Week #5 - Related Rates, Linear Approximation, and Taylor Polynomials Section 4.6
|
|
- Marion Holmes
- 6 years ago
- Views:
Transcription
1 Week #5 - Related Rates, Linear Approximation, and Taylor Polynomials Section 4.6 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 005 by John Wiley & Sons, Inc. This material is used by permission of John Wiley & Sons, Inc. SUGGESTED PROBLEMS. The desired rate is. At the date given, t = 0, so the rate is = 6.4e 0.0 t (0.0) billions of people per year (0) = 6.4 (0.0) billions of people per year = million people per year P = 8R dr = 8( R ) = 8 R 5. (a) How fast is an idea of rate, and is represented by the rate dy. dy = 0.(.5)t0.5 = 0. t At t =, dy () = 0. = 0. cm / hour At t =, dy () = cm / hour (b) Since both y = 0.t.5 and dy = 0.t0.5 increase as t increases, the thickness y and rate dy will both be maximal when t =. 0. (a) The rate of change is with respect to time, treating r as a constant, is = 500e rt/00 r 00 = 5 re rt/00 (i) Initially, (0) = 5 re0 dollars per year
2 (ii) At t =, () = 5 rer/00 = 5 re 0.0r dollars per year (b) If r is a function of t, we need the chain rule to be careful about the derivative: [ ( )] d r(t) t = 500e r(t) t/00 00 = 5e r(t) t/00 [ r (t) t + r(t) ] Substituting in t =, r = 4 and r () = 0., gives = 5e 4 /00 [0. + 4] 4.96 Thus the price is increasing by about $5 per year at when t =.. We are given that V = 9 and constant, R = 5, and dr = 0.. Differentiating the original equation, di = V ( R ) dr = 9(5) (0.) = 0.07 amps per second 5. (a) We are told that the body temperature ops by o F in the first hour. Since the body temperature is T(0) = e 0 = 98.6 at the time of death, this means T() = 98.6 = We use this information to solve for k: T() = 96.6 = e k() 8.6 = 0.6e k = e k ( ) 8.6 Take ln of both sides: ln = ln(e k ) = k 0.6 k (b) Now that we have k, we can solve for t when dt = o /hour: dt = = 0.6e t ( ) (0.6)(0.0676) = e t ( ) Take ln of both sides: ln = ln(e t ) = t (0.6)(0.0676) t 0.75 hours So where t 0.75 hours, the body is cooling at o F/ hour.
3 (c) After 4 hours, the body will have cooled to a temperature of T(4) = e t = o Using the coroner s approximate formula ( o in the first hour, o for each hour afterwards), the body would have cooled by 5 o F in the first 4 hours. This would imply a temperature of = 7.6 o, which is very close to the value of 74 o from the more realistic exponential model.. Let the volume of clay be V. The clay is in the shape of a cylinder, so V = πr L. We know dl = 0. cm/sec and we want to know when r = cm and L = 5 cm. Differentiating both sides of the equation with V with respect to time t gives = πrl + πrdl. However, the amount of clay is unchanged, so = 0 and so Solving for gives rl = rdl, = r dl L. When the radius is cm and the length is 5 cm, and the length is increasing at 0. cm per second, the rate at which the radius is changing is = 0. = 0.0 cm/sec. 5 Thus, the radius is decreasing at 0.0 cm/sec. 5. First, we define the length along the shore as x: We want to calculate dx dθ. We can do this by starting with the relationship x = tan(θ) so x = tan(θ) If we differentiate both sides with respect to θ, we get the desired rate: dx dθ = sec (θ) or = cos (θ)
4 9. Let r be the radius of the rainop. Then its volume V = 4 πr cm and its surface area is S = 4πr cm. It is given that = S = 8πr. Furthermore, differentiating our formula for V gives so from the chain rule, = 4πr, = and thus = = 8πr 4πr =. Since is a constant with value, the radius is increasing at a constant rate of cm/sec. 0. The essential relationship between the volume of a cone and its height and radius is V = πr h We want to find / or dh/. The facts we know are / = 0. m /hr r and h are related through the triangle information about the angle with the vertical (π/6 radians, or 0 degrees). Drawing the triangle, we can compare the cone with the standard 0/60 triangle: h π/6 π/6 r π/ π/ The similar triangles imply that h/r = / =, or that h = r or r = h () Going back to the relationship between V and r and h, we can use () to substitute for r: 4
