Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 2 - Derivatives - 2.1 Derivatives and Rates of Change - 2.1 Exercises: 13

Answer

$v(2) =-24 $ft/s

Work Step by Step

Given the function $s(t) = 40t - 16t^2$ You can get the velocity by evaluating the derivative at $t = 2$ $ v(2) = \lim\limits_{t \to 2}\frac{s(t) - s(2)}{t-2}$ Substitute $s(t)$ and $s(2) (= 80 - 64 = 16)$ $ v(2) = \lim\limits_{t \to 2}\frac{(40t - 16t^2) - 16}{t-2}$ Factor the top $ v(2) = \lim\limits_{t \to 2}\frac{-8(2t^2-5t+2)}{t-2}$ $ v(2) = \lim\limits_{t \to 2}\frac{-8(t-2)(2t-1)}{t-2}$ Simplify the expression ($(t-2)$ cancels) $ v(2) = \lim\limits_{t \to 2}-8(2t-1)$ $ v(2) =-8 \lim\limits_{t \to 2}(2t-1)$ Find the limit $ v(2) =-8 \lim\limits_{t \to 2}(2(2)-1)$ $ v(2) =-8 \lim\limits_{t \to 2}(3)$ $ v(2) =-8 (3)$ $ v(2) =-24$ The instantaneous velocity at 2 seconds is $-24$ft/s
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