Lecture 3. Lecturer: Prof. Sergei Fedotov Calculus and Vectors. Exponential and logarithmic functions
|
|
|
- Dana Robinson
- 5 years ago
- Views:
Transcription
1 Lecture 3 Lecturer: Prof. Sergei Fedotov Calculus and Vectors Exponential and logarithmic functions Sergei Fedotov (University of Manchester) MATH / 7
2 Lecture 3 1 Inverse functions 2 Exponential and logarithmic functions 3 Sketching the graphs 4 Hyperbolic functions Sergei Fedotov (University of Manchester) MATH / 7
3 Inverse functions The graph of f 1 is obtained by reflecting the graph of f about the line y = x. Example: Sketch the graph of f (x) = x and its inverse. Sergei Fedotov (University of Manchester) MATH / 7
4 Inverse functions The graph of f 1 is obtained by reflecting the graph of f about the line y = x. Example: Sketch the graph of f (x) = x and its inverse. The domain of f 1 is the range of f The range of f 1 is the domain of f Sergei Fedotov (University of Manchester) MATH / 7
5 Exponential functions Definition. An exponential function is a function of the form f (x) = a x, where a is a positive constant. Sergei Fedotov (University of Manchester) MATH / 7
6 Exponential functions Definition. An exponential function is a function of the form f (x) = a x, where a is a positive constant. Law of exponent: for any positive number a and real numbers x and y: a x+y = a x a y. Sergei Fedotov (University of Manchester) MATH / 7
7 Exponential functions Definition. An exponential function is a function of the form f (x) = a x, where a is a positive constant. Law of exponent: for any positive number a and real numbers x and y: a x+y = a x a y. Sergei Fedotov (University of Manchester) MATH / 7
8 Logarithmic functions Definition. The exponential function f (x) = a x is one-to-one function. It has an inverse function f 1 called the logarithmic function with base a. Standard notation is log a. Sergei Fedotov (University of Manchester) MATH / 7
9 Logarithmic functions Definition. The exponential function f (x) = a x is one-to-one function. It has an inverse function f 1 called the logarithmic function with base a. Standard notation is log a. Law of logarithms: for any real numbers x and y: Remember: f 1 (f (x)) = x log a (a x ) = x for x R a log a x = x for x > 0 Sergei Fedotov (University of Manchester) MATH / 7
10 Logarithmic functions Definition. The exponential function f (x) = a x is one-to-one function. It has an inverse function f 1 called the logarithmic function with base a. Standard notation is log a. Law of logarithms: for any real numbers x and y: Remember: f 1 (f (x)) = x log a (a x ) = x for x R a log a x = x for x > 0 Sergei Fedotov (University of Manchester) MATH / 7
11 Sketching the graphs Example 1: Sketch the graph of the function 2 2 x and determine the domain and range Sergei Fedotov (University of Manchester) MATH / 7
12 Sketching the graphs Example 1: Sketch the graph of the function 2 2 x and determine the domain and range Example 1: Sketch the graph of the function ln(3 x) + 1 and determine the domain and range Sergei Fedotov (University of Manchester) MATH / 7
13 Hyperbolic functions Hyperbolic functions are defined by using the exponential functions sinhx = ex e x, cosh x = ex + e x 2 2 tanhx = sinhx cosh x, coth x = coshx sinhx Sergei Fedotov (University of Manchester) MATH / 7
14 Hyperbolic functions Hyperbolic functions are defined by using the exponential functions sinhx = ex e x, cosh x = ex + e x 2 2 tanhx = sinhx cosh x, coth x = coshx sinhx sinh x, tanh x and coth x are all one-to-one functions, so that their domains are not restricted in constracting inverses Sergei Fedotov (University of Manchester) MATH / 7
15 Hyperbolic functions Hyperbolic functions are defined by using the exponential functions sinhx = ex e x, cosh x = ex + e x 2 2 tanhx = sinhx cosh x, coth x = coshx sinhx sinh x, tanh x and coth x are all one-to-one functions, so that their domains are not restricted in constracting inverses Example: Sketch the graph of the function cosh x and determine the domain and range Sergei Fedotov (University of Manchester) MATH / 7
Monday, 6 th October 2008
MA211 Lecture 9: 2nd order differential eqns Monday, 6 th October 2008 MA211 Lecture 9: 2nd order differential eqns 1/19 Class test next week... MA211 Lecture 9: 2nd order differential eqns 2/19 This morning
Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions
Chapter 5 Logarithmic, Exponential, an Other Transcenental Functions 5.1 The Natural Logarithmic Function: Differentiation 5.2 The Natural Logarithmic Function: Integration 5.3 Inverse Functions 5.4 Exponential
Hyperbolics. Scott Morgan. Further Mathematics Support Programme - WJEC A-Level Further Mathematics 31st March scott3142.