5 V = πh h V = 9 πh The problem is we know /, and we re looking for dh/, but this equation only involves V and h. To get to the derivatives, we can use the implicit differentiation approach, and simply differentiate both sides of the equation (with respect to t): This allows us to solve for dh/: d (V ) = d ( = 9 π ) 9 πh ( h dh ) = πh dh dh = πh and using the fact that / = 0., dh = 0. πh This tells you that the rate the height of the cone is growing at depends on the current height (or size of the cone). Furthermore, if the current cone is big (large h), the rate of growth will be small. This makes sense, since the sand is being added at a constant rate, and if the cone is already large, any new sand will be distributed thinly over the whole cone. By a similar approach, but using r instead of h, you can arrive at the rate of change of r: 5
6 Differentiating: V = πr ( r) V = πr = π r = πr Solving for /: = πr But / = 0., r = h/ : = 0. πh = 0. πh You could get the same result (more easily) by differentiating the relationship r = h/. 4. (a) We differentiate a (t) + b (t) = c with respect to t to find d (a (t) + b d (t)) = c, a(t) a (t) b(t)b (t) = 0 giving a(t) a (t) = b(t)b (t) (b) (i) If Angela likes Brian, then a(t) > 0, so b (t) < 0. This means that b(t) is decreasing, so Brian s affection decreases when Angela likes him. (ii) If Angela dislikes Brian, then a(t) < 0, so b (t) > 0. This means that b(t) is increasing, so Brian s affection increases when Angela dislikes him. (c) Substituting b (t) = a(t) into (a(t) a (t) = b(t) b (t) gives a(t) a (t) = b(t) b (t) = b(t)( a(t)), so a (t) = b(t) (i) If Brian likes Angela, then b(t) > 0, so a (t) > 0. This means that a(t) is increasing, so Angela s affection increases when Brian likes her. (ii) If Brian dislikes Angela, then b(t) < 0, so a (t) < 0. This means that a(t) is decreasing, so Angela s affection decreases when Brian dislikes her. (d) When t = 0, they both like each other. This means that Angela s affection increases, while Brian s decreases. 6
Unit #5 - Implicit Differentiation, Related Rates Section 4.6
Unit #5 - Implicit Differentiation, Related Rates Section 4.6 Some material from Calculus, Single and MultiVariable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc.
More informationWeek #1 The Exponential and Logarithm Functions Section 1.4
Week #1 The Exponential and Logarithm Functions Section 1.4 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc. This material is used by
More informationWeek #6 - Taylor Series, Derivatives and Graphs Section 10.1
Week #6 - Taylor Series, Derivatives and Graphs Section 10.1 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 005 by John Wiley & Sons, Inc. This material is used by
More informationWeek #8 - Optimization and Newton s Method Section 4.5
Week #8 - Optimization and Newton s Method Section 4.5 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc. This material is used by permission
More informationWeek #1 The Exponential and Logarithm Functions Section 1.2
Week #1 The Exponential and Logarithm Functions Section 1.2 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc. This material is used by
More informationUnit #17 - Differential Equations Section 11.5
Unit #17 - Differential Equations Section 11.5 Some material from Calculus, Single and MultiVariable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc. This material
More informationWeek #8 - Optimization and Newton s Method Section 4.5
Week #8 - Optimization and Newton s Method Section 4.5 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc. This material is used by permission
More informationWeek #16 - Differential Equations (Euler s Method) Section 11.3
Week #16 - Differential Equations (Euler s Method) Section 11.3 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc. This material is used
More informationMath Exam 02 Review
Math 10350 Exam 02 Review 1. A differentiable function g(t) is such that g(2) = 2, g (2) = 1, g (2) = 1/2. (a) If p(t) = g(t)e t2 find p (2) and p (2). (Ans: p (2) = 7e 4 ; p (2) = 28.5e 4 ) (b) If f(t)
More informationUnit #1 - Transformation of Functions; Exponential and Logarithms Section 1.4
Unit #1 - Transformation of Functions; Exponential and Logarithms Section 1.4 Some material from Calculus, Single and MultiVariable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 005 by John Wiley
More informationMath 2250 Final Exam Practice Problem Solutions. f(x) = ln x x. 1 x. lim. lim. x x = lim. = lim 2
Math 5 Final Eam Practice Problem Solutions. What are the domain and range of the function f() = ln? Answer: is only defined for, and ln is only defined for >. Hence, the domain of the function is >. Notice
