+1 /(u + 1) + C. Since this result holds for any fractional power and fractions and irrational numbers can t be isolated from one another, we can show more. The rule above holds for any number other than 1: even a number like 2. But we won t belabor this point now. Proof This is just a statement of all the results we have gone over. Find t 4/5 dt. t 4/5 dt = t 9/5 /(9/5) + C.
11 Find (t 1/2 + 9t 1/3 ) dt. ( ) (t 1/2 + 9t 1/3 ) dt = t 3/2 /(3/2) + 9 t 4/3 /(4/3) + C. Find (6t 5/7 ) dt. ( ) (6t 5/7 ) dt = 6 t 12/7 /(12/7) + C.
12 Find (8t 3/4 + 12t 1/5 ) dt. ( ) (8t 3/4 + 12t 1/5 ) dt = 8 t 1/4 /(1/4) ( ) +12 t 4/5 /(4/5) + C. Homework Find (6 t 2/7 ) dt Find (4 t 3/2 + 5t 1/3 ) dt Find (20 t 12/5 ) dt Find (2 t 2/7 + 14t 5/8 ) dt Find ( 22 t 11/3 + 6 t 1/4 ) dt Find (3 x 9/8 ) dx Find (6 u 4/3 + 5 u 7/2 ) du Find ( 19y 1/6 ) dy.
Antiderivatives! James K. Peterson. January 28, Department of Biological Sciences and Department of Mathematical Sciences Clemson University
! James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 28, 2014 Outline 1 2 Simple Fractional Power Abstract This lecture is going to talk
More informationMathematical Induction
Mathematical Induction James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 12, 2017 Outline Introduction to the Class Mathematical Induction
More informationRiemann Sums. Outline. James K. Peterson. September 15, Riemann Sums. Riemann Sums In MatLab
Riemann Sums James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 15, 2013 Outline Riemann Sums Riemann Sums In MatLab Abstract This
More informationDerivatives and the Product Rule
Derivatives and the Product Rule James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 28, 2014 Outline 1 Differentiability 2 Simple Derivatives
More informationDefining Exponential Functions and Exponential Derivatives and Integrals
Defining Exponential Functions and Exponential Derivatives and Integrals James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 19, 2014
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 informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predator - Prey Model Trajectories and the nonlinear conservation law James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 28, 2013 Outline
More informationProject One: C Bump functions
Project One: C Bump functions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 2, 2018 Outline 1 2 The Project Let s recall what the
More informationRiemann Integration. James K. Peterson. February 2, Department of Biological Sciences and Department of Mathematical Sciences Clemson University
Riemann Integration James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 2, 2017 Outline 1 Riemann Sums 2 Riemann Sums In MatLab 3 Graphing
More informationDifferentiating Series of Functions
Differentiating Series of Functions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 30, 017 Outline 1 Differentiating Series Differentiating
More informationMAT137 - Term 2, Week 2
MAT137 - Term 2, Week 2 This lecture will assume you have watched all of the videos on the definition of the integral (but will remind you about some things). Today we re talking about: More on the definition
More informationMore On Exponential Functions, Inverse Functions and Derivative Consequences
More On Exponential Functions, Inverse Functions and Derivative Consequences James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 10, 2019
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 informationRiemann Integration. Outline. James K. Peterson. February 2, Riemann Sums. Riemann Sums In MatLab. Graphing Riemann Sums
Riemann Integration James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 2, 2017 Outline Riemann Sums Riemann Sums In MatLab Graphing
More informationIntegration by Parts Logarithms and More Riemann Sums!
Integration by Parts Logarithms and More Riemann Sums! James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 16, 2013 Outline 1 IbyP with
More informationHölder s and Minkowski s Inequality
Hölder s and Minkowski s Inequality James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 1, 218 Outline Conjugate Exponents Hölder s
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 informationLogarithm and Exponential Derivatives and Integrals
Logarithm and Exponential Derivatives and Integrals James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 3, 2013 Outline 1 Exponential
More informationThe Method of Undetermined Coefficients.
