A2_f2015_key.pdf

# A2_f2015_key.pdf - MATH 1500 D01/D02 Fall 2015 Assignment 2...

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MATH 1500 D01/D02 Fall 2015 Assignment 2 SHOW ALL WORK to get full marks. Leave answers as exact answers. For example, leave it as 1 { 7 as opposed to 0 . 142857 . Word problems should have sentence answers with units. This assignment covers sections from 2.5–2.8 and 3.1–3.2. Use the definition of derivative ONLY for Q5. Do not use the definition of derivative in any other questions. 1. [7] Use limits to calculate a and b such that the following function is continuous for all real numbers x. f p x q “ \$ & % sin ´ π 2 x ¯ ` ax 2 ` b x ă 2 3 x 2 e p x ´ 2 q ` a ` b ` ln p x ´ 1 q x ą 2 Solution: The function is continuous for x ă 2 since sin ´ π 2 x ¯ ` ax 2 ` b is composed of a polynomial and trigonometric function that are always continuous. The function is continuous for x ą 2 since e p x ´ 2 q ` a ` b ` ln p x ´ 1 q is the sum of logarithmic, exponential and constant functions that are always continuous as long as we take the logarithm of a positive value. Hence we focus at x 2 For the function to be continuous at x 2 we must have that lim x Ñ 2 ´ f p x q “ f p 2 q “ lim x Ñ 2 ` f p x q From lim x Ñ 2 ´ f p x q “ f p 2 q we have that lim x Ñ 2 ´ sin ´ π 2 x ¯ ` ax 2 ` b 3 sin p π q ` 4 a ` b 3 4 a ` b 3 . From lim x Ñ 2 ` f p x q “ f p 2 q we have that 1

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lim x Ñ 2 ` e p x ´ 2 q ` a ` b ` ln p x ´ 1 q “ 3 e 0 ` a ` b ` ln p 1 q “ 12 1 ` a ` b 3 . a ` b 2 . Since 4 a ` b 3 we have that b 3 ´ 4 a. Inserting that into a ` b 2 yields a ` p 3 ´ 4 a q “ 2 ñ ´ 3 a “ ´ 1 ñ a 1 3 . Plugging this value of a back into 4 a ` b 3 yields 4 3 ` b 3 ñ b 3 ´ 4 3 5 3 Hence a 1 3 and b 5 3 . 2. [4] Show that sin ´ πx 2 ¯ 6 x ´ 4 has a solution on the interval p 0 , 1 q . Justify your answer. Solution: Let f p x q “ sin ´ πx 2 ¯ ´ 6 x ` 4. f is continuous on the interval r 0 , 1 s since it is composed of a trigonometric and a polynomial function.
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