Differentiation Shortcuts
|
|
- Madison Nash
- 6 years ago
- Views:
Transcription
1 Differentiation Shortcuts Sections 10-5, 11-2, 11-3, and 11-4 Prof. Nathan Wodarz Math Fall 2008 Contents 1 Basic Properties Notation Constant Functions Power Rule Sum and Difference Properties Constant Multiple Property Basic Property Practice Exponential and Logarithmic Functions Derivative of e x Derivative of ln x Other Exponential and Logarithmic Functions Derivatives of Products and Quotients The Product Rule The Quotient Rule The Chain Rule Composite Functions General Power Rule Chain Rule For Exponential Functions Chain Rule For Logarithmic Functions General Chain Rule
2 1 Basic Properties 1.1 Notation Notation The following are all notations to represent the derivative of y = f (x) f (x) y dy dx 1.2 Constant Functions Constant Functions If y = f (x) = C, then f (x) = 0 Also write as y = 0 dy/dx = 0 C = 0 d dx C = 0 Constant Functions Problem 1. Find y if y = 5 8 A. 0 B. 5 8 C. 1 D. 5 8 x 2
3 1.3 Power Rule Power Rule If y = f (x) = x n (n a real number), then f (x) = nx n 1 Also write as y = nx n 1 dy/dx = nx n 1 Power Rule Problem 2. Find y if y = x 6 A. x 5 B. 6x 4 C. 6x 5 D. 6x Sum and Difference Properties Sum and Difference Properties If y = f (x) = u(x) ± v(x), then f (x) = u (x) ± v (x) Also write as y = u ± v dy dx = du dx + dv dx Use the same sign in the derivative as in the original function 3
4 Sum and Difference Properties Problem 3. Find f (x) if f (x) = x 2 + x 3 + x A. 2x 3 + 3x 2 B. 2x 3 + 3x C. 2x 1 + 3x D. 2x 1 + 3x Constant Multiple Property Constant Multiple Property If y = f (x) = ku(x), then f (x) = ku (x) Also write as y = ku dy = k du dx dx Constant Multiple Property Problem 4. Let f and g be functions with f (4) = 2 and g (4) = 3. Find h (4) for h(x) = 3 f (x) g(x) + 2 A. 2 B. 5 C. 9 D. 11 4
5 1.6 Basic Property Practice Basic Property Practice Problem 5. Find d dv (6v0.7 v 5.8 ) A. 4.2v v 4.7 B. 4.2v v 4.7 C. 4.2v v 4.8 D. 4.2v v 4.8 Basic Property Practice Problem 6. Find f (x) if f (x) = 3π 6 A. 0 B. 3 C. 3π 5 D. 18π 5 Basic Property Practice Problem 7. Find y if y = 4 x x A. 1 x x 4/3 B. 1 4x x 2/3 C. 1 4 x 5 15x 2/3 D. 16x x 2/3 5
6 Basic Property Practice Problem 8. Find an equation of the tangent line at x = 2 for f (x) = 4+x 2x 2 3x 3 A. y = 47x + 68 B. y = 43x + 48 C. y = 43x + 60 D. y = 39x Exponential and Logarithmic Functions 2.1 Derivative of e x Derivative of e x If y = f (x) = e x, then f (x) = e x WARNING! The derivative is not xe x 1 Derivative of e x Problem 9. Find the derivative of f (x) = 5e x 4x 8 and simplify. A. f (x) = 5e x 32x 7 B. f (x) = 5e x 32x 8 C. f (x) = 5e x 4x 8 D. f (x) = 5e 5x 32x 7 6
7 2.2 Derivative of ln x Derivative of ln x If y = f (x) = ln x, then f (x) = 1 x Can use earlier properties to take more complicated derivatives, such as Derivative of ln x d dx ln xn = d dx n ln x = n x Problem 10. Find y for y = ln 6x 2 A. y = 2 x B. y = 12 x C. y = 2x x 2 +6 D. y = 1 2x Other Exponential and Logarithmic Functions Other Exponential Functions If b > 0, b 1: d dx bx = b x ln b Other Logarithmic Functions If b > 0, b 1: d dx log b x = 1 ln b ( 1 x) 7