Hyperbolics Scott Morgan Further Mathematics Support Programme - WJEC A-Level Further Mathematics 3st March 208 scott342.com @Scott342 Topics Hyperbolic Identities Calculus with Hyperbolics - Differentiation
Chapter 3 Differentiation Rules (continued)
Chapter 3 Differentiation Rules (continued) Sec 3.5: Implicit Differentiation (continued) Implicit Differentiation What if you want to find the slope of the tangent line to a curve that is not the graph
6.2. The Hyperbolic Functions. Introduction. Prerequisites. Learning Outcomes
The Hyperbolic Functions 6. Introduction The hyperbolic functions cosh x, sinh x, tanh x etc are certain combinations of the exponential functions e x and e x. The notation implies a close relationship
Lecture Notes for Math 1000
Lecture Notes for Math 1000 Dr. Xiang-Sheng Wang Memorial University of Newfoundland Office: HH-2016, Phone: 864-4321 Office hours: 13:00-15:00 Wednesday, 12:00-13:00 Friday Email: xswang@mun.ca Course
Differential and Integral Calculus
School of science an engineering El Akhawayn University Monay, March 31 st, 2008 Outline 1 Definition of hyperbolic functions: The hyperbolic cosine an the hyperbolic sine of the real number x are enote
Hyperbolic Functions. Notice: this material must not be used as a substitute for attending. the lectures
Hyperbolic Functions Notice: this material must not be use as a substitute for attening the lectures 0. Hyperbolic functions sinh an cosh The hyperbolic functions sinh (pronounce shine ) an cosh are efine
JUST THE MATHS UNIT NUMBER DIFFERENTIATION 4 (Products and quotients) & (Logarithmic differentiation) A.J.Hobson
JUST THE MATHS UNIT NUMBER 104 DIFFERENTIATION 4 (Products and quotients) & (Logarithmic differentiation) by AJHobson 1041 Products 1042 Quotients 1043 Logarithmic differentiation 1044 Exercises 1045 Answers
Hyperbolic Functions: Exercises - Sol'ns (9 pages; 13/5/17)
Hyperbolic Functions: Exercises - Sol'ns (9 pages; 3/5/7) () (i) Prove, using exponential functions, that (a) cosh x sinh x = (b) sinhx = sinhxcoshx (ii) By differentiating the result from (i)(b), obtain
Honors Calculus II [ ] Midterm II
![Honors Calculus II [ ] Midterm II](/thumbs/90/103243723.jpg "Honors Calculus II [ ] Midterm II")
Honors Calculus II [3-00] Midterm II PRINT NAME: SOLUTIONS Q]...[0 points] Evaluate the following expressions and its. Since you don t have a calculator, square roots, fractions etc. are allowed in your
Math 180 Prof. Beydler Homework for Packet #5 Page 1 of 11
Math 180 Prof. Beydler Homework for Packet #5 Page 1 of 11 Due date: Name: Note: Write your answers using positive exponents. Radicals are nice, but not required. ex: Write 1 x 2 not x 2. ex: x is nicer
Throughout this module we use x to denote the positive square root of x; for example, 4 = 2.
