precw_5.pdf - THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MATH1010 University Mathematics(Fall 2019 Pre-Coursework Exercise 5 If you

precw_5.pdf - THE CHINESE UNIVERSITY OF HONG KONG...

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THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MATH1010 University Mathematics (Fall 2019) Pre-Coursework Exercise 5 If you find any typos/mistakes, please email us at [email protected]
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2 Dear Students of MATH1010, The main goal of this note is to provide a list of the facts/results from the teaching materials that you have learned and seen in the past week of lectures. There are a few ways of getting ready for and preparing to do any coursework problems in the tutorial class this week: WeBWork: There are a few assigned problems that can be found in WeBWork. We sincerely encourage you to do them multiple times without any time constraint until you get a correct answer for each question. Worked Examples: Additional Exercise Set: There are six additional exercise sets posted every two weeks that fit in your study plan. Please try to work out all the problems. Extra Exercise Set: There are numerous worked exercises to go along with each lecture topic. These drill-type problems will help you understand the course materials. Many challenge problems and their suggested solutions can be found in the extra problem sets. (You need the username and password) Pre-Coursework Exercise Set: There are many worked examples contained in this file with their suggested solutions. Please work out every step and try to understand the line of reasoning behind this exercise. (You need the username and password) MathGym: If you have any questions, please visit MathGym and click the link below: Kindly reminder : You should expect a few problems that you might never have seen/worked out/thought of before in the coursework problems. That is exactly what we planned to do - we want you to solve some challenging questions in order to enhance your problem- solving skills and test how much you understand the MATH1010 teaching materials every week! Yours truly, The MATH1010 Lecturer Department of Mathematics, CUHK The MATH1010 Course Website
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3 Here are a few useful observations and notes on the topic of limits: Let Δ f = f ( a + Δ x ) - f ( a ). The Linear Approximation is the estimate Δ f f 0 ( a x (for Δ x small) . Differential notation: dx is the change in x , df = f 0 ( a ) dx , Δ y = f ( a + dx ) - f ( a ) . In this notation, the Linear Approximation reads Δ y dy, (for Δ x small) . The linearization of f ( x ) at x = a is the function L ( x ) = f 0 ( a )( x - a ) + f ( a ) . The Linear Approximation is equivalent to the approximation f ( x ) L ( x ) , (for x close to a ) . The error in the Linear Approximation is the quantity Error = | Δ f - f 0 ( a x | . In many cases, the percentage error is more important than the error itself: Percentage error = error actual value × 100% . Extreme Values on an Interval Definition 1. Let f ( x ) be a function on an interval I and let a I. We say that f ( a ) is the Absolute minimum of f ( x ) on I if f ( a ) f ( x ) for all x I .
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