8 Other Exponential and Logarithmic Functions Problem 11. Find f (x) for f (x) = 8 x + 4 log 3 (x 2 ) A. f (x) = 8 x + 8 x B. f (x) = 8 x + 8 x ln 3 C. f (x) = 8 x ln x ln 3 D. f (x) = 8(7 x ) + 4 x 2 3 Derivatives of Products and Quotients 3.1 The Product Rule The Product Rule If y = f (x) = F(x)S (x) where F and S are differentiable, then f (x) = F(x)S (x) + S (x)f (x) Also written as: y = FS + S F The Product Rule Problem 12. Find f (x) for f (x) = (5x 3 + 4)(3x 7 5) A. f (x) = 20x x 6 75x B. f (x) = 20x x 6 75x 2 C. f (x) = 150x x 6 75x D. f (x) = 150x x 6 75x 2 8
9 Constant Multiple Property Problem 13. Let f and g be functions that satisfy f (2) = 1, g(2) = 3, f (2) = 2, and g (2) = 3. Find h (2) for h(x) = f (x)g(x) 2 f (x) + 7 A. 6 B. 5 C. 5 D The Quotient Rule The Quotient Rule If y = f (x) = T(x) where T and B are differentiable, then B(x) Also written as: y = BT T B B 2 The Quotient Rule Problem 14. Find f (t) for f (t) = A. 5 7t 5 B. 5 (7t 5) 2 C. 5t (7t 5) 2 D. 14t 5 (7t 5) 2 f (x) = B(x)T (x) T(x)B (x) [B(x)] 2 t 7t 5 9
10 The Quotient Rule Problem 15. Find f (2) if f (x) = 2x 7 3x 2 A B C D The Quotient Rule Problem 16. Acme Corporation Publishing House has started publishing a new magazine for college students. The monthly sales S (in thousands) are given by S (t) = 800t where t is the number of months since the first issue was published. t+2 Find S (3) and S (3) and interpret the results. A. At three months, monthly sales are 480,000 and decreasing at 64,000 magazines per month B. At three months, monthly sales are 480,000 and increasing at 64,000 magazines per month C. At three months, monthly sales are 2,400,000 and increasing at 64,000 magazines per month D. At three months, monthly sales are 2,400,000 and increasing at 800,000 magazines per month 4 The Chain Rule 4.1 Composite Functions Composite Functions A function like f (x) = 1 + x 2 can be built by chaining two simpler functions together 10
11 Take y = f (u) = u and u = g(x) = 1 + x 2 The composite of f and g expresses y as a function of x: y = f (u) = f (1 + x 2 ) The composite of functions f and g is the function m(x) = f [g(x)] Composite Functions Problem 17. Find the composition f [g(x)] if f (u) = u 5 and g(x) = 2 3x 2 A. (2 3x 2 ) 2 B. (2 3x 2 ) 5 C. (10 15x 2 ) 5 D. 2 3u 10 Composite Functions Problem 18. Choose the answer choice that includes all pairs of functions from 1 the list so that the composite function h(x) = can be written in the form 11 x 2 h(x) = f [g(x)] A. f (x) = 1 x and g(x) = 11 x 2 B. f (x) = 11 x and g(x) = 1 x 2 C. f (x) = 1 11 x and g(x) = x 2 D. Both A and C 11
12 4.2 General Power Rule General Power Rule If u(x) is a differentiable function,n is any real number, and y = f (x) = [u(x)] n, then f (x) = n[u(x)] n 1 u (x) General Power Rule Problem 19. Find d 4 dω (ω 2 + 3) 5 A. 40 (ω 2 +3) 6 B. 40x (ω 2 +3) 6 C. 40 (ω 2 +3) 5 D. 40 (ω 2 +3) 6 General Power Rule Problem 20. Find dy dx if y = 8 8x7 10 A x7 10 B. 448x 6 7 8x7 10 C. 7x 6 (8x 7 10) 7/8 D. 56x 6 (8x 7 10) 7/8 4.3 Chain Rule For Exponential Functions Chain Rule For Exponential Functions If u(x) is a differentiable function and y = f (x) = e u(x), then f (x) = e u(x) u (x) 12
