Authors Gregory Hartman, Ph.D. Brian Heinold, Ph.D. Troy Siemers, Ph.D. Dimplekumar Chalishajar, Ph.D. Editor Jennifer Bowen, Ph.D.
|
|
- Gervais Weaver
- 5 years ago
- Views:
Transcription
1 Section 7 Derivatives of Inverse Functions AP E XC I Version 0 Authors Gregory Hartman, PhD Department of Applie Mathema cs Virginia Military Ins tute Brian Heinol, PhD Department of Mathema cs an Computer Science Mount Saint Mary s University Troy Siemers, PhD Department of Applie Mathema cs Virginia Military Ins tute Dimplekumar Chalishajar, PhD Department of Applie Mathema cs Virginia Military Ins tute Eitor Jennifer Bowen, PhD Department of Mathema cs an Computer Science The College of Wooster
2 Copyright 04 Gregory Hartman License to the public uner Crea ve Commons A ribu on-noncommercial 30 Unite States License
3 Contents Preface Table of Contents iii v Limits An Introuc on To Limits Epsilon-Delta Defini on of a Limit 9 3 Fining Limits Analy cally 6 4 One Sie Limits 7 5 Con nuity 34 6 Limits involving infinity 43 Deriva ves 55 Instantaneous Rates of Change: The Deriva ve 55 Interpreta ons of the Deriva ve 69 3 Basic Differen a on Rules 76 4 The Prouct an Quo ent Rules 83 5 The Chain Rule 93 6 Implicit Differen a on 03 7 Deriva ves of Inverse Func ons 4 3 The Graphical Behavior of Func ons 3 Etreme Values 3 The Mean Value Theorem 9 33 Increasing an Decreasing Func ons Concavity an the Secon Deriva ve 4 35 Curve Sketching 49 4 Applica ons of the Deriva ve 57 4 Newton s Metho 57 4 Relate Rates 64
4 43 Op miza on 7 44 Differen als 78 5 Integra on 85 5 An eriva ves an Inefinite Integra on 85 5 The Definite Integral Riemann Sums The Funamental Theorem of Calculus 55 Numerical Integra on 33 6 Techniques of An ifferen a on 47 6 Subs tu on 47 A Solu ons To Selecte Problems A Ine A
5 Chapter Deriva ves 7 Deriva ves of Inverse Func ons 05, 0375 y, , 05 5, Figure 30: A func on f along with its inverse f Note how it oes not ma er which func on we refer to as f; the other is f a, b y b, a Figure 3: Corresponing tangent lines rawn to f an f Recall that a func on y = f is sai to be one to one if it passes the horizontal line test; that is, for two ifferent values an, we o not have f = f In some cases the omain of f must be restricte so that it is one to one For instance, consier f = Clearly, f = f, so f is not one to one on its regular omain, but by restric ng f to 0,, f is one to one Now recall that one to one func ons have inverses That is, if f is one to one, it has an inverse func on, enote by f, such that if fa = b, then f b = a The omain of f is the range of f, an vice-versa For ease of nota on, we set g = f an treat g as a func on of Since fa = b implies gb = a, when we compose f an g we get a nice result: f gb = fa = b In general, f g = an g f = This gives us a convenient way to check if two func ons are inverses of each other: compose them an if the result is, then they are inverses on the appropriate omains When the point a, b lies on the graph of f, the point b, a lies on the graph of g This leas us to iscover that the graph of g is the reflec on of f across the line y = In Figure 30 we see a func on graphe along with its inverse See how the point, 5 lies on one graph, whereas 5, lies on the other Because of this rela onship, whatever we know about f can quickly be transferre into knowlege about g For eample, consier Figure 3 where the tangent line to f at the point a, b is rawn That line has slope f a Through reflec on across y =, we can see that the tangent line to g at the point b, a shoul have slope f a This then tells us that g b = f a Consier: Informa on about f 05, 0375 lies on f Slope of tangent line to f at = 05 is 3/4 Informa on about g = f 0375, 05 lies on g Slope of tangent line to g at = 0375 is 4/3 f 05 = 3/4 g 0375 = 4/3 We have iscovere a rela onship between f an g in a mostly graphical way We can realize this rela onship analy cally as well Let y = g, where again g = f We want to fin y Since y = g, we know that fy = Using the Chain Rule an Implicit Differen a on, take the eriva ve of both sies of Notes: 4