Throughout this module we use x to denote the positive square root of x; for example, 4 = 2. You may often see (although not in FLAP) the notation sin 1 used in place of arcsin. sinh and cosh are pronounced
6.7 Hyperbolic Functions
6.7 6.7 Hyperbolic Functions Even and Odd Parts of an Exponential Function We recall that a function f is called even if f( x) = f(x). f is called odd if f( x) = f(x). The sine function is odd while the
UNIVERSITI TEKNOLOGI MALAYSIA FACULTY OF SCIENCE... FINAL EXAMINATION SEMESTER I SESSION 2015/2016 COURSE NAME : ENGINEERING MATHEMATICS I
UNIVERSITI TEKNOLOGI MALAYSIA FACULTY OF SCIENCE... FINAL EXAMINATION SEMESTER I SESSION 05/06 COURSE CODE : SSCE 693 COURSE NAME : ENGINEERING MATHEMATICS I PROGRAMME : SKAW/SKEE/SKEL/SKEM/SKMM/SKMV/
Test one Review Cal 2
Name: Class: Date: ID: A Test one Review Cal 2 Short Answer. Write the following expression as a logarithm of a single quantity. lnx 2ln x 2 ˆ 6 2. Write the following expression as a logarithm of a single
Logarithmic and Exponential Equations and Change-of-Base
Logarithmic and Exponential Equations and Change-of-Base MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to solve exponential equations
cosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) = 2 2t 1 + t 2 cos x = 1 t2 We will revisit the double angle identities:
6.4 Integration using tanx/) We will revisit the ouble angle ientities: sin x = sinx/) cosx/) = tanx/) sec x/) = tanx/) + tan x/) cos x = cos x/) sin x/) tan x = = tan x/) sec x/) tanx/) tan x/). = tan
More 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
Exponential Functions:
Exponential Functions: An exponential function has the form f (x) = b x where b is a fixed positive number, called the base. Math 101-Calculus 1 (Sklensky) In-Class Work January 29, 2015 1 / 12 Exponential
Calculus: Early Transcendental Functions Lecture Notes for Calculus 101. Feras Awad Mahmoud
Calculus: Early Transcendental Functions Lecture Notes for Calculus 101 Feras Awad Mahmoud Last Updated: August 2, 2012 1 2 Feras Awad Mahmoud Department of Basic Sciences Philadelphia University JORDAN
7.3 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
ENGI 3425 Review of Calculus Page then
ENGI 345 Review of Calculus Page 1.01 1. Review of Calculus We begin this course with a refresher on ifferentiation an integration from MATH 1000 an MATH 1001. 1.1 Reminer of some Derivatives (review from
Calculus I Announcements
Slide 1 Calculus I Announcements Read sections 3.9-3.10 Do all the homework for section 3.9 and problems 1,3,5,7 from section 3.10. The exam is in Thursday, October 22nd. The exam will cover sections 3.2-3.10,
Math 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
x 2 + y 2 = 1. dx 2y and we conclude that, whichever function we chose, dx y2 = 2x + dy dx x2 + d dy dx = 2x = x y sinh(x) = ex e x 2
Implicit differentiation Suppose we know some relation between x and y, e.g. x + y =. Here, y isn t a function of x. But if we restrict attention to y, then y is a function of x; similarly for y. These
FUNCTIONS OF ONE VARIABLE FUNCTION DEFINITIONS
Page of 6 FUNCTIONS OF ONE VARIABLE FUNCTION DEFINITIONS 6. HYPERBOLIC FUNCTIONS These functions which are defined in terms of e will be seen later to be related to the trigonometic functions via comple
Things you should have learned in Calculus II
Things you should have learned in Calculus II 1 Vectors Given vectors v = v 1, v 2, v 3, u = u 1, u 2, u 3 1.1 Common Operations Operations Notation How is it calculated Other Notation Dot Product v u
L Hôpital s Rule was discovered by Bernoulli but written for the first time in a text by L Hôpital.
7.5. Ineterminate Forms an L Hôpital s Rule L Hôpital s Rule was iscovere by Bernoulli but written for the first time in a text by L Hôpital. Ineterminate Forms 0/0 an / f(x) If f(x 0 ) = g(x 0 ) = 0,
Some functions and their derivatives
Chapter Some functions an their erivatives. Derivative of x n for integer n Recall, from eqn (.6), for y = f (x), Also recall that, for integer n, Hence, if y = x n then y x = lim δx 0 (a + b) n = a n
We define hyperbolic functions cosech, sech and coth in a similar way to the definitions of trigonometric functions cosec, sec and cot respectively:
3 Chapter 5 SECTION F Hyperbolic properties By the end of this section you will be able to: evaluate other hyperbolic functions show hyperbolic identities understand inverse hyperbolic functions F Other
In this chapter we study several functions that are useful in calculus and other areas of mathematics.