13 General Power Rule Problem 21. Find f (x) if f (x) = 3 x 1 A. 3 ln(3) B. 3 x 1 ln x C. 3 x 1 ln 3 D. 3 x 1 ln(3 x 1 ) 4.4 Chain Rule For Logarithmic Functions Chain Rule For Logarithmic Functions If u(x) is a differentiable function and y = f (x) = ln[u(x)], then f (x) = u (x) u(x) General Power Rule Problem 22. Find y if y = log 7 (x 6 + 1) A. 6x 5 (ln 7)(x 6 +1) B. 6x 5 (ln 7) x 6 +1 C. 6x 5 x 6 +1 D. 1 (ln 7)(x 6 +1) + 6x5 4.5 General Chain Rule General Chain Rule If y = f (u) and u = g(x) give a composite function y = m(x) = f [g(x)], then dy = dy du, if both derivatives on the right exist dx du dx m (x) = f [g(x)]g (x), if both derivatives on the right exist 13
14 General Chain Rule Problem 23. Find all values of x where the tangent line to f (x) = horizontal. x (x 2 + 3) 3 is A. x = ± 15 5 B. x = 0, ± 15 5 C. x = ± 3 5 D. x = 0 Summary Summary You should be able to: Use basic properties to compute simple derivatives Find the derivatives of exponential and logarithmic functions Use the product and quotient rules Use the chain rule to differentiate composite functions 14
Big Picture I. MATH 1003 Review: Part 3. The Derivatives of Functions. Big Picture I. Introduction to Derivatives
Big Picture I MATH 1003 Review: Part 3. The Derivatives of Functions Maosheng Xiong Department of Mathematics, HKUST What would the following questions remind you? 1. Concepts: limit, one-sided limit,
More informationBig Picture I. MATH 1003 Review: Part 3. The Derivatives of Functions. Big Picture I. Introduction to Derivatives
Big Picture I MATH 1003 Review: Part 3. The Derivatives of Functions Maosheng Xiong Department of Mathematics, HKUST What would the following questions remind you? 1. Concepts: limit, one-sided limit,
More informationMath 116: Business Calculus Chapter 4 - Calculating Derivatives
Math 116: Business Calculus Chapter 4 - Calculating Derivatives Instructor: Colin Clark Spring 2017 Exam 2 - Thursday March 9. 4.1 Techniques for Finding Derivatives. 4.2 Derivatives of Products and Quotients.
More informationChapter 4 Notes, Calculus I with Precalculus 3e Larson/Edwards
4.1 The Derivative Recall: For the slope of a line we need two points (x 1,y 1 ) and (x 2,y 2 ). Then the slope is given by the formula: m = y x = y 2 y 1 x 2 x 1 On a curve we can find the slope of a
More informationPRE-LEAVING CERTIFICATE EXAMINATION, 2010
L.7 PRE-LEAVING CERTIFICATE EXAMINATION, 00 MATHEMATICS HIGHER LEVEL PAPER (300 marks) TIME : ½ HOURS Attempt SIX QUESTIONS (50 marks each). WARNING: Marks will be lost if all necessary work is not clearly
More information3.9 Derivatives of Exponential and Logarithmic Functions
322 Chapter 3 Derivatives 3.9 Derivatives of Exponential and Logarithmic Functions Learning Objectives 3.9.1 Find the derivative of exponential functions. 3.9.2 Find the derivative of logarithmic functions.