6 7 Deriva ves of Inverse Func ons this last equality fy = f y y = y = f y y = f g This leas us to the following theorem Theorem Deriva ves of Inverse Func ons Let f be ifferen able an one to one on an open interval I, where f 0 for all in I, let J be the range of f on I, let g be the inverse func on of f, an let fa = b for some a in I Then g is a ifferen able func on on J, an in par cular, f b = g b = f an f = g = a f g The results of Theorem are not trivial; the nota on may seem confusing at first Careful consiera on, along with eamples, shoul earn unerstaning In the net eample we apply Theorem to the arcsine func on Eample 73 Fining the eriva ve of an inverse trigonometric func on Let y = arcsin = sin Fin y using Theorem S Aop ng our previously efine nota on, let g = arcsin an f = sin Thus f = cos Applying the theorem, we have g = f g = cosarcsin This last epression is not immeiately illumina ng Drawing a figure will help, as shown in Figure 33 Recall that the sine func on can be viewe as taking in an angle an returning a ra o of sies of a right triangle, specifically, the ra o opposite over hypotenuse This means that the arcsine func on takes as input a ra o of sies an returns an angle The equa on y = arcsin can be rewri en as y = arcsin/; that is, consier a right triangle where the Notes: 5
7 Chapter Deriva ves hypotenuse has length an the sie opposite of the angle with measure y has length This means the final sie has length, using the Pythagorean Theorem y Therefore cossin = cos y = / =, resul ng in Figure 33: A right triangle efine by y = sin / with the length of the thir leg foun using the Pythagorean Theorem arcsin = g = π π 4 y = sin π π 4 y π 3, 3 π 4 3, π 3 π Remember that the input of the arcsine func on is a ra o of a sie of a right triangle to its hypotenuse; the absolute value of this ra o will never be greater than Therefore the insie of the square root will never be nega ve In orer to make y = sin one to one, we restrict its omain to [ π/, π/]; on this omain, the range is [, ] Therefore the omain of y = arcsin is [, ] an the range is [ π/, π/] When = ±, note how the eriva ve of the arcsine func on is unefine; this correspons to the fact that as ±, the tangent lines to arcsine approach ver cal lines with unefine slopes In Figure 34 we see f = sin an f = sin graphe on their respec ve omains The line tangent to sin at the point π/3, 3/ has slope cos π/3 = / The slope of the corresponing point on sin, the point 3/, π/3, is y = sin π 4 = = = 3/ 3/4 /4 / =, π Figure 34: Graphs of sin an sin along with corresponing tangent lines verifying yet again that at corresponing points, a func on an its inverse have reciprocal slopes Using similar techniques, we can fin the eriva ves of all the inverse trigonometric func ons In Figure 3 we show the restric ons of the omains of the stanar trigonometric func ons that allow them to be inver ble Notes: 6
8 7 Deriva ves of Inverse Func ons Func on Domain Range Inverse Func on Domain Range sin [ π/, π/] [, ] sin [, ] [ π/, π/] cos [0, π] [, ] cos [, ] [0, π] tan π/, π/, tan, π/, π/ csc [ π/, 0 0, π/], ] [, csc, ] [, [ π/, 0 0, π/] sec [0, π/ π/, π], ] [, sec, ] [, [0, π/ π/, π] cot 0, π, cot, 0, π Figure 3: Domains an ranges of the trigonometric an inverse trigonometric func ons Theorem 3 Deriva ves of Inverse Trigonometric Func ons The inverse trigonometric func ons are ifferen able on their omains as liste in Figure 3 an their eriva ves are as follows: sin = sec = 3 tan = + 4 cos = 5 csc = 6 cot = + Note how the last three eriva ves are merely the opposites of the first three, respec vely Because of this, the first three are use almost eclusively throughout this tet In Sec on 3, we state without proof or eplana on that ln = We can jus fy that now using Theorem, as shown in the eample Eample 74 Fining the eriva ve of y = ln Use Theorem to compute ln S View y = ln as the inverse of y = e Therefore, using our stanar nota on, let f = e an g = ln We wish to fin g Theorem Notes: 7
9 Chapter Deriva ves gives: g = f g = e ln = In this chapter we have efine the eriva ve, given rules to facilitate its computa on, an given the eriva ves of a number of stanar func ons We restate the most important of these in the following theorem, intene to be a reference for further work Theorem 4 Glossary of Deriva ves of Elementary Func ons Let u an v be ifferen able func ons, an let a, c an n be a real numbers, a > 0, n 0 cu = cu 3 u v = uv + u v uv = u vv 5 7 = e = e 9 ln = 3 sin = cos 5 csc = csc cot 7 tan = sec sin = 9 csc = 3 tan = + u ± v = u ± v = u v uv 4 u v v c = 0 n = n n a = ln a a loga = ln a 4 cos = sin 6 sec = sec tan 8 cot = csc cos = 0 4 sec = cot = + Notes: 8
