Calculus Concepts: An Informal Approach to the Mathematics of Change 5th Edition

Published by Brooks Cole
ISBN 10: 1-43904-957-2
ISBN 13: 978-1-43904-957-0

Chapter 5 - Accumulating Change: Limits of Sums and the Definite Integral - 5.4 Activities - Page 364: 19

Answer

$$F\left( t \right) = \frac{{{t^3}}}{3} + {t^2} - 20$$

Work Step by Step

$$\eqalign{ & f\left( t \right) = {t^2} + 2t;\,\,\,F\left( {12} \right) = 700 \cr & {\text{Write a formula }}F\left( t \right){\text{ for the antiderivative of }}f\left( t \right) \cr & F\left( t \right) = \int {\left( {{t^2} + 2t} \right)} dt \cr & {\text{integrate by using the power rule}} \cr & F\left( t \right) = \frac{{{t^3}}}{3} + {t^2} + C \cr & \cr & {\text{Use the condition }}F\left( {12} \right) = 700{\text{ to find }}C \cr & 700 = \frac{{{{\left( {12} \right)}^3}}}{3} + {\left( {12} \right)^2} + C \cr & 700 = 576 + 144 + C \cr & 700 = 720 + C \cr & C = - 20 \cr & \cr & {\text{The specific antiderivative of }}f{\text{ is}} \cr & F\left( t \right) = \frac{{{t^3}}}{3} + {t^2} - 20 \cr} $$
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