More informationCalculus I - Lecture 14 - Related Rates
Calculus I - Lecture 14 - Related Rates Lecture Notes: http://www.math.ksu.edu/ gerald/math220d/ Course Syllabus: http://www.math.ksu.edu/math220/spring-2014/indexs14.html Gerald Hoehn (based on notes
More informationChapter 3 Practice Test
-- 0 4W0Cu gkujtda UScohfwtKwcaZrYe0 LBLTCT.W V CATlrlZ wrdigthhtmsg yrbeysjetrhvede.r l kmhasdfel YwEi9tqh8 vikncfminoirtkeb WCAa8lnc8uPlXuusA.4 Worksheet by Kuta Software LLC Calculus BC 0 Name Chapter
More informationUnit #17 - Differential Equations Section 11.6
Unit #7 - Differential Equations Section.6 Some material from Calculus, Single and MultiVariable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 005 by John Wiley & Sons, Inc. This material is used
More information3. On the grid below, sketch and label graphs of the following functions: y = sin x, y = cos x, and y = sin(x π/2). π/2 π 3π/2 2π 5π/2
AP Physics C Calculus C.1 Name Trigonometric Functions 1. Consider the right triangle to the right. In terms of a, b, and c, write the expressions for the following: c a sin θ = cos θ = tan θ =. Using
More informationSolutions to Math 41 Final Exam December 10, 2012
Solutions to Math 4 Final Exam December,. ( points) Find each of the following limits, with justification. If there is an infinite limit, then explain whether it is or. x ln(t + ) dt (a) lim x x (5 points)
More informationSolve for an unknown rate of change using related rates of change.
Objectives: Solve for an unknown rate of change using related rates of change. 1. Draw a diagram. 2. Label your diagram, including units. If a quantity in the diagram is not changing, label it with a number.
More informationWeek #1 The Exponential and Logarithm Functions Section 1.3
Week #1 The Exponential and Logarithm Functions Section 1.3 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc. This material is used by
More informationRelated Rates. MATH 151 Calculus for Management. J. Robert Buchanan. Department of Mathematics. J. Robert Buchanan Related Rates
Related Rates MATH 151 Calculus for Management J. Robert Buchanan Department of Mathematics 2014 Related Rates Problems Another common application of the derivative involved situations in which two or
More informationMATH 10550, EXAM 2 SOLUTIONS. 1. Find an equation for the tangent line to. f(x) = sin x cos x. 2 which is the slope of the tangent line at
MATH 100, EXAM SOLUTIONS 1. Find an equation for the tangent line to at the point ( π 4, 0). f(x) = sin x cos x f (x) = cos(x) + sin(x) Thus, f ( π 4 ) = which is the slope of the tangent line at ( π 4,
More informationMATH 18.01, FALL PROBLEM SET # 6 SOLUTIONS
MATH 181, FALL 17 - PROBLEM SET # 6 SOLUTIONS Part II (5 points) 1 (Thurs, Oct 6; Second Fundamental Theorem; + + + + + = 16 points) Let sinc(x) denote the sinc function { 1 if x =, sinc(x) = sin x if
More informationSecond Midterm Exam Name: Practice Problems Septmber 28, 2015
Math 110 4. Treibergs Second Midterm Exam Name: Practice Problems Septmber 8, 015 1. Use the limit definition of derivative to compute the derivative of f(x = 1 at x = a. 1 + x Inserting the function into
More informationMath 2250 Final Exam Practice Problem Solutions. f(x) = ln x x. 1 x. lim. lim. x x = lim. = lim 2
Math 5 Final Eam Practice Problem Solutions. What are the domain and range of the function f() = ln? Answer: is only defined for, and ln is only defined for >. Hence, the domain of the function is >. Notice
More informationMATH1013 Calculus I. Derivatives II (Chap. 3) 1
MATH1013 Calculus I Derivatives II (Chap. 3) 1 Edmund Y. M. Chiang Department of Mathematics Hong Kong University of Science & Technology October 16, 2013 2013 1 Based on Briggs, Cochran and Gillett: Calculus
More informationGuidelines for implicit differentiation
Guidelines for implicit differentiation Given an equation with x s and y s scattered, to differentiate we use implicit differentiation. Some informal guidelines to differentiate an equation containing
More informationFunctions Modeling Change A Preparation for Calculus Third Edition
Powerpoint slides copied from or based upon: Functions Modeling Change A Preparation for Calculus Third Edition Connally, Hughes-Hallett, Gleason, Et Al. Copyright 2007 John Wiley & Sons, Inc. 1 Section
More information) # ( 281). Give units with your answer.