10 Eercises 7 Terms an Concepts T/F: Every func on has an inverse In your own wors eplain what it means for a func on to be one to one 3 If, 0 lies on the graph of y = f, what can be sai about the graph of y = f? 4 If, 0 lies on the graph of y = f an f = 5, what can be sai about y = f? Problems In Eercises 5 8, verify that the given func ons are inverses 5 f = + 6 an g = 3 6 f = + 6 +, 3 an g = 3, 7 f = 3 5, 5 an g = 3 + 5, 0 8 f = +, an g = f In Eercises 9 4, an inver ble func on f is given along with a point that lies on its graph Using Theorem, evaluate f at the inicate value 9 f = Point=, 0 Evaluate f 0 0 f = + 4, Point= 3, 7 Evaluate f 7 f = sin, π/4 π/4 Point= π/6, 3/ Evaluate f 3/ f = Point=, 8 Evaluate f 8 3 f = +, 0 Point=, / Evaluate f / 4 f = 6e 3 Point= 0, 6 Evaluate f 6 In Eercises 5 4, compute the eriva ve of the given func- on 5 ht = sin t 6 ft = sec t 7 g = tan 8 f = sin 9 gt = sin t cos t 0 ft = ln te t h = sin cos g = tan 3 f = sec / 4 f = sinsin In Eercises 5 7, compute the eriva ve of the given func- on in two ways: a By simplifying first, then taking the eriva ve, an b by using the Chain Rule first then simplifying Verify that the two answers are the same 5 f = sinsin 6 f = tan tan 7 f = sincos In Eercises 8 9, fin the equa on of the line tangent to the graph of f at the inicate value 8 f = sin at = 9 f = cos at = 3 4 Review 30 Fin y, where y y = 3 Fin the equa on of the line tangent to the graph of + y + y = 7 at the point, 3 Let f = 3 + f + s f Evaluate lim s 0 s
11
12 Solutions to O Eercises 3 y y = y siny cosy y+ cos+cosy+sin+y sinysin+y++cosy y 0 +y y+ 3 a y = 0 b y = a y = 4 b y = Sec on 5 T 3 F 5 T 7 f t = 53t 4 9 h t = 6t + e 3t +t f = 3 sin3 3 h t = 8 sin 3 t cost 5 f = tan 7 f = / 9 g t = ln 5 5 cos t sin t m w = ln3/3/ w 3 f = ln ln 5 g t = 5 cost +3t cos5t 7 t+3 sint +3t sin5t 7 7 Tangent line: y = 0 Normal line: = 0 9 Tangent line: y = 3θ π/ + Normal line: y = /3θ π/ + 3 In both cases the eriva ve is the same: / 33 a F/mph b The sign woul be nega ve; when the win is blowing at 0 mph, any increase in win spee will make it feel coler, ie, a lower number on the Fahrenheit scale Sec on 6 Answers will vary 3 T 5 f = 3 /3 = g t = t cos t + sin t t 9 y = 43 y+ y = sin secy 7 a y = b y = y = 3 y 3/ /5 5 y 6/5 y = 0 33 y = ln + Tangent line: y = + 4 ln + 35 y = sin+ sin + cos ln + Tangent line: y = 3π /4 π/ + π/ 3 37 y = Tangent line: y = /7 + /6 Sec on 7 F 3 The point 0, lies on the graph of y = f assuming f is inver ble 5 Compose fg an gf to confirm that each equals 7 Compose fg an gf to confirm that each equals 9 f 0 = f = /5 f 3/ = f π/6 = 3 f / = f = 5 h t = 4t 7 g = +4 9 g t = cos t cost sint t h = sin +cos cos 3 f = 5 a f =, so f = b f = cossin = 7 a f =, so f = b f = coscos 9 y = 4 3/4 + π/6 3 y = 4/5 + =
Section 6.2 Integration by Parts
Section 6 Integration by Parts AP E XC II Version 0 Authors Gregory Hartman, PhD Department of Applied Mathema cs Virginia Military Ins tute Brian Heinold, PhD Department of Mathema cs and Computer Science
More informationUniversity of North Dakota
AP E X C II Late Transcendentals University of North Dakota Adapted from AP E X Calculus by Gregory Hartman, Ph.D. Department of Applied Mathema cs Virginia Military Ins tute Revised July 7, 07 Contribu
More informationUnit #6 - Families of Functions, Taylor Polynomials, l Hopital s Rule
Unit # - Families of Functions, Taylor Polynomials, l Hopital s Rule Some problems an solutions selecte or aapte from Hughes-Hallett Calculus. Critical Points. Consier the function f) = 54 +. b) a) Fin
More informationLecture 5: Inverse Trigonometric Functions
Lecture 5: Inverse Trigonometric Functions 5 The inverse sine function The function f(x = sin(x is not one-to-one on (,, but is on [ π, π Moreover, f still has range [, when restricte to this interval
More informationTOTAL NAME DATE PERIOD AP CALCULUS AB UNIT 4 ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT /6 10/8 10/9 10/10 X X X X 10/11 10/12
NAME DATE PERIOD AP CALCULUS AB UNIT ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT 0 0 0/6 0/8 0/9 0/0 X X X X 0/ 0/ 0/5 0/6 QUIZ X X X 0/7 0/8 0/9 0/ 0/ 0/ 0/5 UNIT EXAM X X X TOTAL AP Calculus
More informationSection The Chain Rule and Implicit Differentiation with Application on Derivative of Logarithm Functions