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES 3. The Product and Quotient Rules In this section, we will learn about: Formulas that enable us to differentiate new functions formed from old functions by
More informationSection 11.3 Rates of Change:
Section 11.3 Rates of Change: 1. Consider the following table, which describes a driver making a 168-mile trip from Cleveland to Columbus, Ohio in 3 hours. t Time (in hours) 0 0.5 1 1.5 2 2.5 3 f(t) Distance
More informationCALCULUS. Berkant Ustaoğlu CRYPTOLOUNGE.NET
CALCULUS Berkant Ustaoğlu CRYPTOLOUNGE.NET Secant 1 Definition Let f be defined over an interval I containing u. If x u and x I then f (x) f (u) Q = x u is the difference quotient. Also if h 0, such that
More informationChapter 2: Differentiation
Chapter 2: Differentiation Winter 2016 Department of Mathematics Hong Kong Baptist University 1 / 75 2.1 Tangent Lines and Their Slopes This section deals with the problem of finding a straight line L
More informationCalculating the Derivative Using Derivative Rules Implicit Functions Higher-Order Derivatives
Topic 4 Outline 1 Derivative Rules Calculating the Derivative Using Derivative Rules Implicit Functions Higher-Order Derivatives D. Kalajdzievska (University of Manitoba) Math 1500 Fall 2015 1 / 32 Topic
More informationChapter 4. Section Derivatives of Exponential and Logarithmic Functions
Chapter 4 Section 4.2 - Derivatives of Exponential and Logarithmic Functions Objectives: The student will be able to calculate the derivative of e x and of lnx. The student will be able to compute the
More informationChapter 2: Differentiation
Chapter 2: Differentiation Spring 2018 Department of Mathematics Hong Kong Baptist University 1 / 82 2.1 Tangent Lines and Their Slopes This section deals with the problem of finding a straight line L
More informationMath 1071 Final Review Sheet The following are some review questions to help you study. They do not
Math 1071 Final Review Sheet The following are some review questions to help you study. They do not They do The exam represent the entirety of what you could be expected to know on the exam; reflect distribution
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 informationMath 1501 Calc I Summer 2015 QUP SOUP w/ GTcourses
Math 1501 Calc I Summer 2015 QUP SOUP w/ GTcourses Instructor: Sal Barone School of Mathematics Georgia Tech May 22, 2015 (updated May 22, 2015) Covered sections: 3.3 & 3.5 Exam 1 (Ch.1 - Ch.3) Thursday,
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 informationTHE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK. Summer Examination 2009.
OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK Summer Examination 2009 First Engineering MA008 Calculus and Linear Algebra
More informationMath Practice Exam 3 - solutions
Math 181 - Practice Exam 3 - solutions Problem 1 Consider the function h(x) = (9x 2 33x 25)e 3x+1. a) Find h (x). b) Find all values of x where h (x) is zero ( critical values ). c) Using the sign pattern
More informationDifferentiation Review, Part 1 (Part 2 follows; there are answers at the end of each part.)
Differentiation Review 1 Name Differentiation Review, Part 1 (Part 2 follows; there are answers at the end of each part.) Derivatives Review: Summary of Rules Each derivative rule is summarized for you
More information1 + x 2 d dx (sec 1 x) =
Page This exam has: 8 multiple choice questions worth 4 points each. hand graded questions worth 4 points each. Important: No graphing calculators! Any non-graphing, non-differentiating, non-integrating
More informationMath 142 Week-in-Review #4 (Sections , 4.1, and 4.2)
Math 142 WIR, copyright Angie Allen, Fall 2018 1 Math 142 Week-in-Review #4 (Sections 3.1-3.3, 4.1, and 4.2) Note: This collection of questions is intended to be a brief overview of the exam material (with
More informationLimit. Chapter Introduction
Chapter 9 Limit Limit is the foundation of calculus that it is so useful to understand more complicating chapters of calculus. Besides, Mathematics has black hole scenarios (dividing by zero, going to
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 informationy+2 x 1 is in the range. We solve x as x =
Dear Students, Here are sample solutions. The most fascinating thing about mathematics is that you can solve the same problem in many different ways. The correct answer will always be the same. Be creative
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 informationCalculus I: Practice Midterm II
Calculus I: Practice Mierm II April 3, 2015 Name: Write your solutions in the space provided. Continue on the back for more space. Show your work unless asked otherwise. Partial credit will be given for
More information3.4 The Chain Rule. F (x) = f (g(x))g (x) Alternate way of thinking about it: If y = f(u) and u = g(x) where both are differentiable functions, then
3.4 The Chain Rule To find the derivative of a function that is the composition of two functions for which we already know the derivatives, we can use the Chain Rule. The Chain Rule: Suppose F (x) = f(g(x)).
