Chapter 2 The Derivative Business Calculus 155
|
|
- Vernon Reeves
- 6 years ago
- Views:
Transcription
1 Chapter The Derivative Business Calculus 155 Section 11: Implicit Differentiation an Relate Rates In our work up until now, the functions we neee to ifferentiate were either given explicitly, x such as y = x + e, or it was possible to get an explicit formula for them, such as solving 3 3 = 3 y x 5 to get y = 5 + 3x. Sometimes, however, we will have an equation relating x an y which is either ifficult or impossible to solve explicitly for y, such as y + e y = x. In any case, we can still fin y ' = f '(x) by using implicit ifferentiation. The key iea behin implicit ifferentiation is to assume that y is a function of x even if we cannot explicitly solve for y. This assumption oes not require any work, but we nee to be very careful to treat y as a function when we ifferentiate an to use the Chain Rule. Example 1 Assume that y is a function of x. Calculate 3 (a) ( y ) (b) ( x 3 y ) an (c) ln(y) x x x (a) We nee the chain rule since y is a function of x: 3 ( y ) = 3 y ( y) = 3y y x x (b) We nee to use the prouct rule an the Chain Rule: y 3 x y = x ( y ) + y ( x ) = x y + y 3x = x yy + 3y x x x x x (c) We know x ( ) ln( y) = 1 y y ln( x) = 1, so we use that an the Chain Rule: x x IMPLICIT DIFFERENTIATION: To etermine y ', ifferentiate each sie of the efining equation, treating y as a function of x, an then algebraically solve for y '. This chapter is (c) 013. It was remixe by Davi Lippman from Shana Calaway's remix of Contemporary Calculus by Dale Hoffman. It is license uner the Creative Commons Attribution license.
2 Chapter The Derivative Business Calculus 156 Example Fin the slope of the tangent line to the circle x + y = 5 at the point (3,4) using implicit ifferentiation. We ifferentiate each sie of the equation x + y = 5 an then solve for y' ( x + y ) = (5) x x x + yy = 0 x x Solving for y', we have y = =, an, at the point (3,4), y y y' = 3/4. In the previous example, it woul have been easy to explicitly solve for y, an then we coul ifferentiate y to get y '. Because we coul explicitly solve for y, we ha a choice of methos for calculating y '. Sometimes, however, we can not explicitly solve for y, an the only way of etermining y ' is implicit ifferentiation. Relate Rates If several variables or quantities are relate to each other an some of the variables are changing at a known rate, then we can use erivatives to etermine how rapily the other variables must be changing. Example 3 Suppose the borer of a town is roughly circular, an the raius of that circle has been increasing at a rate of 0.1 miles each year. Fin how fast the area of the town has been increasing when the raius is 5 miles. We coul get an approximate answer by calculating the area of the circle when the raius is 5 miles ( A = πr = π(5 miles) 78.6 miles ) an 1 year later when the raius is 0.1 feet larger than before ( A = πr = π(5.1 miles) 81.7 miles ) an then fining Area/ time = (81.7 mi 78.6 mi )/(1 year) = 3.1 mi /yr. This approximate answer represents the average change in area uring the 1 year perio when the raius increase from 5 miles to 5.1 miles, an woul correspon to the secant slope on the area graph. To fin the exact answer, though, we nee erivatives. In this case both raius an area are functions of time: r(t) = raius at time t A(t) = area at time t
3 Chapter The Derivative Business Calculus 157 We know how fast the raius is changing, which is a statement about the erivative: r mile = 0.1. We also know that r = 5 at our moment of interest. t year A We are looking for how fast the area is increasing, which is. t Now we nee an equation relating our variables, which is the area equation: A = πr. Taking the erivative of both sies of that equation with respect to t, we can use implicit ifferentiation: ( A) = ( πr ) t t A r = π r t t Plugging in the values we know for r an r/t, A miles miles = π (5miles).1 = 3.14 t year year The area of the town is increasing by 3.14 square miles per year when the raius is 5 miles. Relate Rates When working with a relate rates problem, 1. Ientify the quantities that are changing, an assign them variables. Fin an equation that relates those quantities 3. Differentiate both sies of that equation with respect to time 4. Plug in any known values for the variables or rates of change 5. Solve for the esire rate. Example 4 A company has etermine the eman curve for their prouct is q = 5000 p, where p is the price in ollars, an q is the quantity in millions. If weather conitions are riving the price up $ a week, fin the rate at which eman is changing when the price is $40. The quantities changing are p an q, an we assume they are both functions of time, t, in weeks. We alreay have an equation relating the quantities, so we can implicitly ifferentiate it. t ( q) = ( 5000 p ) t
4 Chapter The Derivative Business Calculus 158 t q t ( q) = ( 5000 p ) 1/ 1 = t q t ( ) 1/ 5000 p ( 5000 p ) t p p t ( 1/ 5000 p ) 1 = Using the given information, we know the price is increasing by $/week when the price is $40, p giving = when p = 40. Plugging in these values, t q 1 1/ = ( ) ( 40 ) t Deman is falling by 1.37 million items per week.