Math 120 Winter 2009 Handout 17: In-Class Review for Exam 2 The topics covered by Exam 2 in the course include the following: Implicit differentiation. Finding formulas for tangent lines using implicit
More information10.1 Curves Defined by Parametric Equation
10.1 Curves Defined by Parametric Equation 1. Imagine that a particle moves along the curve C shown below. It is impossible to describe C by an equation of the form y = f (x) because C fails the Vertical
More information2011 Form B Solution. Jim Rahn
Form B Solution By Jim Rahn Form B AB 6 6 S'( t) dt 7.8 mm 6 S '( t) dt.86 mm or.864 mm c) S '(7).96998 dv d( r h) dh dh r r dt dt dt dt dr since r is constant, dt dv dh r.96998 6.8 mm dt dt day d) D()=M
More informationFunctions Modeling Change A Preparation for Calculus Third Edition
Powerpoint slides copied from or based upon: Functions Modeling Change A Preparation for Calculus Third Edition Connally, Hughes-Hallett, Gleason, Et Al. Copyright 2007 John Wiley & Sons, Inc. Chapter
More informationSection 3.8 Related Rates
Section 3.8 Related Rates Read and re-read the problem until you understand it. Draw and label a picture which gives the relevant information (if possible). Introduce notation. Assign a symbol to every
More informationQuestions Q1. The function f is defined by. (a) Show that (5) The function g is defined by. (b) Differentiate g(x) to show that g '(x) = (3)
Questions Q1. The function f is defined by (a) Show that The function g is defined by (b) Differentiate g(x) to show that g '(x) = (c) Find the exact values of x for which g '(x) = 1 (Total 12 marks) Q2.
More informationSee animations and interactive applets of some of these at. Fall_2009/Math123/Notes
MA123, Chapter 7 Word Problems (pp. 125-153) Chapter s Goal: In this chapter we study the two main types of word problems in Calculus. Optimization Problems. i.e., max - min problems Related Rates See
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 informationMULTIVARIABLE INTEGRATION
MULTIVARIABLE INTEGRATION (PLANE & CYLINDRICAL POLAR COORDINATES) PLANE POLAR COORDINATES Question 1 The finite region on the x-y plane satisfies 1 x + y 4, y 0. Find, in terms of π, the value of I. I
More informationMultiple Choice. at the point where x = 0 and y = 1 on the curve
Multiple Choice 1.(6 pts.) A particle is moving in a straight line along a horizontal axis with a position function given by s(t) = t 3 12t, where distance is measured in feet and time is measured in seconds.