More informationMath 180 Written Homework Solutions Assignment #4 Due Tuesday, September 23rd at the beginning of your discussion class.
Math 180 Written Homework Solutions Assignment #4 Due Tuesday, September 23rd at the beginning of your discussion class. Directions. You are welcome to work on the following problems with other MATH 180
More informationChapter 3 Differentiation Rules
Chapter 3 Differentiation Rules Derivative constant function if c is any real number, then Example: The Power Rule: If n is a positive integer, then Example: Extended Power Rule: If r is any real number,
More information1. Compute the derivatives of the following functions, by any means necessary. f (x) = (1 x3 )(1/2)(x 2 1) 1/2 (2x) x 2 1( 3x 2 ) (1 x 3 ) 2
Math 51 Exam Nov. 4, 009 SOLUTIONS Directions 1. SHOW YOUR WORK and be thorough in your solutions. Partial credit will only be given for work shown.. Any numerical answers should be left in exact form,
More informationMAT137 Calculus! Lecture 6
MAT137 Calculus! Lecture 6 Today: 3.2 Differentiation Rules; 3.3 Derivatives of higher order. 3.4 Related rates 3.5 Chain Rule 3.6 Derivative of Trig. Functions Next: 3.7 Implicit Differentiation 4.10
More informationMarch 5, 2009 Name The problems count as marked. The total number of points available is 131. Throughout this test, show your work.
March 5, 2009 Name The problems count as marked. The total number of points available is 131. Throughout this test, show your work. 1. (12 points) Consider the cubic curve f(x) = 2x 3 + 3x + 2. (a) What
More informationMath 106 Answers to Exam 1a Fall 2015
Math 06 Answers to Exam a Fall 05.. Find the derivative of the following functions. Do not simplify your answers. (a) f(x) = ex cos x x + (b) g(z) = [ sin(z ) + e z] 5 Using the quotient rule on f(x) and
More information11.6: Ratio and Root Tests Page 1. absolutely convergent, conditionally convergent, or divergent?
.6: Ratio and Root Tests Page Questions ( 3) n n 3 ( 3) n ( ) n 5 + n ( ) n e n ( ) n+ n2 2 n Example Show that ( ) n n ln n ( n 2 ) n + 2n 2 + converges for all x. Deduce that = 0 for all x. Solutions
More informationMATH 425, FINAL EXAM SOLUTIONS
MATH 425, FINAL EXAM SOLUTIONS Each exercise is worth 50 points. Exercise. a The operator L is defined on smooth functions of (x, y by: Is the operator L linear? Prove your answer. L (u := arctan(xy u
More informationDifferentiation 2. The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996.
Differentiation 2 The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996. 1 Launch Mathematica. Type
More informationIntegration - Past Edexcel Exam Questions
Integration - Past Edexcel Exam Questions 1. (a) Given that y = 5x 2 + 7x + 3, find i. - ii. - (b) ( 1 + 3 ) x 1 x dx. [4] 2. Question 2b - January 2005 2. The gradient of the curve C is given by The point
More information2.1 The derivative. Rates of change. m sec = y f (a + h) f (a)
2.1 The derivative Rates of change 1 The slope of a secant line is m sec = y f (b) f (a) = x b a and represents the average rate of change over [a, b]. Letting b = a + h, we can express the slope of the
More informationQuiz 4A Solutions. Math 150 (62493) Spring Name: Instructor: C. Panza
Math 150 (62493) Spring 2019 Quiz 4A Solutions Instructor: C. Panza Quiz 4A Solutions: (20 points) Neatly show your work in the space provided, clearly mark and label your answers. Show proper equality,
More informationMath 131 Exam 2 Spring 2016
Math 3 Exam Spring 06 Name: ID: 7 multiple choice questions worth 4.7 points each. hand graded questions worth 0 points each. 0. free points (so the total will be 00). Exam covers sections.7 through 3.0
More informationCore Mathematics 3 Differentiation
http://kumarmaths.weebly.com/ Core Mathematics Differentiation C differentiation Page Differentiation C Specifications. By the end of this unit you should be able to : Use chain rule to find the derivative
More informationMathematics for Economics ECON MA/MSSc in Economics-2017/2018. Dr. W. M. Semasinghe Senior Lecturer Department of Economics
Mathematics for Economics ECON 53035 MA/MSSc in Economics-2017/2018 Dr. W. M. Semasinghe Senior Lecturer Department of Economics MATHEMATICS AND STATISTICS LERNING OUTCOMES: By the end of this course unit
More informationTo take the derivative of x raised to a power, you multiply in front by the exponent and subtract 1 from the exponent.