Calculus 5 7 Special functions In this chapter we study several functions that are useful in calculus and other areas of mathematics. 7. Hyperbolic trigonometric functions The functions we study in this
Hyperbolic functions
Roberto s Notes on Differential Calculus Chapter 5: Derivatives of transcendental functions Section Derivatives of Hyperbolic functions What you need to know already: Basic rules of differentiation, including
90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y.
90 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions Test Form A Chapter 5 Name Class Date Section. Find the derivative: f ln. 6. Differentiate: y. ln y y y y. Find dy d if ey y. y
cos t 2 sin 2t (vi) y = cosh t sinh t (vii) y sin x 2 = x sin y 2 (viii) xy = cot(xy) (ix) 1 + x = sin(xy 2 ) (v) g(t) =
MATH1003 REVISION 1. Differentiate the following functions, simplifying your answers when appropriate: (i) f(x) = (x 3 2) tan x (ii) y = (3x 5 1) 6 (iii) y 2 = x 2 3 (iv) y = ln(ln(7 + x)) e 5x3 (v) g(t)
2 Recollection of elementary functions. II
Recollection of elementary functions. II Last updated: October 5, 08. In this section we continue recollection of elementary functions. In particular, we consider exponential, trigonometric and hyperbolic
CHAPTER 1. DIFFERENTIATION 18. As x 1, f(x). At last! We are now in a position to sketch the curve; see Figure 1.4.
CHAPTER. DIFFERENTIATION 8 and similarly for x, As x +, fx), As x, fx). At last! We are now in a position to sketch the curve; see Figure.4. Figure.4: A sketch of the function y = fx) =/x ). Observe the
2.6 Logarithmic Functions. Inverse Functions. Question: What is the relationship between f(x) = x 2 and g(x) = x?
Inverse Functions Question: What is the relationship between f(x) = x 3 and g(x) = 3 x? Question: What is the relationship between f(x) = x 2 and g(x) = x? Definition (One-to-One Function) A function f
Chapter 13: General Solutions to Homogeneous Linear Differential Equations
Worked Solutions 1 Chapter 13: General Solutions to Homogeneous Linear Differential Equations 13.2 a. Verifying that {y 1, y 2 } is a fundamental solution set: We have y 1 (x) = cos(2x) y 1 (x) = 2 sin(2x)
cosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) 2t 1 + t 2 cos x = 1 t2 sin x =
6.4 Integration using tan/ We will revisit the ouble angle ientities: sin = sin/ cos/ = tan/ sec / = tan/ + tan / cos = cos / sin / tan = = tan / sec / tan/ tan /. = tan / + tan / So writing t = tan/ we
Math Spring 2014 Solutions to Assignment # 6 Completion Date: Friday May 23, 2014
Math 11 - Spring 014 Solutions to Assignment # 6 Completion Date: Friday May, 014 Question 1. [p 109, #9] With the aid of expressions 15) 16) in Sec. 4 for sin z cos z, namely, sin z = sin x + sinh y cos
YOU CAN BACK SUBSTITUTE TO ANY OF THE PREVIOUS EQUATIONS
The two methods we will use to solve systems are substitution and elimination. Substitution was covered in the last lesson and elimination is covered in this lesson. Method of Elimination: 1. multiply
Goal: Simplify and solve exponential expressions and equations
Pre- Calculus Mathematics 12 4.1 Exponents Part 1 Goal: Simplify and solve exponential expressions and equations Logarithms involve the study of exponents so is it vital to know all the exponent laws.
MIDTERM 1. Name-Surname: 15 pts 20 pts 15 pts 10 pts 10 pts 10 pts 15 pts 20 pts 115 pts Total. Overall 115 points.
Name-Surname: Student No: Grade: 15 pts 20 pts 15 pts 10 pts 10 pts 10 pts 15 pts 20 pts 115 pts 1 2 3 4 5 6 7 8 Total Overall 115 points. Do as much as you can. Write your answers to all of the questions.