More informationCopyright 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Chapter 5 Review 95 (c) f( ) f ( 7) ( 7) 7 6 + ( 6 7) 7 6. 96 Chapter 5 Review Eercises (pp. 60 6). y y ( ) + ( )( ) + ( ) The first derivative has a zero at. 6 Critical point value: y 9 Endpoint values:
More informationAP Calculus BC: Syllabus 3
AP Calculus BC: Syllabus 3 Scoring Components SC1 SC2 SC3 SC4 The course teaches Functions, Graphs, and Limits as delineated in the Calculus BC Topic The course teaches Derivatives as delineated The course
More informationSOLUTIONS FOR PRACTICE FINAL EXAM
SOLUTIONS FOR PRACTICE FINAL EXAM ANDREW J. BLUMBERG. Solutions () Short answer questions: (a) State the mean value theorem. Proof. The mean value theorem says that if f is continuous on (a, b) and differentiable
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 informationMATHEMATICS: SPECIALIST UNITS 3C AND 3D FORMULA SHEET 2015
MATHEMATICS: SPECIALIST UNITS 3C AND 3D FORMULA SHEET 05 Copyright School Curriculum and Standards Authority, 05 This document apart from any third party copyright material contained in it may be freely
More information8.1 Supplement: Differential Equations and Slope Fields
Math 131 -copyright Angela Allen, Spring 2008 1 8.1 Supplement: Differential Equations and Slope Fields Note: Several of these examples come from your textbook Calculus Concepts: An Applied Approach to
More informationFinal Exam 2011 Winter Term 2 Solutions
. (a Find the radius of convergence of the series: ( k k+ x k. Solution: Using the Ratio Test, we get: L = lim a k+ a k = lim ( k+ k+ x k+ ( k k+ x k = lim x = x. Note that the series converges for L
More informationUnit #13 - Integration to Find Areas and Volumes, Volumes of Revolution
Unit #1 - Integration to Find Areas and Volumes, Volumes of Revolution Some problems and solutions selected or adapted from Hughes-Hallett Calculus. Areas In Questions #1-8, find the area of one strip
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES If we are pumping air into a balloon, both the volume and the radius of the balloon are increasing and their rates of increase are related to each other. However,
More informationLecture 10. (2) Functions of two variables. Partial derivatives. Dan Nichols February 27, 2018
Lecture 10 Partial derivatives Dan Nichols nichols@math.umass.edu MATH 233, Spring 2018 University of Massachusetts February 27, 2018 Last time: functions of two variables f(x, y) x and y are the independent
More informationFunctions Modeling Change A Preparation for Calculus Third Edition
Powerpoint slides copied from or based upon: Functions Modeling Change A Preparation for Calculus Third Edition Connally, Hughes-Hallett, Gleason, Et Al. Copyright 2007 John Wiley & Sons, Inc. 1 Chapter
More information1. The accumulated net change function or area-so-far function
Name: Section: Names of collaborators: Main Points: 1. The accumulated net change function ( area-so-far function) 2. Connection to antiderivative functions: the Fundamental Theorem of Calculus 3. Evaluating
More informationCalculus I Sample Final exam
Calculus I Sample Final exam Solutions [] Compute the following integrals: a) b) 4 x ln x) Substituting u = ln x, 4 x ln x) = ln 4 ln u du = u ln 4 ln = ln ln 4 Taking common denominator, using properties
More informationCalculus 1: Sample Questions, Final Exam
Calculus : Sample Questions, Final Eam. Evaluate the following integrals. Show your work and simplify your answers if asked. (a) Evaluate integer. Solution: e 3 e (b) Evaluate integer. Solution: π π (c)
More informationLevel 3 Calculus, 2014
91578 915780 3SUPERVISOR S Level 3 Calculus, 2014 91578 Apply differentiation methods in solving problems 9.30 am Tuesday 18 November 2014 Credits: Six Achievement Achievement with Merit Achievement with
More informationCONNECTED RATE OF CHANGE PACK
C4 CONNECTED RATE OF CHANGE PACK 1. A vase with a circular cross-section is shown in. Water is flowing into the vase. When the depth of the water is h cm, the volume of water V cm 3 is given by V = 4 πh(h
More informationSolutions to Tutorial Sheet 10 Topics: Applications: Optimization + Modelling