MA123, Chapter 5: Formulas for derivatives (pp. 83-102) Date: Chapter Goals: Know and be able to apply the formulas for derivatives. Understand the chain rule and be able to apply it. Know how to compute
More informationCalculus Overview. f(x) f (x) is slope. I. Single Variable. A. First Order Derivative : Concept : measures slope of curve at a point.
Calculus Overview I. Single Variable A. First Order Derivative : Concept : measures slope of curve at a point. Notation : Let y = f (x). First derivative denoted f ʹ (x), df dx, dy dx, f, etc. Example
More information1 Review of di erential calculus
Review of di erential calculus This chapter presents the main elements of di erential calculus needed in probability theory. Often, students taking a course on probability theory have problems with concepts
More informationand verify that it satisfies the differential equation:
MOTIVATION: Chapter One: Basic and Review Why study differential equations? Suppose we know how a certain quantity changes with time (for example, the temperature of coffee in a cup, the number of people
More informationFebruary 21 Math 1190 sec. 63 Spring 2017
February 21 Math 1190 sec. 63 Spring 2017 Chapter 2: Derivatives Let s recall the efinitions an erivative rules we have so far: Let s assume that y = f (x) is a function with c in it s omain. The erivative
More informationCampus Academic Resource Program Chain Rule
This handout will: Provide a strategy to identify composite functions Provide a strategy to find chain rule by using a substitution method. Identifying Composite Functions This section will provide a strategy
More informationSection 3.6 The chain rule 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 3.6 The chain rule 1 Lecture College of Science MATHS 101: Calculus I (University of Bahrain) Logarithmic Differentiation 1 / 23 Motivation Goal: We want to derive rules to find the derivative
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES We have: Seen how to interpret derivatives as slopes and rates of change Seen how to estimate derivatives of functions given by tables of values Learned how
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 informationMath 222 Spring 2013 Exam 3 Review Problem Answers
. (a) By the Chain ule, Math Spring 3 Exam 3 eview Problem Answers w s w x x s + w y y s (y xy)() + (xy x )( ) (( s + 4t) (s 3t)( s + 4t)) ((s 3t)( s + 4t) (s 3t) ) 8s 94st + 3t (b) By the Chain ule, w
More informationChapter 1. Functions, Graphs, and Limits
Review for Final Exam Lecturer: Sangwook Kim Office : Science & Tech I, 226D math.gmu.eu/ skim22 Chapter 1. Functions, Graphs, an Limits A function is a rule that assigns to each objects in a set A exactly
More informationMath Camp II. Calculus. Yiqing Xu. August 27, 2014 MIT
Math Camp II Calculus Yiqing Xu MIT August 27, 2014 1 Sequence and Limit 2 Derivatives 3 OLS Asymptotics 4 Integrals Sequence Definition A sequence {y n } = {y 1, y 2, y 3,..., y n } is an ordered set
More informationFunctions. A function is a rule that gives exactly one output number to each input number.
Functions A function is a rule that gives exactly one output number to each input number. Why it is important to us? The set of all input numbers to which the rule applies is called the domain of the function.