Practice Questions From Calculus II. 0. State the following calculus rules (these are many of the key rules from Test 1 topics).
Math 132. Practice Questions From Calculus II I. Topics Covered in Test I 0. State the following calculus rules (these are many of the key rules from Test 1 topics). (Trapezoidal Rule) b a f(x) dx (Fundamental
MSM120 1M1 First year mathematics for civil engineers Revision notes 3
MSM0 M First year mathematics for civil engineers Revision notes Professor Robert. Wilson utumn 00 Functions Definition of a function: it is a rule which, given a value of the independent variable (often
Hyperbolic Discrete Ricci Curvature Flows
Hyperbolic Discrete Ricci Curvature Flows 1 1 Mathematics Science Center Tsinghua University Tsinghua University 010 Unified Framework of Discrete Curvature Flow Unified framework for both Discrete Ricci
School of Distance Education UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION. B Sc Mathematics. (2011 Admission Onwards) IV Semester.
School of Dtance Education UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc Mathematics 0 Admsion Onwards IV Semester Core Course CALCULUS AND ANALYTIC GEOMETRY QUESTION BANK The natural logarithm
Bishop Kelley High School Summer Math Program Course: Honors Pre-Calculus
017 018 Summer Math Program Course: Honors Pre-Calculus NAME: DIRECTIONS: Show all work in the packet. Make sure you are aware of the calculator policy for this course. No matter when you have math, this
7.1. Calculus of inverse functions. Text Section 7.1 Exercise:
Contents 7. Inverse functions 1 7.1. Calculus of inverse functions 2 7.2. Derivatives of exponential function 4 7.3. Logarithmic function 6 7.4. Derivatives of logarithmic functions 7 7.5. Exponential
CALCULUS Exercise Set 2 Integration
CALCULUS Exercise Set Integration 1 Basic Indefinite Integrals 1. R = C. R x n = xn+1 n+1 + C n 6= 1 3. R 1 =ln x + C x 4. R sin x= cos x + C 5. R cos x=sinx + C 6. cos x =tanx + C 7. sin x = cot x + C
7.4* General logarithmic and exponential functions
7.4* General logarithmic and exponential functions Mark Woodard Furman U Fall 2010 Mark Woodard (Furman U) 7.4* General logarithmic and exponential functions Fall 2010 1 / 9 Outline 1 General exponential
MA 242 Review Exponential and Log Functions Notes for today s class can be found at
MA 242 Review Exponential and Log Functions Notes for today s class can be found at www.xecu.net/jacobs/index242.htm Example: If y = x n If y = x 2 then then dy dx = nxn 1 dy dx = 2x1 = 2x Power Function
Friday 09/15/2017 Midterm I 50 minutes
Fa 17: MATH 2924 040 Differential and Integral Calculus II Noel Brady Friday 09/15/2017 Midterm I 50 minutes Name: Student ID: Instructions. 1. Attempt all questions. 2. Do not write on back of exam sheets.
Replacing the a in the definition of the derivative of the function f at a with a variable x, gives the derivative function f (x).