The Method of Undetermined Coefficients. James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 24, 2017 Outline 1 Annihilators 2 Finding The
More informationHölder s and Minkowski s Inequality
Hölder s and Minkowski s Inequality James K. Peterson Deartment of Biological Sciences and Deartment of Mathematical Sciences Clemson University Setember 10, 2018 Outline 1 Conjugate Exonents 2 Hölder
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 informationInfinite series, improper integrals, and Taylor series
Chapter 2 Infinite series, improper integrals, and Taylor series 2. Introduction to series In studying calculus, we have explored a variety of functions. Among the most basic are polynomials, i.e. functions
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 informationSin, Cos and All That
Sin, Cos and All That James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University March 9, 2017 Outline 1 Sin, Cos and all that! 2 A New Power Rule 3
More informationDirchlet s Function and Limit and Continuity Arguments
Dirchlet s Function and Limit and Continuity Arguments James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 23, 2018 Outline 1 Dirichlet
More informationRelationship Between Integration and Differentiation
Relationship Between Integration and Differentiation Fundamental Theorem of Calculus Philippe B. Laval KSU Today Philippe B. Laval (KSU) FTC Today 1 / 16 Introduction In the previous sections we defined
More informationLinear Systems of ODE: Nullclines, Eigenvector lines and trajectories
Linear Systems of ODE: Nullclines, Eigenvector lines and trajectories James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 6, 203 Outline
More informationConvergence of Sequences
James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 5, 2018 Outline 1 2 Homework Definition Let (a n ) n k be a sequence of real numbers.
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 informationAxiomatic systems. Revisiting the rules of inference. Example: A theorem and its proof in an abstract axiomatic system:
Axiomatic systems Revisiting the rules of inference Material for this section references College Geometry: A Discovery Approach, 2/e, David C. Kay, Addison Wesley, 2001. In particular, see section 2.1,
More informationLinear Systems of ODE: Nullclines, Eigenvector lines and trajectories
Linear Systems of ODE: Nullclines, Eigenvector lines and trajectories James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 6, 2013 Outline
More informationThe Derivative of a Function
The Derivative of a Function James K Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University March 1, 2017 Outline A Basic Evolutionary Model The Next Generation
More informationVariation of Parameters
Variation of Parameters James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University April 13, 218 Outline Variation of Parameters Example One We eventually
More informationMAT 122 Homework 7 Solutions
MAT 1 Homework 7 Solutions Section 3.3, Problem 4 For the function w = (t + 1) 100, we take the inside function to be z = t + 1 and the outside function to be z 100. The derivative of the inside function
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 informationProofs Not Based On POMI
s Not Based On POMI James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 12, 2018 Outline 1 Non POMI Based s 2 Some Contradiction s 3
More informationFourier Sin and Cos Series and Least Squares Convergence
Fourier and east Squares Convergence James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 7, 28 Outline et s look at the original Fourier sin
More informationMathematical Induction Again
Mathematical Induction Again James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 12, 2017 Outline Mathematical Induction Simple POMI Examples
More informationMathematical Induction Again
Mathematical Induction Again James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 2, 207 Outline Mathematical Induction 2 Simple POMI Examples
More informationThe First Derivative and Second Derivative Test
The First Derivative and Second Derivative Test James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University April 9, 2018 Outline 1 Extremal Values 2
More informationConvergence of Sequences
Convergence of Sequences James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 12, 2018 Outline Convergence of Sequences Definition Let
More informationMath 126: Course Summary
Math 126: Course Summary Rich Schwartz August 19, 2009 General Information: Math 126 is a course on complex analysis. You might say that complex analysis is the study of what happens when you combine calculus
More informationThe First Derivative and Second Derivative Test
The First Derivative and Second Derivative Test James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 8, 2017 Outline Extremal Values The
More information3 Algebraic Methods. we can differentiate both sides implicitly to obtain a differential equation involving x and y:
3 Algebraic Methods b The first appearance of the equation E Mc 2 in Einstein s handwritten notes. So far, the only general class of differential equations that we know how to solve are directly integrable
More 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 informationCHAPTER 7: TECHNIQUES OF INTEGRATION