5 Chapter The Derivative Business Calculus Exercises In problems 1 10 fin y/x by ifferentiating implicitly then fin the value of y/x at the given point. 1. x + y = 100, point (6, 8). x + 5y = 45, point (5, ) 3. x 3xy + 7y = 5, point (,1) 4. x + y = 5, point (4,9) 5. x 9 + y 16 = 1, point (0,4) 6. x 9 + y 16 = 1, point (3,0) 7. ln(y) + 3x 7 = 0, point (,e) 8. x y = 16, point (5,3) 9. x y = 16, point (5, 3) 10. y + 7x 3 3x = 8, point (1,) 11. Fin the slopes of the lines tangent to the graph in shown at the points (3,1), (3,3), an (4,). 1. Fin the slopes of the lines tangent to the graph in shown where the graph crosses the y axis. 13. Fin the slopes of the lines tangent to the graph in graph shown at the points ((5,0), (5,6), an ( 4,3). 14. Fin the slopes of the lines tangent to the graph in the graph shown where the graph crosses the y axis. In problems 15 16, fin y/x using implicit ifferentiation an then fin the slope of the line tangent to the graph of the equation at the given point. 15. y 3 5y = 5x + 7, point (1,3) 16. y 5xy + x + 1 = 0, point (,5)
6 Chapter The Derivative Business Calculus An expanable sphere is being fille with liqui at a constant rate from a tap (imagine a water balloon connecte to a faucet). When the raius of the sphere is 3 inches, the raius is increasing at inches per minute. How fast is the liqui coming out of the tap? ( V = 4 3 π r3 ) 18. The 1 inch base of a right triangle is growing at 3 inches per hour, an the 16 inch height is shrinking at 3 inches per hour. (a) Is the area increasing or ecreasing? (b) Is the perimeter increasing or ecreasing? (c) Is the hypotenuse increasing or ecreasing? 19. One hour later the right triangle in Problem is 15 inches long an 13 inches high, an the base an height are changing at the same rate as in Problem 18. (a) Is the area increasing or ecreasing now? (b) Is the hypotenuse increasing or ecreasing now? (c) Is the perimeter increasing or ecreasing now? 0. A young woman an her boyfrien plan to elope, but she must rescue him from his mother who has locke him in his room. The young woman has place a 0 foot long laer against his house an is knocking on his winow when his mother begins pulling the bottom of the laer away from the house at a rate of 3 feet per secon. How fast is the top of the laer (an the young couple) falling when the bottom of the laer is (a) 1 feet from the bottom of the wall? (b) 16 feet from the bottom of the wall? (c) 19 feet from the bottom of the wall? 1. The length of a 1 foot by 8 foot rectangle is increasing at a rate of 3 feet per secon an the with is ecreasing at feet per secon. (a) How fast is the perimeter changing? (b) How fast is the area changing?. An oil tanker in Puget Soun has sprung a leak, an a circular oil slick is forming. The oil slick is 4 inches thick everywhere, is 100 feet in iameter, an the iameter is increasing at 1 feet per hour. Your job, as the Coast Guar commaner or the tanker's captain, is to etermine how fast the oil is leaking from the tanker.
7 Chapter The Derivative Business Calculus 161
23 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 informationThe derivative of a constant function is 0. That is,
NOTES 3: DIFFERENTIATION RULES Name: Date: Perio: LESSON 3. DERIVATIVE OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS Eample : Prove f ( ) 6 is not ifferentiable at 4. Practice Problems: Fin f '( ) using the
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 informationImplicit Differentiation
Implicit Differentiation Implicit Differentiation Using the Chain Rule In the previous section we focuse on the erivatives of composites an saw that THEOREM 20 (Chain Rule) Suppose that u = g(x) is ifferentiable
More information2.6 Related Rates. The Derivative as a Rate of Change. = ft related rates 175
2.6 related rates 175 2.6 Related Rates Throughout the next several tions we ll look at a variety of applications of derivatives. Probably no single application will be of interest or use to everyone,
More informationImplicit Differentiation. Lecture 16.