The University of Sydney School of Mathematics and Statistics Solutions to Tutorial Sheet 10 Topics: Applications: Optimization + Modelling MATH1111: Introduction to Calculus Semester 1, 2012 Web Page:
More informationApril 23, 2009 SOLUTIONS SECTION NUMBER:
MATH 5 FINAL EXAM April, 9 NAME: SOLUTIONS INSTRUCTOR: SECTION NUMBER:. Do not open this exam until you are told to begin.. This exam has pages including this cover. There are 9 questions.. Do not separate
More informationFinal Exam Solutions
Final Exam Solutions Laurence Field Math, Section March, Name: Solutions Instructions: This exam has 8 questions for a total of points. The value of each part of each question is stated. The time allowed
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 information(x + 3)(x 1) lim(x + 3) = 4. lim. (x 2)( x ) = (x 2)(x + 2) x + 2 x = 4. dt (t2 + 1) = 1 2 (t2 + 1) 1 t. f(x) = lim 3x = 6,
Math 140 MT1 Sample C Solutions Tyrone Crisp 1 (B): First try direct substitution: you get 0. So try to cancel common factors. We have 0 x 2 + 2x 3 = x 1 and so the it as x 1 is equal to (x + 3)(x 1),
More informationI II III IV V VI VII VIII IX Total
DEPARTMENT OF MATHEMATICS AND STATISTICS QUEEN S UNIVERSITY AT KINGSTON MATH 121 - DEC 2014 CDS/Section 700 Students ONLY INSTRUCTIONS: Answer all questions, writing clearly in the space provided. If you
More informationWeek #15 - Word Problems & Differential Equations Section 8.2
Week #1 - Word Problems & Differential Equations Section 8. From Calculus, Single Variable by Huges-Hallett, Gleason, McCallum et. al. Copyrigt 00 by Jon Wiley & Sons, Inc. Tis material is used by permission
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 information1 What is a differential equation
Math 10B - Calculus by Hughes-Hallett, et al. Chapter 11 - Differential Equations Prepared by Jason Gaddis 1 What is a differential equation Remark 1.1. We have seen basic differential equations already
More informationStudent s Printed Name:
Student s Printed Name: Instructor: CUID: Section # : You are not permitted to use a calculator on any part of this test. You are not allowed to use any textbook, notes, cell phone, laptop, PDA, or any
More informationUnit #17 - Differential Equations Section 11.7
Unit #17 - Differential Equations Section 11.7 Some material from Calculus, Single and MultiVariable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc. This material
More informationa Write down the coordinates of the point on the curve where t = 2. b Find the value of t at the point on the curve with coordinates ( 5 4, 8).
Worksheet A 1 A curve is given by the parametric equations x = t + 1, y = 4 t. a Write down the coordinates of the point on the curve where t =. b Find the value of t at the point on the curve with coordinates
More informationImplicit Differentiation and Related Rates
Math 3A Discussion Notes Week 5 October 7 and October 9, 05 Because of the mierm, we re a little behind lecture, but this week s topics will help prepare you for the quiz. Implicit Differentiation and
More informationMATH 104 MID-TERM EXAM SOLUTIONS. (1) Find the area of the region enclosed by the curves y = x 1 and y = x 1
MATH MID-TERM EXAM SOLUTIONS CLAY SHONKWILER ( Find the area of the region enclosed by the curves y and y. Answer: First, we find the points of intersection by setting the two functions equal to eachother:.
More information( ) ( ). ( ) " d#. ( ) " cos (%) " d%
Math 22 Fall 2008 Solutions to Homework #6 Problems from Pages 404-407 (Section 76) 6 We will use the technique of Separation of Variables to solve the differential equation: dy d" = ey # sin 2 (") y #
More informationMATH 280 Multivariate Calculus Fall Integration over a curve
dr dr y MATH 28 Multivariate Calculus Fall 211 Integration over a curve Given a curve C in the plane or in space, we can (conceptually) break it into small pieces each of which has a length ds. In some
More informationUnit #24 - Lagrange Multipliers Section 15.3
Unit #24 - Lagrange Multipliers Section 1.3 Some material from Calculus, Single and MultiVariable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 200 by John Wiley & Sons, Inc. This material is
More informationName (please print) π cos(θ) + sin(θ)dθ