CHAPTER 7: TECHNIQUES OF INTEGRATION DAVID GLICKENSTEIN. Introduction This semester we will be looking deep into the recesses of calculus. Some of the main topics will be: Integration: we will learn how
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 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 informationThe Integral of a Function. The Indefinite Integral
The Integral of a Function. The Indefinite Integral Undoing a derivative: Antiderivative=Indefinite Integral Definition: A function is called an antiderivative of a function on same interval,, if differentiation
More informationMath Lecture 18 Notes
Math 1010 - Lecture 18 Notes Dylan Zwick Fall 2009 In our last lecture we talked about how we can add, subtract, and multiply polynomials, and we figured out that, basically, if you can add, subtract,
More informationComplex Numbers. Outline. James K. Peterson. September 19, Complex Numbers. Complex Number Calculations. Complex Functions
Complex Numbers James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 19, 2013 Outline Complex Numbers Complex Number Calculations Complex
More informationComplex Numbers. James K. Peterson. September 19, Department of Biological Sciences and Department of Mathematical Sciences Clemson University
Complex Numbers James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 19, 2013 Outline 1 Complex Numbers 2 Complex Number Calculations
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 informationDifferential Forms. Introduction
ifferential Forms A 305 Kurt Bryan Aesthetic pleasure needs no justification, because a life without such pleasure is one not worth living. ana Gioia, Can Poetry atter? Introduction We ve hit the big three
More informationConvergence of Fourier Series
MATH 454: Analysis Two James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University April, 8 MATH 454: Analysis Two Outline The Cos Family MATH 454: Analysis
More informationMath Lecture 23 Notes
Math 1010 - Lecture 23 Notes Dylan Zwick Fall 2009 In today s lecture we ll expand upon the concept of radicals and radical expressions, and discuss how we can deal with equations involving these radical
More informationProofs Not Based On POMI
s Not Based On POMI James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 1, 018 Outline Non POMI Based s Some Contradiction s Triangle
More informationQuadratic Equations Part I
Quadratic Equations Part I Before proceeding with this section we should note that the topic of solving quadratic equations will be covered in two sections. This is done for the benefit of those viewing
More informationIntroduction to Algebra: The First Week
Introduction to Algebra: The First Week Background: According to the thermostat on the wall, the temperature in the classroom right now is 72 degrees Fahrenheit. I want to write to my friend in Europe,
More informationExtreme Values and Positive/ Negative Definite Matrix Conditions
Extreme Values and Positive/ Negative Definite Matrix Conditions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 8, 016 Outline 1
More informationTaylor Polynomials. James K. Peterson. Department of Biological Sciences and Department of Mathematical Sciences Clemson University
James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 24, 2013 Outline 1 First Order Approximation s Second Order Approximations 2 Approximation
More informationLecture 7 - Separable Equations
Lecture 7 - Separable Equations Separable equations is a very special type of differential equations where you can separate the terms involving only y on one side of the equation and terms involving only
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 1: Preliminaries and Error Analysis
Chapter 1: Error Analysis Peter W. White white@tarleton.edu Department of Tarleton State University Summer 2015 / Numerical Analysis Overview We All Remember Calculus Derivatives: limit definition, sum
More informationMath Calculus I
Math 165 - Calculus I Christian Roettger 382 Carver Hall Mathematics Department Iowa State University www.iastate.edu/~roettger November 13, 2011 4.1 Introduction to Area Sigma Notation 4.2 The Definite
More informationDirchlet s Function and Limit and Continuity Arguments
Dirchlet s Function and Limit and Continuity Arguments James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 2, 2018 Outline Dirichlet
More informationMore Protein Synthesis and a Model for Protein Transcription Error Rates
More Protein Synthesis and a Model for Protein James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 3, 2013 Outline 1 Signal Patterns Example
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 information( ) = f(x) 6 INTEGRATION. Things to remember: n + 1 EXERCISE A function F is an ANTIDERIVATIVE of f if F'(x) = f(x).
6 INTEGRATION EXERCISE 6-1 Things to remember: 1. A function F is an ANTIDERIVATIVE of f if F() = f().. THEOREM ON ANTIDERIVATIVES If the derivatives of two functions are equal on an open interval (a,
More informationThe total differential
The total differential The total differential of the function of two variables The total differential gives the full information about rates of change of the function in the -direction and in the -direction.