Definition of The Derivative Function Definition (The Derivative Function) Replacing the a in the definition of the derivative of the function f at a with a variable x, gives the derivative function f
Solution Sheet 1.4 Questions 26-31
Solution Sheet 1.4 Questions 26-31 26. Using the Limit Rules evaluate i) ii) iii) 3 2 +4+1 0 2 +4+3, 3 2 +4+1 2 +4+3, 3 2 +4+1 1 2 +4+3. Note When using a Limit Rule you must write down which Rule you
ARAB ACADEMY FOR SCIENCE TECHNOLOGY AND MARITIME TRANSPORT
ARAB ACADEMY FOR SCIENCE TECHNOLOGY AND MARITIME TRANSPORT Course: Math For Engineering Winter 8 Lecture Notes By Dr. Mostafa Elogail Page Lecture [ Functions / Graphs of Rational Functions] Functions
18.01 Final Exam. 8. 3pm Hancock Total: /250
18.01 Final Exam Name: Please circle the number of your recitation. 1. 10am Tyomkin 2. 10am Kilic 3. 12pm Coskun 4. 1pm Coskun 5. 2pm Hancock Problem 1: /25 Problem 6: /25 Problem 2: /25 Problem 7: /25
Math 1A Midterm 2 Fall 2015 Riverside City College (Use this as a Review)
Name Date Miterm Score Overall Grae Math A Miterm 2 Fall 205 Riversie City College (Use this as a Review) Instructions: All work is to be shown, legible, simplifie an answers are to be boxe in the space
APPM 1350 Final Exam Fall 2017
APPM 350 Final Exam Fall 207. (26 pts) Evaluate the following. (a) Let g(x) cos 3 (π 2x). Find g (π/3). (b) Let y ( x) x. Find y (4). (c) lim r 0 e /r ln(r) + (a) (9 pt) g (x) 3 cos 2 (π 2x)( sin(π 2x))(
Getting Ready to Teach Online course Core Pure
Getting Ready to Teach Online course Core Pure GCE Further Mathematics (2017) Poll 1 Which boards do you have experience of teaching Further Maths with? GCE Further Mathematics (2017) Poll 2 Which Edexcel
Your signature: (1) (Pre-calculus Review Set Problems 80 and 124.)
(1) (Pre-calculus Review Set Problems 80 an 14.) (a) Determine if each of the following statements is True or False. If it is true, explain why. If it is false, give a counterexample. (i) If a an b are
Section 6.1: Composite Functions
Section 6.1: Composite Functions Def: Given two function f and g, the composite function, which we denote by f g and read as f composed with g, is defined by (f g)(x) = f(g(x)). In other words, the function
Solutions to Tutorial for Week 4
The University of Sydney School of Mathematics and Statistics Solutions to Tutorial for Week 4 MATH191/1931: Calculus of One Variable (Advanced) Semester 1, 018 Web Page: sydneyeduau/science/maths/u/ug/jm/math191/
Welcome to Math 104. D. DeTurck. January 16, University of Pennsylvania. D. DeTurck Math A: Welcome 1 / 44
Welcome to Math 104 D. DeTurck University of Pennsylvania January 16, 2018 D. DeTurck Math 104 002 2018A: Welcome 1 / 44 Welcome to the course Math 104 Calculus I Topics: Quick review of Math 103 topics,
Complex Trigonometric and Hyperbolic Functions (7A)
Complex Trigonometric and Hyperbolic Functions (7A) 07/08/015 Copyright (c) 011-015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
CALCULUS PROBLEMS Courtesy of Prof. Julia Yeomans. Michaelmas Term
CALCULUS PROBLEMS Courtesy of Prof. Julia Yeomans Michaelmas Term The problems are in 5 sections. The first 4, A Differentiation, B Integration, C Series and limits, and D Partial differentiation follow
CALCULUS II MATH Dr. Hyunju Ban
CALCULUS II MATH 2414 Dr. Hyunju Ban Introduction Syllabus Chapter 5.1 5.4 Chapters To Be Covered: Chap 5: Logarithmic, Exponential, and Other Transcendental Functions (2 week) Chap 7: Applications of
MATH 312 Section 7.1: Definition of a Laplace Transform
MATH 312 Section 7.1: Definition of a Laplace Transform Prof. Jonathan Duncan Walla Walla University Spring Quarter, 2008 Outline 1 The Laplace Transform 2 The Theory of Laplace Transforms 3 Conclusions
Transcendental Functions
9 Transcendental Functions º½ ÁÒÚ Ö ÙÒØ ÓÒ Informally, two functions f and g are inverses if each reverses, or undoes, the other More precisely: DEFINITION 9 Two functions f and g are inverses if for all
PRELIM 2 REVIEW QUESTIONS Math 1910 Section 205/209
PRELIM 2 REVIEW QUESTIONS Math 9 Section 25/29 () Calculate the following integrals. (a) (b) x 2 dx SOLUTION: This is just the area under a semicircle of radius, so π/2. sin 2 (x) cos (x) dx SOLUTION:
