Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 14 - Calculus of Vector-Valued Functions - 14.2 Calculus of Vector-Valued Functions - Exercises - Page 720: 20


$$\frac{d}{dt}(r(t)\cdot r_1(t))|_2 =62$$

Work Step by Step

Since $$ r_1(t)=\langle t^2,t^3,t \rangle $$ then, we have $$\frac{d}{dt}(r(t)\cdot r_1(t))=\frac{d}{dt}(r(t))\cdot r_1(t) +r(t)\cdot \frac{d}{dt}( r_1(t))\\ =\frac{d}{dt}(r(t))\cdot r_1(t) +r(t)\cdot \langle 2t,3t^2,1\rangle.$$ Now, at $ t=2$, we have $$\frac{d}{dt}(r(t)\cdot r_1(t))|_2=\frac{d}{dt}(r(2))\cdot r_1(2) +r(2)\cdot \frac{d}{dt}( r_1(2))\\ =\langle 1,4,3 \rangle\cdot \langle 4,8,2 \rangle+\langle 2,1,0 \rangle\cdot \langle 4,12,1 \rangle\\ =62$$
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