More informationSection 3.6 The chain rule 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 3.6 The chain rule 1 Lecture College of Science MATHS 101: Calculus I (University of Bahrain) Logarithmic Differentiation 1 / 1 Motivation Goal: We want to derive rules to find the derivative of
More informationFind the indicated derivative. 1) Find y(4) if y = 3 sin x. A) y(4) = 3 cos x B) y(4) = 3 sin x C) y(4) = - 3 cos x D) y(4) = - 3 sin x
Assignment 5 Name Find the indicated derivative. ) Find y(4) if y = sin x. ) A) y(4) = cos x B) y(4) = sin x y(4) = - cos x y(4) = - sin x ) y = (csc x + cot x)(csc x - cot x) ) A) y = 0 B) y = y = - csc
More informationLogarithmic 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
More informationDRAFT - Math 101 Lecture Note - Dr. Said Algarni
3 Differentiation Rules 3.1 The Derivative of Polynomial and Exponential Functions In this section we learn how to differentiate constant functions, power functions, polynomials, and exponential functions.
More informationMath 106 Answers to Test #1 11 Feb 08
Math 06 Answers to Test # Feb 08.. A projectile is launched vertically. Its height above the ground is given by y = 9t 6t, where y is the height in feet and t is the time since the launch, in seconds.
More information, find the value(s) of a and b which make f differentiable at bx 2 + x if x 2 x = 2 or explain why no such values exist.
Math 171 Exam II Summary Sheet and Sample Stuff (NOTE: The questions posed here are not necessarily a guarantee of the type of questions which will be on Exam II. This is a sampling of questions I have
More informationMA 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
More informationb n x n + b n 1 x n b 1 x + b 0
Math Partial Fractions Stewart 7.4 Integrating basic rational functions. For a function f(x), we have examined several algebraic methods for finding its indefinite integral (antiderivative) F (x) = f(x)
More informationMATH 236 ELAC FALL 2017 CA 9 NAME: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
MATH 236 ELAC FALL 207 CA 9 NAME: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. ) 27 p 3 27 p 3 ) 2) If 9 t 3 4t 9-2t = 3, find t. 2) Solve the equation.
More informationMath 141: Lecture 11
Math 141: Lecture 11 The Fundamental Theorem of Calculus and integration methods Bob Hough October 12, 2016 Bob Hough Math 141: Lecture 11 October 12, 2016 1 / 36 First Fundamental Theorem of Calculus
More informationSection 11.7 The Chain Rule
Section.7 The Chain Rule Composition of Functions There is another way of combining two functions to obtain a new function. For example, suppose that y = fu) = u and u = gx) = x 2 +. Since y is a function
More informationLecture 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
More informationDifferentiation. Timur Musin. October 10, University of Freiburg 1 / 54
Timur Musin University of Freiburg October 10, 2014 1 / 54 1 Limit of a Function 2 2 / 54 Literature A. C. Chiang and K. Wainwright, Fundamental methods of mathematical economics, Irwin/McGraw-Hill, Boston,
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 informationSkill 6 Exponential and Logarithmic Functions
Skill 6 Exponential and Logarithmic Functions Skill 6a: Graphs of Exponential Functions Skill 6b: Solving Exponential Equations (not requiring logarithms) Skill 6c: Definition of Logarithms Skill 6d: Graphs
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 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 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 informationMath 2a Prac Lectures on Differential Equations
Math 2a Prac Lectures on Differential Equations Prof. Dinakar Ramakrishnan 272 Sloan, 253-37 Caltech Office Hours: Fridays 4 5 PM Based on notes taken in class by Stephanie Laga, with a few added comments
More informationSection 4.2 Logarithmic Functions & Applications
34 Section 4.2 Logarithmic Functions & Applications Recall that exponential functions are one-to-one since every horizontal line passes through at most one point on the graph of y = b x. So, an exponential
More informationUNIT 2 DERIVATIVES 2.1 EXPONENTIAL AND LOGARITHMIC FUNCTION APPLICATIONS. Pre-Class:
1830 UNIT 2 DERIVATIVES 2.1 EXPONENTIAL AND LOGARITHMIC FUNCTION APPLICATIONS Pre-Class: Take notes on the videos and readings (use the space below). Work and check problem #1 in the 2.1 NOTES section.