Section 3.4-3.6 The Chain Rule an Implicit Differentiation with Application on Derivative of Logarithm Functions Ruipeng Shen September 3r, 5th Ruipeng Shen MATH 1ZA3 September 3r, 5th 1 / 3 The Chain
More informationd dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1
Lecture 5 Some ifferentiation rules Trigonometric functions (Relevant section from Stewart, Seventh Eition: Section 3.3) You all know that sin = cos cos = sin. () But have you ever seen a erivation of
More information3.7 Implicit Differentiation -- A Brief Introduction -- Student Notes
Fin these erivatives of these functions: y.7 Implicit Differentiation -- A Brief Introuction -- Stuent Notes tan y sin tan = sin y e = e = Write the inverses of these functions: y tan y sin How woul we
More informationSECTION 3.2 THE PRODUCT AND QUOTIENT RULES 1 8 3
SECTION 3.2 THE PRODUCT AND QUOTIENT RULES 8 3 L P f Q L segments L an L 2 to be tangent to the parabola at the transition points P an Q. (See the figure.) To simplify the equations you ecie to place the
More informationSection 2.1 The Derivative and the Tangent Line Problem
Chapter 2 Differentiation Course Number Section 2.1 The Derivative an the Tangent Line Problem Objective: In this lesson you learne how to fin the erivative of a function using the limit efinition an unerstan
More informationChapter 1 Overview: Review of Derivatives
Chapter Overview: Review of Derivatives The purpose of this chapter is to review the how of ifferentiation. We will review all the erivative rules learne last year in PreCalculus. In the net several chapters,
More informationCHAPTER 3 DERIVATIVES (continued)
CHAPTER 3 DERIVATIVES (continue) 3.3. RULES FOR DIFFERENTIATION A. The erivative of a constant is zero: [c] = 0 B. The Power Rule: [n ] = n (n-1) C. The Constant Multiple Rule: [c *f()] = c * f () D. The
More informationMath 1271 Solutions for Fall 2005 Final Exam
Math 7 Solutions for Fall 5 Final Eam ) Since the equation + y = e y cannot be rearrange algebraically in orer to write y as an eplicit function of, we must instea ifferentiate this relation implicitly
More informationNOTES ON INVERSE TRIGONOMETRIC FUNCTIONS
NOTES ON INVERSE TRIGONOMETRIC FUNCTIONS MATH 5 (S). Definitions of Inverse Trigonometric Functions () y = sin or y = arcsin is the inverse function of y = sin on [, ]. The omain of y = sin = arcsin is
More information1 Applications of the Chain Rule
November 7, 08 MAT86 Week 6 Justin Ko Applications of the Chain Rule We go over several eamples of applications of the chain rule to compute erivatives of more complicate functions. Chain Rule: If z =
More informationx f(x) x f(x) approaching 1 approaching 0.5 approaching 1 approaching 0.
Engineering Mathematics 2 26 February 2014 Limits of functions Consier the function 1 f() = 1. The omain of this function is R + \ {1}. The function is not efine at 1. What happens when is close to 1?
More informationx f(x) x f(x) approaching 1 approaching 0.5 approaching 1 approaching 0.
Engineering Mathematics 2 26 February 2014 Limits of functions Consier the function 1 f() = 1. The omain of this function is R + \ {1}. The function is not efine at 1. What happens when is close to 1?
More informationMath Implicit Differentiation. We have discovered (and proved) formulas for finding derivatives of functions like
Math 400 3.5 Implicit Differentiation Name We have iscovere (an prove) formulas for fining erivatives of functions like f x x 3x 4x. 3 This amounts to fining y for 3 y x 3x 4x. Notice that in this case,
More informationCalculus BC Section II PART A A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS
Calculus BC Section II PART A A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS. An isosceles triangle, whose base is the interval from (0, 0) to (c, 0), has its verte on the graph
More informationImplicit Differentiation and Inverse Trigonometric Functions
Implicit Differentiation an Inverse Trigonometric Functions MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Summer 2018 Explicit vs. Implicit Functions 0.5 1 y 0.0 y 2 0.5 3 4 1.0 0.5
More informationImplicit Differentiation
Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,
More informationChapter Primer on Differentiation
Capter 0.01 Primer on Differentiation After reaing tis capter, you soul be able to: 1. unerstan te basics of ifferentiation,. relate te slopes of te secant line an tangent line to te erivative of a function,.