Implicit Differentiation. Lecture 16. We are use to working only with functions that are efine explicitly. That is, ones like f(x) = 5x 3 + 7x x 2 + 1 or s(t) = e t5 3, in which the function is escribe
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 informationSection 2.7 Derivatives of powers of functions
Section 2.7 Derivatives of powers of functions (3/19/08) Overview: In this section we iscuss the Chain Rule formula for the erivatives of composite functions that are forme by taking powers of other functions.
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 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 informationWorksheet 8, Tuesday, November 5, 2013, Answer Key
Math 105, Fall 2013 Worksheet 8, Tuesay, November 5, 2013, Answer Key Reminer: This worksheet is a chance for you not to just o the problems, but rather unerstan the problems. Please iscuss ieas with your
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 informationRelated Rates. Introduction. We are familiar with a variety of mathematical or quantitative relationships, especially geometric ones.
Relate Rates Introuction We are familiar with a variety of mathematical or quantitative relationships, especially geometric ones For example, for the sies of a right triangle we have a 2 + b 2 = c 2 or
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 information1 Lecture 18: The chain rule
1 Lecture 18: The chain rule 1.1 Outline Comparing the graphs of sin(x) an sin(2x). The chain rule. The erivative of a x. Some examples. 1.2 Comparing the graphs of sin(x) an sin(2x) We graph f(x) = sin(x)
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 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 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 informationFurther Differentiation and Applications
Avance Higher Notes (Unit ) Prerequisites: Inverse function property; prouct, quotient an chain rules; inflexion points. Maths Applications: Concavity; ifferentiability. Real-Worl Applications: Particle
More informationThe Explicit Form of a Function
Section 3 5 Implicit Differentiation The Eplicit Form of a Function Function Notation requires that we state a function with f () on one sie of an equation an an epression in terms of on the other sie
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 information2.5 SOME APPLICATIONS OF THE CHAIN RULE
2.5 SOME APPLICATIONS OF THE CHAIN RULE The Chain Rule will help us etermine the erivatives of logarithms an exponential functions a x. We will also use it to answer some applie questions an to fin slopes
More informationCalculus I Homework: Related Rates Page 1
Calculus I Homework: Relate Rates Page 1 Relate Rates in General Relate rates means relate rates of change, an since rates of changes are erivatives, relate rates really means relate erivatives. The only
More informationMath 142 (Summer 2018) Business Calculus 5.8 Notes
Math 142 (Summer 2018) Business Calculus 5.8 Notes Implicit Differentiation and Related Rates Why? We have learned how to take derivatives of functions, and we have seen many applications of this. However
More informationChapter 3 Notes, Applied Calculus, Tan
Contents 3.1 Basic Rules of Differentiation.............................. 2 3.2 The Prouct an Quotient Rules............................ 6 3.3 The Chain Rule...................................... 9 3.4
More informationMath Test #2 Info and Review Exercises
Math 180 - Test #2 Info an Review Exercises Spring 2019, Prof. Beyler Test Info Date: Will cover packets #7 through #16. You ll have the entire class to finish the test. This will be a 2-part test. Part
More informationSection 2: Limits and Continuity
Chapter 2 The Derivative Business Calculus 79 Section 2: Limits and Continuity In the last section, we saw that as the interval over which we calculated got smaller, the secant slopes approached the tangent
More informationMathematics 1210 PRACTICE EXAM II Fall 2018 ANSWER KEY
Mathematics 1210 PRACTICE EXAM II Fall 2018 ANSWER KEY 1. Calculate the following: a. 2 x, x(t) = A sin(ωt φ) t2 Solution: Using the chain rule, we have x (t) = A cos(ωt φ)ω = ωa cos(ωt φ) x (t) = ω 2
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 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 informationImplicit Differentiation
Implicit Differentiation Much of our algebraic study of mathematics has dealt with functions. In pre-calculus, we talked about two different types of equations that relate x and y explicit and implicit.
