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: 17


$$\frac{d}{dt}(r_1(t)\cdot r_2(t)) =(2t+3t^2)e^{3t}+(3t^2+2t^3)e^{2t}+(1+t)e^{t}.$$

Work Step by Step

Since $$ r_1(t)=\langle t^2,t^3,t \rangle, \quad r_2(t)=\langle e^{3t},e^{2t},e^{ t} \rangle $$ then, we have $$\frac{d}{dt}(r_1(t)\cdot r_2(t))=\frac{d}{dt}(r_1(t))\cdot r_2(t) +r_1(t)\cdot \frac{d}{dt}( r_2(t))\\ =\langle 2 t,3t^2,1 \rangle\cdot \langle e^{3t},e^{2t},e^{ t} \rangle+\langle t^2,t^3,t \rangle\cdot\langle 3e^{3t},2e^{2t},e^{ t} \rangle\\ =(2t+3t^2)e^{3t}+(3t^2+2t^3)e^{2t}+(1+t)e^{t}.$$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.