More informationModule FP2. Further Pure 2. Cambridge University Press Further Pure 2 and 3 Hugh Neill and Douglas Quadling Excerpt More information
5548993 - Further Pure an 3 Moule FP Further Pure 5548993 - Further Pure an 3 Differentiating inverse trigonometric functions Throughout the course you have graually been increasing the number of functions
More informationMath Chapter 2 Essentials of Calculus by James Stewart Prepared by Jason Gaddis
Math 231 - Chapter 2 Essentials of Calculus by James Stewart Prepare by Jason Gais Chapter 2 - Derivatives 21 - Derivatives an Rates of Change Definition A tangent to a curve is a line that intersects
More informationTHEOREM: THE CONSTANT RULE
MATH /MYERS/ALL FORMULAS ON THIS REVIEW MUST BE MEMORIZED! DERIVATIVE REVIEW THEOREM: THE CONSTANT RULE The erivative of a constant function is zero. That is, if c is a real number, then c 0 Eample 1:
More informationTable of Common Derivatives By David Abraham
Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec
More informationRecapitulation of Mathematics
Unit I Recapitulation of Mathematics Basics of Differentiation Rolle s an Lagrange s Theorem Tangent an Normal Inefinite an Definite Integral Engineering Mathematics I Basics of Differentiation CHAPTER
More informationChapter 2 Derivatives
Chapter Derivatives Section. An Intuitive Introuction to Derivatives Consier a function: Slope function: Derivative, f ' For each, the slope of f is the height of f ' Where f has a horizontal tangent line,
More informationTutorial 1 Differentiation
Tutorial 1 Differentiation What is Calculus? Calculus 微積分 Differential calculus Differentiation 微分 y lim 0 f f The relation of very small changes of ifferent quantities f f y y Integral calculus Integration
More informationIntegration: Using the chain rule in reverse
Mathematics Learning Centre Integration: Using the chain rule in reverse Mary Barnes c 999 University of Syney Mathematics Learning Centre, University of Syney Using the Chain Rule in Reverse Recall that
More information1 Definition of the derivative
Math 20A - Calculus by Jon Rogawski Chapter 3 - Differentiation Prepare by Jason Gais Definition of the erivative Remark.. Recall our iscussion of tangent lines from way back. We now rephrase this in terms
More informationIntegration Review. May 11, 2013
Integration Review May 11, 2013 Goals: Review the funamental theorem of calculus. Review u-substitution. Review integration by parts. Do lots of integration eamples. 1 Funamental Theorem of Calculus In
More information23 Implicit differentiation
23 Implicit ifferentiation 23.1 Statement The equation y = x 2 + 3x + 1 expresses a relationship between the quantities x an y. If a value of x is given, then a corresponing value of y is etermine. For
More information1 Lecture 20: Implicit differentiation
Lecture 20: Implicit ifferentiation. Outline The technique of implicit ifferentiation Tangent lines to a circle Derivatives of inverse functions by implicit ifferentiation Examples.2 Implicit ifferentiation
More information18 EVEN MORE CALCULUS
8 EVEN MORE CALCULUS Chapter 8 Even More Calculus Objectives After stuing this chapter you shoul be able to ifferentiate an integrate basic trigonometric functions; unerstan how to calculate rates of change;
More information5.4 Fundamental Theorem of Calculus Calculus. Do you remember the Fundamental Theorem of Algebra? Just thought I'd ask
5.4 FUNDAMENTAL THEOREM OF CALCULUS Do you remember the Funamental Theorem of Algebra? Just thought I' ask The Funamental Theorem of Calculus has two parts. These two parts tie together the concept of
More informationDifferentiation ( , 9.5)
Chapter 2 Differentiation (8.1 8.3, 9.5) 2.1 Rate of Change (8.2.1 5) Recall that the equation of a straight line can be written as y = mx + c, where m is the slope or graient of the line, an c is the
More informationFlash Card Construction Instructions
Flash Car Construction Instructions *** THESE CARDS ARE FOR CALCULUS HONORS, AP CALCULUS AB AND AP CALCULUS BC. AP CALCULUS BC WILL HAVE ADDITIONAL CARDS FOR THE COURSE (IN A SEPARATE FILE). The left column
More informationcosh 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
More informationMATH 13200/58: Trigonometry
MATH 00/58: Trigonometry Minh-Tam Trinh For the trigonometry unit, we will cover the equivalent of 0.7,.4,.4 in Purcell Rigon Varberg.. Right Triangles Trigonometry is the stuy of triangles in the plane
More informationDefine each term or concept.