More informationMath 180, Exam 2, Practice Fall 2009 Problem 1 Solution. f(x) = arcsin(2x + 1) = sin 1 (3x + 1), lnx
Math 80, Exam, Practice Fall 009 Problem Solution. Differentiate the functions: (do not simplify) f(x) = x ln(x + ), f(x) = xe x f(x) = arcsin(x + ) = sin (3x + ), f(x) = e3x lnx Solution: For the first
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 informationSection 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point
Section 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point Line through P and Q approaches to the tangent line at P as Q approaches P. That is as a + h a = h gets smaller. Slope of the
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 informationx C) y = - A) $20000; 14 years B) $28,000; 14 years C) $28,000; 28 years D) $30,000; 15 years
Dr. Lee - Math 35 - Calculus for Business - Review of 3 - Show Complete Work for Each Problem MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find
More informationAP Calculus Summer Homework
Class: Date: AP Calculus Summer Homework Show your work. Place a circle around your final answer. 1. Use the properties of logarithms to find the exact value of the expression. Do not use a calculator.
More informationMath RE - Calculus II Antiderivatives and the Indefinite Integral Page 1 of 5
Math 201-203-RE - Calculus II Antiderivatives and the Indefinite Integral Page 1 of 5 What is the Antiderivative? In a derivative problem, a function f(x) is given and you find the derivative f (x) using
More 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 informationLeamy Maths Community
Leaving Certificate Examination, 213 Sample paper prepared by Mathematics Project Maths - Phase 2 Paper 1 Higher Level Saturday 18 May Paper written by J.P.F. Charpin and S. King 3 marks http://www.leamymaths.com/
More informationReview of Integration Techniques
A P P E N D I X D Brief Review of Integration Techniques u-substitution The basic idea underlying u-substitution is to perform a simple substitution that converts the intergral into a recognizable form
More informationMath 147 Exam II Practice Problems
Math 147 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture, all homework problems, all lab
More informationTo take the derivative of x raised to a power, you multiply in front by the exponent and subtract 1 from the exponent.
MA123, Chapter 5: Formulas for Derivatives (pp. 83-102, Gootman) Chapter Goals: Know and be able to apply the formulas for derivatives. Understand the chain rule and be able to apply it. Know how to compute
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 I Review Solutions
Calculus I Review Solutions. Compare and contrast the three Value Theorems of the course. When you would typically use each. The three value theorems are the Intermediate, Mean and Extreme value theorems.
More informationA. Evaluate log Evaluate Logarithms
A. Evaluate log 2 16. Evaluate Logarithms Evaluate Logarithms B. Evaluate. C. Evaluate. Evaluate Logarithms D. Evaluate log 17 17. Evaluate Logarithms Evaluate. A. 4 B. 4 C. 2 D. 2 A. Evaluate log 8 512.
More informationChapter 7: Techniques of Integration
Chapter 7: Techniques of Integration MATH 206-01: Calculus II Department of Mathematics University of Louisville last corrected September 14, 2013 1 / 43 Chapter 7: Techniques of Integration 7.1. Integration
More informationMATH 019: Final Review December 3, 2017
Name: MATH 019: Final Review December 3, 2017 1. Given f(x) = x 5, use the first or second derivative test to complete the following (a) Calculate f (x). If using the second derivative test, calculate
More informationComputing Derivatives With Formulas Some More (pages 14-15), Solutions
Computing Derivatives With Formulas Some More pages 14-15), Solutions This worksheet focuses on computing derivatives using the shortcut formulas, including the power rule, product rule, quotient rule,
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 informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Calculus I - Homework Chapter 2 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the graph is the graph of a function. 1) 1)
More information2009 A-level Maths Tutor All Rights Reserved
2 This book is under copyright to A-level Maths Tutor. However, it may be distributed freely provided it is not sold for profit. Contents the derivative formula 3 tangents & normals 7 maxima & minima 10
More informationReview For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: = 0 : homogeneous equation.
Review For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: y y x y2 = 0 : homogeneous equation. x2 v = y dy, y = vx, and x v + x dv dx = v + v2. dx =
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 information