Mathematics 2443-3 Final Eamination Form B December 2, 27 Instructions: Give brief, clear answers. I. Evaluate by changing to polar coordinates: 2 + y 2 3 and above the -ais. + y d 23 3 )/3. π 3 Name please
More informationAverage rates of change May be used to estimate the derivative at a point
Derivatives Big Ideas Rule of Four: Numerically, Graphically, Analytically, and Verbally Average rate of Change: Difference Quotient: y x f( a+ h) f( a) f( a) f( a h) f( a+ h) f( a h) h h h Average rates
More informationSolutions to Second Midterm(pineapple)
Math 125 Solutions to Second Midterm(pineapple) 1. Compute each of the derivatives below as indicated. 4 points (a) f(x) = 3x 8 5x 4 + 4x e 3. Solution: f (x) = 24x 7 20x + 4. Don t forget that e 3 is
More informationMATH 162. FINAL EXAM ANSWERS December 17, 2006
MATH 6 FINAL EXAM ANSWERS December 7, 6 Part A. ( points) Find the volume of the solid obtained by rotating about the y-axis the region under the curve y x, for / x. Using the shell method, the radius
More informationMath. 151, WebCalc Sections December Final Examination Solutions
Math. 5, WebCalc Sections 507 508 December 00 Final Examination Solutions Name: Section: Part I: Multiple Choice ( points each) There is no partial credit. You may not use a calculator.. Another word for
More informationIntroduction to Differentials
Introduction to Differentials David G Radcliffe 13 March 2007 1 Increments Let y be a function of x, say y = f(x). The symbol x denotes a change or increment in the value of x. Note that a change in the
More information10550 PRACTICE FINAL EXAM SOLUTIONS. x 2 4. x 2 x 2 5x +6 = lim x +2. x 2 x 3 = 4 1 = 4.
55 PRACTICE FINAL EXAM SOLUTIONS. First notice that x 2 4 x 2x + 2 x 2 5x +6 x 2x. This function is undefined at x 2. Since, in the it as x 2, we only care about what happens near x 2 an for x less than
More informationMULTIVARIABLE CALCULUS
MULTIVARIABLE CALCULUS Summer Assignment Welcome to Multivariable Calculus, Multivariable Calculus is a course commonly taken by second and third year college students. The general concept is to take the
More informationMath 106: Calculus I, Spring 2018: Midterm Exam II Monday, April Give your name, TA and section number:
Math 106: Calculus I, Spring 2018: Midterm Exam II Monday, April 6 2018 Give your name, TA and section number: Name: TA: Section number: 1. There are 6 questions for a total of 100 points. The value of
More information4.6 Related Rates 8 b o lk0uct5ai FSHopfitcwkadr9ee MLBL1Cv.0 h ca5lrlx 8rzi8gThzt Zs9 2rJejsqeprTvCeVdy.w Solve each related rate problem.
-- g 52P0l33e 5Ktu3tlaY tswobfrtcwsawrkeq mlzlzcd.u 2 7AklGlf lrbiegkhjtbsa 9rlewsSeIr2vPeVdW.L 2 7Mza5dWeI gwbimtmhn bimnff0ieneistuet SCDallJcrulsuTsG.k Calculus 4.6 Related Rates 8 b230593 o lk0uct5ai
More informationThe University of Sydney Math1003 Integral Calculus and Modelling. Semester 2 Exercises and Solutions for Week
The University of Sydney Math3 Integral Calculus and Modelling Semester 2 Exercises and Solutions for Week 2 2 Assumed Knowledge Sigma notation for sums. The ideas of a sequence of numbers and of the limit
More informationExam A. Exam 3. (e) Two critical points; one is a local maximum, the other a local minimum.
1.(6 pts) The function f(x) = x 3 2x 2 has: Exam A Exam 3 (a) Two critical points; one is a local minimum, the other is neither a local maximum nor a local minimum. (b) Two critical points; one is a local
More informationAbsolute Extrema and Constrained Optimization
Calculus 1 Lia Vas Absolute Extrema and Constrained Optimization Recall that a function f (x) is said to have a relative maximum at x = c if f (c) f (x) for all values of x in some open interval containing
More informationMath 113 Final Exam Practice Problem Solutions. f(x) = ln x x. lim. lim. x x = lim. = lim 2
Math 3 Final Eam Practice Problem Solutions. What are the domain and range of the function f() = ln? Answer: is only defined for, and ln is only defined for >. Hence, the domain of the function is >. Notice
More informationImplicit Differentiation and Related Rates
Math 31A Discussion Session Week 5 Notes February 2 and 4, 2016 This week we re going to learn how to find tangent lines to curves which aren t necessarily graphs of functions, using an approach called
More informatione x3 dx dy. 0 y x 2, 0 x 1.