More information102 Problems Calculus AB Students Should Know: Solutions. 18. product rule d. 19. d sin x. 20. chain rule d e 3x2) = e 3x2 ( 6x) = 6xe 3x2
Problems Calculus AB Stuents Shoul Know: Solutions. + ) = + =. chain rule ) e = e = e. ) =. ) = ln.. + + ) = + = = +. ln ) =. ) log ) =. sin ) = cos. cos ) = sin. tan ) = sec. cot ) = csc. sec ) = sec
More informationby using the derivative rules. o Building blocks: d
Calculus for Business an Social Sciences - Prof D Yuen Eam Review version /9/01 Check website for any poste typos an upates Eam is on Sections, 5, 6,, 1,, Derivatives Rules Know how to fin the formula
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 informationFall 2016: Calculus I Final
Answer the questions in the spaces provie on the question sheets. If you run out of room for an answer, continue on the back of the page. NO calculators or other electronic evices, books or notes are allowe
More informationExam 2 Review Solutions
Exam Review Solutions 1. True or False, an explain: (a) There exists a function f with continuous secon partial erivatives such that f x (x, y) = x + y f y = x y False. If the function has continuous secon
More informationRelated Rates. Introduction
Relate Rates Introuction We are familiar with a variet of mathematical or quantitative relationships, especiall geometric ones For eample, for the sies of a right triangle we have a 2 + b 2 = c 2 or the
More informationChapter 3 Definitions and Theorems
Chapter 3 Definitions an Theorems (from 3.1) Definition of Tangent Line with slope of m If f is efine on an open interval containing c an the limit Δy lim Δx 0 Δx = lim f (c + Δx) f (c) = m Δx 0 Δx exists,
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 informationImplicit Differentiation and Related Rates
Implicit Differentiation an Relate Rates Up until now ou have been fining the erivatives of functions that have alrea been solve for their epenent variable. However, there are some functions that cannot
More informationMore from Lesson 6 The Limit Definition of the Derivative and Rules for Finding Derivatives.
Math 1314 ONLINE More from Lesson 6 The Limit Definition of the Derivative an Rules for Fining Derivatives Eample 4: Use the Four-Step Process for fining the erivative of the function Then fin f (1) f(
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 informationCalculus I Sec 2 Practice Test Problems for Chapter 4 Page 1 of 10
Calculus I Sec 2 Practice Test Problems for Chapter 4 Page 1 of 10 This is a set of practice test problems for Chapter 4. This is in no way an inclusive set of problems there can be other types of problems
More informationAP Calculus AB Chapter 4 Packet Implicit Differentiation. 4.5: Implicit Functions
4.5: Implicit Functions We can employ implicit differentiation when an equation that defines a function is so complicated that we cannot use an explicit rule to find the derivative. EXAMPLE 1: Find dy
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 informationf(x) f(a) Limit definition of the at a point in slope notation.
Lesson 9: Orinary Derivatives Review Hanout Reference: Brigg s Calculus: Early Transcenentals, Secon Eition Topics: Chapter 3: Derivatives, p. 126-235 Definition. Limit Definition of Derivatives at a point
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 informationSolutions to Practice Problems Tuesday, October 28, 2008
Solutions to Practice Problems Tuesay, October 28, 2008 1. The graph of the function f is shown below. Figure 1: The graph of f(x) What is x 1 + f(x)? What is x 1 f(x)? An oes x 1 f(x) exist? If so, what
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 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 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 informationMA119-A Applied Calculus for Business Fall Homework 5 Solutions Due 10/4/ :30AM
MA9-A Applie Calculus for Business 2006 Fall Homework 5 Solutions Due 0/4/2006 0:0AM. #0 Fin the erivative of the function by using the rules of i erentiation. r f (r) r. #4 Fin the erivative of the function
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 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 informationLinear First-Order Equations
5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)
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 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 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 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 informationImplicit Differentiation
Week 6. Implicit Differentiation Let s say we want to differentiate the equation of a circle: y 2 + x 2 =9 Using the techniques we know so far, we need to write the equation as a function of one variable
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 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 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 informationRules of Differentiation. Lecture 12. Product and Quotient Rules.
Rules of Differentiation. Lecture 12. Prouct an Quotient Rules. We warne earlier that we can not calculate the erivative of a prouct as the prouct of the erivatives. It is easy to see that this is so.
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 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 informationFinal Exam: Sat 12 Dec 2009, 09:00-12:00
MATH 1013 SECTIONS A: Professor Szeptycki APPLIED CALCULUS I, FALL 009 B: Professor Toms C: Professor Szeto NAME: STUDENT #: SECTION: No ai (e.g. calculator, written notes) is allowe. Final Exam: Sat 1
More informationThe Exact Form and General Integrating Factors
7 The Exact Form an General Integrating Factors In the previous chapters, we ve seen how separable an linear ifferential equations can be solve using methos for converting them to forms that can be easily
More informationAP Calculus AB One Last Mega Review Packet of Stuff. Take the derivative of the following. 1.) 3.) 5.) 7.) Determine the limit of the following.
AP Calculus AB One Last Mega Review Packet of Stuff Name: Date: Block: Take the erivative of the following. 1.) x (sin (5x)).) x (etan(x) ) 3.) x (sin 1 ( x3 )) 4.) x (x3 5x) 4 5.) x ( ex sin(x) ) 6.)