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 informationTangent Lines Sec. 2.1, 2.7, & 2.8 (continued)
Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Prove this Result How Can a Derivative Not Exist? Remember that the derivative at a point (or slope of a tangent line) is a LIMIT, so it doesn t exist whenever
More informationHow to write maths (well)
How to write maths (well) Dr Euan Spence 29 September 2017 These are the slides from a talk I gave to the new first-year students at Bath, annotated with some of the things I said (which appear in boxes
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 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 informationMath101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2:
Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2: 03 17 08 3 All about lines 3.1 The Rectangular Coordinate System Know how to plot points in the rectangular coordinate system. Know 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 informationBoolean circuits. Lecture Definitions
Lecture 20 Boolean circuits In this lecture we will discuss the Boolean circuit model of computation and its connection to the Turing machine model. Although the Boolean circuit model is fundamentally
More informationThe Existence of the Riemann Integral
The Existence of the Riemann Integral James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 18, 2018 Outline The Darboux Integral Upper
More informationMATH 408N PRACTICE FINAL
2/03/20 Bormashenko MATH 408N PRACTICE FINAL Show your work for all the problems. Good luck! () Let f(x) = ex e x. (a) [5 pts] State the domain and range of f(x). Name: TA session: Since e x is defined
More informationMA1131 Lecture 15 (2 & 3/12/2010) 77. dx dx v + udv dx. (uv) = v du dx dx + dx dx dx
MA3 Lecture 5 ( & 3//00) 77 0.3. Integration by parts If we integrate both sides of the proct rule we get d (uv) dx = dx or uv = d (uv) = dx dx v + udv dx v dx dx + v dx dx + u dv dx dx u dv dx dx This
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 informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Basic Concepts Paul Dawkins Table of Contents Preface... Basic Concepts... 1 Introduction... 1 Definitions... Direction Fields... 8 Final Thoughts...19 007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx
More information7.3 Hyperbolic Functions Hyperbolic functions are similar to trigonometric functions, and have the following
Math 2-08 Rahman Week3 7.3 Hyperbolic Functions Hyperbolic functions are similar to trigonometric functions, and have the following definitions: sinh x = 2 (ex e x ) cosh x = 2 (ex + e x ) tanh x = sinh
More informationMath Lecture 3 Notes
Math 1010 - Lecture 3 Notes Dylan Zwick Fall 2009 1 Operations with Real Numbers In our last lecture we covered some basic operations with real numbers like addition, subtraction and multiplication. This
More informationMAT137 - Term 2, Week 5
MAT137 - Term 2, Week 5 Test 3 is tomorrow, February 3, at 4pm. See the course website for details. Today we will: Talk more about integration by parts. Talk about integrating certain combinations of trig
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 informationReview for Final Exam, MATH , Fall 2010
Review for Final Exam, MATH 170-002, Fall 2010 The test will be on Wednesday December 15 in ILC 404 (usual class room), 8:00 a.m - 10:00 a.m. Please bring a non-graphing calculator for the test. No other
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 informationOne-to-one functions and onto functions
MA 3362 Lecture 7 - One-to-one and Onto Wednesday, October 22, 2008. Objectives: Formalize definitions of one-to-one and onto One-to-one functions and onto functions At the level of set theory, there are
More informationUniform Convergence and Series of Functions
Uniform Convergence and Series of Functions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 7, 017 Outline Uniform Convergence Tests
More informationNotes on Quadratic Extension Fields
Notes on Quadratic Extension Fields 1 Standing notation Q denotes the field of rational numbers. R denotes the field of real numbers. F always denotes a subfield of R. The symbol k is always a positive
More informationMath 110 (S & E) Textbook: Calculus Early Transcendentals by James Stewart, 7 th Edition
Math 110 (S & E) Textbook: Calculus Early Transcendentals by James Stewart, 7 th Edition 1 Appendix A : Numbers, Inequalities, and Absolute Values Sets A set is a collection of objects with an important
More informationArea. A(2) = sin(0) π 2 + sin(π/2)π 2 = π For 3 subintervals we will find
Area In order to quantify the size of a -dimensional object, we use area. Since we measure area in square units, we can think of the area of an object as the number of such squares it fills up. Using this
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 informationDerivatives in 2D. Outline. James K. Peterson. November 9, Derivatives in 2D! Chain Rule
Derivatives in 2D James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 9, 2016 Outline Derivatives in 2D! Chain Rule Let s go back to
More informationSin, Cos and All That
Sin, Cos and All That James K Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 9, 2014 Outline Sin, Cos and all that! A New Power Rule Derivatives
More informationFinal Exam Study Guide
Final Exam Study Guide Final Exam Coverage: Sections 10.1-10.2, 10.4-10.5, 10.7, 11.2-11.4, 12.1-12.6, 13.1-13.2, 13.4-13.5, and 14.1 Sections/topics NOT on the exam: Sections 10.3 (Continuity, it definition
More information7.1 Indefinite Integrals Calculus
7.1 Indefinite Integrals Calculus Learning Objectives A student will be able to: Find antiderivatives of functions. Represent antiderivatives. Interpret the constant of integration graphically. Solve differential
More informationMath F15 Rahman
Math - 9 F5 Rahman Week3 7.3 Hyperbolic Functions Hyperbolic functions are similar to trigonometric functions, and have the following definitions: sinh x = (ex e x ) cosh x = (ex + e x ) tanh x = sinh
More information