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MATH 163 EXAM 3 1) (10 points) Draw any four level curves of the surface given by 2? + 3?2 − 2? = 0. Describe the shape of the surface in words. 2) The table below represents the wind chill ?(?, ?) where ? represents the air temperature in Fahrenheit, and ? represents the wind speed in mph. ? ? = 20 ? = 25 ? = 30 ? = 20 4 11 17 ? = 25 3 9 16 ? = 30 1 8 15 a. (5 points) Estimate ??(25, 25). What does it mean in this context? b. (5 points) Estimate ??(25, 25). What does it mean in this context? 3) (10 points) On a continuous surface ?(?, ?), assume the tangent slope at point ?, in the direction ?, has value ?. Prove that in the direction opposite ?, the tangent slope at point ? must have the value −?. EXTRA CREDIT (up to 4 points) – Why is the surface being continuous necessary for this seemingly obvious result? 4) (18 points) Let ℎ(?, ?) = 2?2 − 2?2? + ?2. Find the absolute minimum and maximum values of the surface on the region −2 ≤ ? ≤ 2 and −2 ≤ ? ≤ 2. 5) (3 points each) For the following situations, explain which concept, method, and/or formula from chapter 14 you would need to use to solve the problem, and why: a. I need to measure a container full of radioactive material in order to calculate the volume of material. But because of the danger, I must measure it from a distance and my measurements may be incorrect. b. I’m diving to do emergency repair on an underwater component of an oil rig. I sent a drone first and calculated the pressure at various locations and depths in the water, since the machinery can affect the normal water pressure. I need to make sure that I don’t follow a dive path where the pressure increases too fast, or else I might get injured. c. I have a raw spherical diamond that I am cutting using computer-aided tools, in order to put into jewelry. I need to program the computer to make sure that the facets that it cuts always face directly away from the center. d. I run a business, and the business makes money from selling two types of products. I want to make the most profit I can, but my factory and the global market have limitations and restrictions that I must consider. Calculus Early Transcendentals 8th Edition ebook pdf calculus Early TranscEndEnTals EighTh EdiTion JamEs sTEwarT McMaster University and University of toronto Australia • Brazil • Mexico • Singapore • United Kingdom • United States Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. 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[email protected]. Library of Congress Control Number: 2014951195 Student Edition: ISBN: 978-1-285-74155-0 Loose-leaf Edition: ISBN: 978-1-305-27235-4 Cengage Learning 20 Channel Center Street Boston, MA 02210 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at www.cengage.com/global. Cengage Learning products are represented in Canada by Nelson Education, Ltd. To learn more about Cengage Learning Solutions, visit www.cengage.com. Purchase any of our products at your local college store or at our pre- ferred online store www.cengagebrain.com. WCN: 02-200-203 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. iii PrEfacE xi To ThE sTudEnT xxiii calculaTors, comPuTErs, and oThEr graPhing dEvicEs xxiv diagnosTic TEsTs xxvi a Preview of calculus 1 1 1.1 Four Ways to Represent a Function 10 1.2 Mathematical Models: A Catalog of Essential Functions 23 1.3 New Functions from Old Functions 36 1.4 Exponential Functions 45 1.5 Inverse Functions and Logarithms 55 Review 68 Principles of Problem solving 71 2 2.1 The Tangent and Velocity Problems 78 2.2 The Limit of a Function 83 2.3 Calculating Limits Using the Limit Laws 95 2.4 The Precise Definition of a Limit 104 2.5 Continuity 114 2.6 Limits at Infinity; Horizontal Asymptotes 126 2.7 Derivatives and Rates of Change 140 Writing Project • Early Methods for Finding Tangents 152 2.8 The Derivative as a Function 152 Review 165 Problems Plus 169 contents Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 3 3.1 Derivatives of Polynomials and Exponential Functions 172 Applied Project • Building a Better Roller Coaster 182 3.2 The Product and Quotient Rules 183 3.3 Derivatives of Trigonometric Functions 190 3.4 The Chain Rule 197 Applied Project • Where Should a Pilot Start Descent? 208 3.5 Implicit Differentiation 208 Laboratory Project • Families of Implicit Curves 217 3.6 Derivatives of Logarithmic Functions 218 3.7 Rates of Change in the Natural and Social Sciences 224 3.8 Exponential Growth and Decay 237 Applied Project • Controlling Red Blood Cell Loss During Surgery 244 3.9 Related Rates 245 3.10 Linear Approximations and Differentials 251 Laboratory Project • Taylor Polynomials 258 3.11 Hyperbolic Functions 259 Review 266 Problems Plus 270 4 4.1 Maximum and Minimum Values 276 Applied Project • The Calculus of Rainbows 285 4.2 The Mean Value Theorem 287 4.3 How Derivatives Affect the Shape of a Graph 293 4.4 Indeterminate Forms and l’Hospital’s Rule 304 Writing Project • The Origins of l’Hospital’s Rule 314 4.5 Summary of Curve Sketching 315 4.6 Graphing with Calculus and Calculators 323 4.7 Optimization Problems 330 Applied Project • The Shape of a Can 343 Applied Project • Planes and Birds: Minimizing Energy 344 4.8 Newton’s Method 345 4.9 Antiderivatives 350 Review 358 Problems Plus 363 iv Contents Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Contents v 5 5.1 Areas and Distances