Problem 1. Evaluate by changing the order of integration y e x3 dx dy. Solution:We change the order of integration over the region y x 1. We find and x e x3 dy dx = y x, x 1. x e x3 dx = 1 x=1 3 ex3 x=
More informationExponential, Logarithmic &Trigonometric Derivatives
1 U n i t 9 12CV Date: Name: Exponential, Logarithmic &Trigonometric Derivatives Tentative TEST date Big idea/learning Goals The world s population experiences exponential growth the rate of growth becomes
More informationMAC 2311 Calculus I Spring 2004
MAC 2 Calculus I Spring 2004 Homework # Some Solutions.#. Since f (x) = d dx (ln x) =, the linearization at a = is x L(x) = f() + f ()(x ) = ln + (x ) = x. The answer is L(x) = x..#4. Since e 0 =, 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 informationRelated Rates - Introduction
Related Rates - Introduction Related rates problems involve finding the rate of change of one quantity, based on the rate of change of a related quantity. Related Rates - Introduction Related rates problems
More informationMA 113 Calculus I Fall 2016 Exam Final Wednesday, December 14, True/False 1 T F 2 T F 3 T F 4 T F 5 T F. Name: Section:
MA 113 Calculus I Fall 2016 Exam Final Wednesday, December 14, 2016 Name: Section: Last 4 digits of student ID #: This exam has five true/false questions (two points each), ten multiple choice questions
More informationI.G.C.S.E. Volume & Surface Area. You can access the solutions from the end of each question
I.G.C.S.E. Volume & Surface Area Index: Please click on the question number you want Question 1 Question Question Question 4 Question 5 Question 6 Question 7 Question 8 You can access the solutions from
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 informationWeek #6 - Taylor Series, Derivatives and Graphs Section 4.1
Week #6 - Talor Series, Derivatives and Graphs Section 4.1 From Calculus, Single Variable b Hughes-Hallett, Gleason, McCallum et. al. Copright 2005 b John Wile & Sons, Inc. This material is used b permission
More informationWeek #7 Maxima and Minima, Concavity, Applications Section 4.2
Week #7 Maima and Minima, Concavit, Applications Section 4.2 From Calculus, Single Variable b Hughes-Hallett, Gleason, McCallum et. al. Copright 2005 b John Wile & Sons, Inc. This material is used b permission
More informationMath 251 Final Exam Review Fall 2016
Below are a set of review problems that are, in general, at least as hard as the problems you will see on the final eam. You should know the formula for area of a circle, square, and triangle. All other
More informationMath 234. What you should know on day one. August 28, You should be able to use general principles like. x = cos t, y = sin t, 0 t π.
Math 234 What you should know on day one August 28, 2001 1 You should be able to use general principles like Length = ds, Area = da, Volume = dv For example the length of the semi circle x = cos t, y =
More informationAP Calculus Chapter 4 Testbank (Mr. Surowski)
AP Calculus Chapter 4 Testbank (Mr. Surowski) Part I. Multiple-Choice Questions 1. Let f(x) = x 3 + 3x 2 45x + 4. Then the local extrema of f are (A) a local minimum of 179 at x = 5 and a local maximum
More informationCalculus & Analytic Geometry I
TQS 124 Autumn 2008 Quinn Calculus & Analytic Geometry I The Derivative: Analytic Viewpoint Derivative of a Constant Function. For c a constant, the derivative of f(x) = c equals f (x) = Derivative of
More informationMATH 1241 Common Final Exam Fall 2010
MATH 1241 Common Final Exam Fall 2010 Please print the following information: Name: Instructor: Student ID: Section/Time: The MATH 1241 Final Exam consists of three parts. You have three hours for the
More information( ) as a fraction. If both numerator and denominator are
A. Limits and Horizontal Asymptotes What you are finding: You can be asked to find lim f x x a (H.A.) problem is asking you find lim f x x ( ) and lim f x x ( ). ( ) or lim f x x ± ( ). Typically, a horizontal
More information