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A2_f2015.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 2. [4] Show that sin ´ πx 2 ¯ 6 x ´ 4 has a solution on the interval p 0 , 1 q . Justify your answer. 3. Evaluate the following limits: (a) [4] lim x Ñ´8 ˆ e x ` 1 ` x 2 9 x 2 ´ 7 ˙ . (b) [4] lim x Ñ´8 ? x 6 ´ x ` 1 2 x 3 ´ 1 . (c) [5] lim x Ñ8 ` 2 x ´ ? x 2 ` 1 ˘ . 4. [6] Use limits to determine all horizontal and vertical asymptotes for the function f p x q “ ? x 2 ` 1 ´ x ` 1 . 5. [7] Use the definition of the derivative to calculate the derivative of the function f p x q “ 1 x 2 ` x ` 1 . 1
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6. Calculate the derivative of the following functions. Do not simplify your answers.
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