Hyperbolic Geometry Solutions
Hyperbolic Geometry Solutions LAMC November 11, 2018 Problem 1.3. b) By Power of a Point (Problem 1.1) we have ON OM = OA 2 = r 2, so each orthogonal circle is fixed under inversion. c) Lines through O,
MATHEMATICAL FORMULAS AND INTEGRALS
MATHEMATICAL FORMULAS AND INTEGRALS ALAN JEFFREY Department of Engineering Mathematics University of Newcastle upon Tyne Newcastle upon Tyne United Kingdom Academic Press San Diego New York Boston London
Practice Differentiation Math 120 Calculus I Fall 2015
. x. Hint.. (4x 9) 4x + 9. Hint. Practice Differentiation Math 0 Calculus I Fall 0 The rules of differentiation are straightforward, but knowing when to use them and in what order takes practice. Although
Exponential and Logarithmic Functions
Lesson 6 Eponential and Logarithmic Fu tions Lesson 6 Eponential and Logarithmic Functions Eponential functions are of the form y = a where a is a constant greater than zero and not equal to one and is
Tuesday, Feb 12. These slides will cover the following. [cos(x)] = sin(x) 1 d. 2 higher-order derivatives. 3 tangent line problems
![Tuesday, Feb 12. These slides will cover the following. [cos(x)] = sin(x) 1 d. 2 higher-order derivatives. 3 tangent line problems](/thumbs/96/127321199.jpg "Tuesday, Feb 12. These slides will cover the following. [cos(x)] = sin(x) 1 d. 2 higher-order derivatives. 3 tangent line problems")
Tuesday, Feb 12 These slides will cover the following. 1 d dx [cos(x)] = sin(x) 2 higher-order derivatives 3 tangent line problems 4 basic differential equations Proof First we will go over the following
Math 180 Chapter 4 Lecture Notes. Professor Miguel Ornelas
Math 80 Chapter 4 Lecture Notes Professor Miguel Ornelas M. Ornelas Math 80 Lecture Notes Section 4. Section 4. Inverse Functions Definition of One-to-One Function A function f with domain D and range
MATHEMATICAL FORMULAS AND INTEGRALS
HANDBOOK OF MATHEMATICAL FORMULAS AND INTEGRALS Second Edition ALAN JEFFREY Department of Engineering Mathematics University of Newcastle upon Tyne Newcastle upon Tyne United Kingdom ACADEMIC PRESS A Harcourt
Exponential and Logarithmic Functions. Copyright Cengage Learning. All rights reserved.
3 Exponential and Logarithmic Functions Copyright Cengage Learning. All rights reserved. 3.2 Logarithmic Functions and Their Graphs Copyright Cengage Learning. All rights reserved. What You Should Learn
cosh 2 x sinh 2 x = 1 sin 2 x = 1 2 cos 2 x = 1 2 dx = dt r 2 = x 2 + y 2 L =
Integrals Volume: Suppose A(x) is the cross-sectional area of the solid S perpendicular to the x-axis, then the volume of S is given by V = b a A(x) dx Work: Suppose f(x) is a force function. The work
HOMEWORK SET 4 SOLUTIONS. 1. Problem (8ex each) Prove or disprove the following identities. (a)2 3 n n = n
HOMEWORK SET 4 SOLUTIONS MATH 1113 1. Problem (8ex each) Prove or disprove the following identities. (a)2 3 n+1 2 3 n = 2 2 3 n (b)4 n+1 5 n 4 n 5 n+1 = 4 n 5 n (c)(e t ) 2 = e 2t (e) em+t e = e 2t m t
10.7. DIFFERENTIATION 7 (Inverse hyperbolic functions) A.J.Hobson
JUST THE MATHS SLIDES NUMBER 0.7 DIFFERENTIATION 7 (Inverse hyperbolic functions) by A.J.Hobson 0.7. Summary of results 0.7.2 The erivative of an inverse hyperbolic sine 0.7.3 The erivative of an inverse
Practice Test - Chapter 3
Sketch and analyze the graph of each function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. 1. f (x) = e x + 7 Evaluate the function
AP CALCULUS AB Study Guide for Midterm Exam 2017
AP CALCULUS AB Study Guide for Midterm Exam 2017 CHAPTER 1: PRECALCULUS REVIEW 1.1 Real Numbers, Functions and Graphs - Write absolute value as a piece-wise function - Write and interpret open and closed
FLC Ch 9. Ex 2 Graph each function. Label at least 3 points and include any pertinent information (e.g. asymptotes). a) (# 14) b) (# 18) c) (# 24)
Math 5 Trigonometry Sec 9.: Exponential Functions Properties of Exponents a = b > 0, b the following statements are true: b x is a unique real number for all real numbers x f(x) = b x is a function with
Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function.