More informationA = 1 2 ab da dt = 1 da. We can find how fast the area is growing at 3 seconds by plugging everything into that differentiated equation: da
1 Related Rates In most related rates problems, we have an equation that relates a bunch of quantities that are changing over time. For example, suppose we have a right triangle whose base and height are
More informationMath 251 Notes. Part I.
Math 251 Notes. Part I. F. Patricia Meina May 6, 2013 Growth Moel.Consumer price inex. [Problem 20, page 172] The U.S. consumer price inex (CPI) measures the cost of living base on a value of 100 in the
More informationDays 3 & 4 Notes: Related Rates
AP Calculus Unit 4 Applications of the Derivative Part 1 Days 3 & 4 Notes: Related Rates Implicitly differentiate the following formulas with respect to time. State what each rate in the differential equation
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 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 informationMathematical Review Problems
Fall 6 Louis Scuiero Mathematical Review Problems I. Polynomial Equations an Graphs (Barrante--Chap. ). First egree equation an graph y f() x mx b where m is the slope of the line an b is the line's intercept
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 informationChapter 3 The Integral Business Calculus 197
Chapter The Integral Business Calculus 97 Chapter Exercises. Let A(x) represent the area bounded by the graph and the horizontal axis and vertical lines at t=0 and t=x for the graph in Fig.. Evaluate A(x)
More informationIMPLICIT DIFFERENTIATION
IMPLICIT DIFFERENTIATION CALCULUS 3 INU0115/515 (MATHS 2) Dr Arian Jannetta MIMA CMath FRAS Implicit Differentiation 1/ 11 Arian Jannetta Explicit an implicit functions Explicit functions An explicit function
More informationDecember 2, 1999 Multiple choice section. Circle the correct choice. You do not need to show your work on these problems.
December 2, 1999 Multiple choice section Circle the correct choice You o not nee to show your work on these problems 1 Let f(x) = e 2x+1 What is f (0)? (A) 0 (B) e (C) 2e (D) e 2 (E) 2e 2 1b Let f(x) =
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 informationCollege Calculus Final Review
College Calculus Final Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Determine the following limit. (Hint: Use the graph to calculate the limit.)
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 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 informationMath 190 Chapter 3 Lecture Notes. Professor Miguel Ornelas
Math 190 Chapter 3 Lecture Notes Professor Miguel Ornelas 1 M. Ornelas Math 190 Lecture Notes Section 3.1 Section 3.1 Derivatives of Polynomials an Exponential Functions Derivative of a Constant Function
More informationYORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics. MATH A Test #2. June 25, 2014 SOLUTIONS
YORK UNIVERSITY Faculty of Science Department of Mathematics an Statistics MATH 505 6.00 A Test # June 5, 04 SOLUTIONS Family Name (print): Given Name: Stuent No: Signature: INSTRUCTIONS:. Please write
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 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 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 information(a) 82 (b) 164 (c) 81 (d) 162 (e) 624 (f) 625 None of these. (c) 12 (d) 15 (e)
Math 2 (Calculus I) Final Eam Form A KEY Multiple Choice. Fill in the answer to each problem on your computer-score answer sheet. Make sure your name, section an instructor are on that sheet.. Approimate
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 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 informationStrauss PDEs 2e: Section Exercise 6 Page 1 of 5
Strauss PDEs 2e: Section 4.3 - Exercise 6 Page 1 of 5 Exercise 6 If a 0 = a l = a in the Robin problem, show that: (a) There are no negative eigenvalues if a 0, there is one if 2/l < a < 0, an there are
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 informationMA 2232 Lecture 08 - Review of Log and Exponential Functions and Exponential Growth
MA 2232 Lecture 08 - Review of Log an Exponential Functions an Exponential Growth Friay, February 2, 2018. Objectives: Review log an exponential functions, their erivative an integration formulas. Exponential
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 1B, lecture 8: Integration by parts
Math B, lecture 8: Integration by parts Nathan Pflueger 23 September 2 Introuction Integration by parts, similarly to integration by substitution, reverses a well-known technique of ifferentiation an explores
More informationIMPLICIT DIFFERENTIATION
Mathematics Revision Guies Implicit Differentiation Page 1 of Author: Mark Kulowski MK HOME TUITION Mathematics Revision Guies Level: AS / A Level AQA : C4 Eecel: C4 OCR: C4 OCR MEI: C3 IMPLICIT DIFFERENTIATION
More informationKramers Relation. Douglas H. Laurence. Department of Physical Sciences, Broward College, Davie, FL 33314
Kramers Relation Douglas H. Laurence Department of Physical Sciences, Browar College, Davie, FL 333 Introuction Kramers relation, name after the Dutch physicist Hans Kramers, is a relationship between
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 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 information