Chapter Differentiation Course Number Section.1 The Derivative an the Tangent Line Problem Objective: In this lesson you learne how to fin the erivative of a function using the limit efinition an unerstan
More informationInverse Trig Functions
Inverse Trig Functions -7-08 If you restrict fx) = sinx to the interval π x π, the function increases: y = sin x - / / This implies that the function is one-to-one, an hence it has an inverse. The inverse
More informationCalculus in the AP Physics C Course The Derivative
Limits an Derivatives Calculus in the AP Physics C Course The Derivative In physics, the ieas of the rate change of a quantity (along with the slope of a tangent line) an the area uner a curve are essential.
More informationDifferentiability, Computing Derivatives, Trig Review. Goals:
Secants vs. Derivatives - Unit #3 : Goals: Differentiability, Computing Derivatives, Trig Review Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an
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 informationLecture 6: Calculus. In Song Kim. September 7, 2011
Lecture 6: Calculus In Song Kim September 7, 20 Introuction to Differential Calculus In our previous lecture we came up with several ways to analyze functions. We saw previously that the slope of a linear
More informationYou should also review L Hôpital s Rule, section 3.6; follow the homework link above for exercises.
BEFORE You Begin Calculus II If it has been awhile since you ha Calculus, I strongly suggest that you refresh both your ifferentiation an integration skills. I woul also like to remin you that in Calculus,
More informationUnit #3 Rules of Differentiation Homework Packet
Unit #3 Rules of Differentiation Homework Packet In the table below, a function is given. Show the algebraic analysis that leads to the derivative of the function. Find the derivative by the specified
More information3.2 Differentiability
Section 3 Differentiability 09 3 Differentiability What you will learn about How f (a) Might Fail to Eist Differentiability Implies Local Linearity Numerical Derivatives on a Calculator Differentiability
More informationDay 4: Motion Along a Curve Vectors
Day 4: Motion Along a Curve Vectors I give my stuents the following list of terms an formulas to know. Parametric Equations, Vectors, an Calculus Terms an Formulas to Know: If a smooth curve C is given
More informationInverse Functions. Review from Last Time: The Derivative of y = ln x. [ln. Last time we saw that
Inverse Functions Review from Last Time: The Derivative of y = ln Last time we saw that THEOREM 22.0.. The natural log function is ifferentiable an More generally, the chain rule version is ln ) =. ln
More informationMATH 205 Practice Final Exam Name:
MATH 205 Practice Final Eam Name:. (2 points) Consier the function g() = e. (a) (5 points) Ientify the zeroes, vertical asymptotes, an long-term behavior on both sies of this function. Clearly label which
More informationThe Natural Logarithm
The Natural Logarithm -28-208 In earlier courses, you may have seen logarithms efine in terms of raising bases to powers. For eample, log 2 8 = 3 because 2 3 = 8. In those terms, the natural logarithm
More informationInverse Trig Functions
Inverse Trig Functions -8-006 If you restrict fx) = sinx to the interval π x π, the function increases: y = sin x - / / This implies that the function is one-to-one, an hence it has an inverse. The inverse
More informationTrigonometric Functions
72 Chapter 4 Trigonometric Functions 4 Trigonometric Functions To efine the raian measurement system, we consier the unit circle in the y-plane: (cos,) A y (,0) B So far we have use only algebraic functions
More informationWith the Chain Rule. y x 2 1. and. with respect to second axle. dy du du dx. Rate of change of first axle. with respect to third axle
0 CHAPTER Differentiation Section The Chain Rule Fin the erivative of a composite function using the Chain Rule Fin the erivative of a function using the General Power Rule Simplif the erivative of a function
More informationThe Explicit Form of a Function
Section 3 5 Implicit Differentiation The Eplicit Form of a Function The normal way we see function notation has f () on one sie of an equation an an epression in terms of on the other sie. We know the
More informationx = c of N if the limit of f (x) = L and the right-handed limit lim f ( x)
Limit We say the limit of f () as approaches c equals L an write, lim L. One-Sie Limits (Left an Right-Hane Limits) Suppose a function f is efine near but not necessarily at We say that f has a left-hane
More informationMath 115 Section 018 Course Note
Course Note 1 General Functions Definition 1.1. A function is a rule that takes certain numbers as inputs an assigns to each a efinite output number. The set of all input numbers is calle the omain of
More informationMATH2231-Differentiation (2)
-Differentiation () The Beginnings of Calculus The prime occasion from which arose my iscovery of the metho of the Characteristic Triangle, an other things of the same sort, happene at a time when I ha
More informationMath 210 Midterm #1 Review
Math 20 Miterm # Review This ocument is intene to be a rough outline of what you are expecte to have learne an retaine from this course to be prepare for the first miterm. : Functions Definition: A function
More informationLectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs
Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent
More informationCHAPTER SEVEN. Solutions for Section x x t t4 4. ) + 4x = 7. 6( x4 3x4
CHAPTER SEVEN 7. SOLUTIONS 6 Solutions for Section 7.. 5.. 4. 5 t t + t 5 5. 5. 6. t 8 8 + t4 4. 7. 6( 4 4 ) + 4 = 4 + 4. 5q 8.. 9. We break the antierivative into two terms. Since y is an antierivative
More informationDifferentiability, Computing Derivatives, Trig Review
Unit #3 : Differentiability, Computing Derivatives, Trig Review Goals: Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an original function Compute
More informationDerivative of a Constant Multiple of a Function Theorem: If f is a differentiable function and if c is a constant, then
Bob Brown Math 51 Calculus 1 Chapter 3, Section Complete 1 Review of the Limit Definition of the Derivative Write the it efinition of the erivative function: f () Derivative of a Constant Multiple of a
More informationMath Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors
Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+
More informationFormulas From Calculus
Formulas You Shoul Memorize (an I o mean Memorize!) S 997 Pat Rossi Formulas From Calculus. [sin ()] = cos () 2. [cos ()] = sin () 3. [tan ()] = sec2 () 4. [cot ()] = csc2 () 5. [sec ()] = sec () tan ()
More informationJUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson
JUST THE MATHS UNIT NUMBER 10.2 DIFFERENTIATION 2 (Rates of change) by A.J.Hobson 10.2.1 Introuction 10.2.2 Average rates of change 10.2.3 Instantaneous rates of change 10.2.4 Derivatives 10.2.5 Exercises
More informationChapter 4 Trigonometric Functions
SECTION 4.1 Special Right Triangles and Trigonometric Ratios Chapter 4 Trigonometric Functions Section 4.1: Special Right Triangles and Trigonometric Ratios Special Right Triangles Trigonometric Ratios
More informationIntegration by Parts
Integration by Parts 6-3-207 If u an v are functions of, the Prouct Rule says that (uv) = uv +vu Integrate both sies: (uv) = uv = uv + u v + uv = uv vu, vu v u, I ve written u an v as shorthan for u an
More informationQF101: Quantitative Finance September 5, Week 3: Derivatives. Facilitator: Christopher Ting AY 2017/2018. f ( x + ) f(x) f(x) = lim
QF101: Quantitative Finance September 5, 2017 Week 3: Derivatives Facilitator: Christopher Ting AY 2017/2018 I recoil with ismay an horror at this lamentable plague of functions which o not have erivatives.
More informationMath 180, Exam 2, Fall 2012 Problem 1 Solution. (a) The derivative is computed using the Chain Rule twice. 1 2 x x
. Fin erivatives of the following functions: (a) f() = tan ( 2 + ) ( ) 2 (b) f() = ln 2 + (c) f() = sin() Solution: Math 80, Eam 2, Fall 202 Problem Solution (a) The erivative is compute using the Chain
More information6.5 Trigonometric Equations
6. Trigonometric Equations In this section, we discuss conditional trigonometric equations, that is, equations involving trigonometric functions that are satisfied only by some values of the variable (or
More informationLinear and quadratic approximation
Linear an quaratic approximation November 11, 2013 Definition: Suppose f is a function that is ifferentiable on an interval I containing the point a. The linear approximation to f at a is the linear function
More informationAP Calculus I Summer Packet
AP Calculus I Summer Packet This will be your first grade of AP Calculus and due on the first day of class. Please turn in ALL of your work and the attached completed answer sheet. I. Intercepts The -intercept
More informationSolutions to Math 41 Second Exam November 4, 2010
Solutions to Math 41 Secon Exam November 4, 2010 1. (13 points) Differentiate, using the metho of your choice. (a) p(t) = ln(sec t + tan t) + log 2 (2 + t) (4 points) Using the rule for the erivative of
More informationDifferentiation Rules Derivatives of Polynomials and Exponential Functions
Derivatives of Polynomials an Exponential Functions Differentiation Rules Derivatives of Polynomials an Exponential Functions Let s start with the simplest of all functions, the constant function f(x)
More informationSection Inverse Trigonometry. In this section, we will define inverse since, cosine and tangent functions. x is NOT one-to-one.
Section 5.4 - Inverse Trigonometry In this section, we will define inverse since, cosine and tangent functions. RECALL Facts about inverse functions: A function f ) is one-to-one if no two different inputs
More informationMake graph of g by adding c to the y-values. on the graph of f by c. multiplying the y-values. even-degree polynomial. graph goes up on both sides
Reference 1: Transformations of Graphs an En Behavior of Polynomial Graphs Transformations of graphs aitive constant constant on the outsie g(x) = + c Make graph of g by aing c to the y-values on the graph
More informationSome 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
More informationExam 3 Review. Lesson 19: Concavity, Inflection Points, and the Second Derivative Test. Lesson 20: Absolute Extrema on an Interval
Exam 3 Review Lessons 17-18: Relative Extrema, Critical Numbers, an First Derivative Test (from exam 2 review neee for curve sketching) Critical Numbers: where the erivative of a function is zero or unefine.