Test Instructions Objectives Section 5.1 Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function. Form a polynomial whose zeros and degree are given. Graph
Use a graphing utility to approximate the real solutions, if any, of the equation rounded to two decimal places. 4) x3-6x + 3 = 0 (-5,5) 4)
Advanced College Prep Pre-Calculus Midyear Exam Review Name Date Per List the intercepts for the graph of the equation. 1) x2 + y - 81 = 0 1) Graph the equation by plotting points. 2) y = -x2 + 9 2) List
Math 132 Information for Test 2
Math 13 Information for Test Test will cover material from Sections 5.6, 5.7, 5.8, 6.1, 6., 6.3, 7.1, 7., and 7.3. The use of graphing calculators will not be allowed on the test. Some practice questions
Further Mathematics SAMPLE. Marking Scheme
Further Mathematics SAMPLE Marking Scheme This marking scheme has been prepared as a guide only to markers. This is not a set of model answers, or the exclusive answers to the questions, and there will
5.4 Variation of Parameters
202 5.4 Variation of Parameters The method of variation of parameters applies to solve (1) a(x)y + b(x)y + c(x)y = f(x). Continuity of a, b, c and f is assumed, plus a(x) 0. The method is important because
MTH 133 Solutions to Exam 1 October 11, Without fully opening the exam, check that you have pages 1 through 11.
MTH 33 Solutions to Exam October, 7 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show
Further Pure Mathematics 3 GCE Further Mathematics GCE Pure Mathematics and Further Mathematics (Additional) A2 optional unit
Unit FP3 Further Pure Mathematics 3 GCE Further Mathematics GCE Pure Mathematics and Further Mathematics (Additional) A optional unit FP3.1 Unit description Further matrix algebra; vectors, hyperbolic
Hyperbolic Functions and the Twin Paradox
Hyperbolic Functions and the Twin Paradox Basics of hyperbolic functions The basic hyperbolic functions are defined as cosh x 1 2 (ex + e x ), sinh 1 2 (ex e x ). The notation ch, sh is also used (especially
ECM Calculus and Geometry. Revision Notes
ECM1702 - Calculus and Geometry Revision Notes Joshua Byrne Autumn 2011 Contents 1 The Real Numbers 1 1.1 Notation.................................................. 1 1.2 Set Notation...............................................
2.12: 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
Rewrite logarithmic equations 2 3 = = = 12
EXAMPLE 1 Rewrite logarithmic equations Logarithmic Form a. log 2 8 = 3 Exponential Form 2 3 = 8 b. log 4 1 = 0 4 0 = 1 log 12 = 1 c. 12 12 1 = 12 log 4 = 1 d. 1/4 1 4 1 = 4 GUIDED PRACTICE for Example
( ) = 1 t + t. ( ) = 1 cos x + x ( sin x). Evaluate y. MTH 111 Test 1 Spring Name Calculus I
MTH Test Spring 209 Name Calculus I Justify all answers by showing your work or by proviing a coherent eplanation. Please circle your answers.. 4 z z + 6 z 3 ez 2 = 4 z + 2 2 z2 2ez Rewrite as 4 z + 6
Mock Final Exam Name. Solve and check the linear equation. 1) (-8x + 8) + 1 = -7(x + 3) A) {- 30} B) {- 6} C) {30} D) {- 28}
Mock Final Exam Name Solve and check the linear equation. 1) (-8x + 8) + 1 = -7(x + 3) 1) A) {- 30} B) {- 6} C) {30} D) {- 28} First, write the value(s) that make the denominator(s) zero. Then solve the
MAT137 Calculus! Lecture 20
official website http://uoft.me/mat137 MAT137 Calculus! Lecture 20 Today: 4.6 Concavity 4.7 Asypmtotes Next: 4.8 Curve Sketching Indeterminate Forms for Limits Which of the following are indeterminate