More informationL 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,
More informationSummary: Differentiation
Techniques of Differentiation. Inverse Trigonometric functions The basic formulas (available in MF5 are: Summary: Differentiation ( sin ( cos The basic formula can be generalize as follows: Note: ( sin
More informationMA Midterm Exam 1 Spring 2012
MA Miterm Eam Spring Hoffman. (7 points) Differentiate g() = sin( ) ln(). Solution: We use the quotient rule: g () = ln() (sin( )) sin( ) (ln()) (ln()) = ln()(cos( ) ( )) sin( )( ()) (ln()) = ln() cos(
More informationFinal Exam Study Guide and Practice Problems Solutions
Final Exam Stuy Guie an Practice Problems Solutions Note: These problems are just some of the types of problems that might appear on the exam. However, to fully prepare for the exam, in aition to making
More informationCalculus I Practice Test Problems for Chapter 3 Page 1 of 9
Calculus I Practice Test Problems for Chapter 3 Page of 9 This is a set of practice test problems for Chapter 3. This is in no wa an inclusive set of problems there can be other tpes of problems on the
More informationAP CALCULUS AB Summer Work. The following are guidelines for completing the summer work packet
Name: Perio: AP CALCULUS AB Summer Work For stuents to successfully complete the objectives of the AP Calculus curriculum, the stuent must emonstrate a high level of inepenence, capability, eication, an
More informationDerivatives of Trigonometric Functions
Derivatives of Trigonometric Functions 9-8-28 In this section, I ll iscuss its an erivatives of trig functions. I ll look at an important it rule first, because I ll use it in computing the erivative of
More information11.7. Implicit Differentiation. Introduction. Prerequisites. Learning Outcomes
Implicit Differentiation 11.7 Introuction This Section introuces implicit ifferentiation which is use to ifferentiate functions expresse in implicit form (where the variables are foun together). Examples
More informationy. ( sincos ) (sin ) (cos ) + (cos ) (sin ) sin + cos cos. 5. 6.. y + ( )( ) ( + )( ) ( ) ( ) s [( t )( t + )] t t [ t ] t t s t + t t t ( t )( t) ( t + )( t) ( t ) t ( t ) y + + / / ( + + ) / / /....
More informationThe Chain Rule. This is a generalization of the (general) power rule which we have already met in the form: then f (x) = r [g(x)] r 1 g (x).
The Chain Rule This is a generalization of the general) power rule which we have already met in the form: If f) = g)] r then f ) = r g)] r g ). Here, g) is any differentiable function and r is any real
More informationThe derivative of a constant function is 0. That is,
NOTES : DIFFERENTIATION RULES Name: LESSON. DERIVATIVE OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS Date: Perio: Mrs. Nguyen s Initial: Eample : Prove f ( ) 4 is not ifferentiable at. Practice Problems: Fin
More informationChapter 1 Prerequisites for Calculus
Section. Chapter Prerequisites for Calculus Section. Lines (pp. 9) Quick Review.. ( ) (). ( ). m 5. m ( ) 5 ( ) 5. (a) () 5 Section. Eercises.. (). 8 () 5. 6 5. (a, c) 5 B A 5 6 5 Yes (b) () () 5 5 No
More informationcosh 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 informationLogarithmic, Exponential and Other Transcendental Functions
Logarithmic, Eponential an Other Transcenental Fnctions 5: The Natral Logarithmic Fnction: Differentiation The Definition First, yo mst know the real efinition of the natral logarithm: ln= t (where > 0)
More informationLecture 16: The chain rule
Lecture 6: The chain rule Nathan Pflueger 6 October 03 Introuction Toay we will a one more rule to our toolbo. This rule concerns functions that are epresse as compositions of functions. The iea of a composition
More informationA. Incorrect! The letter t does not appear in the expression of the given integral
AP Physics C - Problem Drill 1: The Funamental Theorem of Calculus Question No. 1 of 1 Instruction: (1) Rea the problem statement an answer choices carefully () Work the problems on paper as neee (3) Question
More informationCalculus with business applications, Lehigh U, Lecture 05 notes Summer
Calculus with business applications, Lehigh U, Lecture 0 notes Summer 0 Trigonometric functions. Trigonometric functions often arise in physical applications with periodic motion. They do not arise often
More informationThe derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)
Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)
More informationA.P. Calculus Summer Assignment
A.P. Calculus Summer Assignment This assignment is due the first day of class at the beginning of the class. It will be graded and counts as your first test grade. This packet contains eight sections and
More information