QuestiondGiven r(t) = 5t8i – 9 sin (t)j + e³tk and s(t) = -4ti + 8tj + 4tk, finddtSorry, that's incorrect. Try again?d[2/²r(t) + 2t³s(t)] = ( ___])i+(\)j + (¯)kDi[2t² r(t) + 2t²s(t)].

Answered: Given r(t) = 5t8i - 9 sin (t)j + e³tk…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Step 1

Given that $r (t) = 5 t^{8} i - 9 \sin (t) j + e^{3 t} k$ and $s (t) = - 4 t i + 8 t j + 4 t k$ .

The objective is to find the value of $\frac{d}{d t} [2 t^{2} r (t) + 2 t^{7} s (t)]$ .

To find the value of $\frac{d}{d t} [2 t^{2} r (t) + 2 t^{7} s (t)]$ , first find $2 t^{2} r (t)$ and $2 t^{7} s (t)$ and add the two obtained expressions.

Use the differentiation rules, addition/subtraction rule $\frac{d}{d x} (f (x) \pm g (x)) = \frac{d}{d x} (f (x)) \pm \frac{d}{d x} (g (x))$ , constant multiplication rule $\frac{d}{d x} (c f (x)) = c \frac{d}{d x} (f (x))$ , product rule $\frac{d}{d x} (u v) = u \frac{d v}{d x} + v \frac{d u}{d x}$ , chain rule $\frac{d y}{d x} = \frac{d y}{d t} \cdot \frac{d t}{d x}$ , exponential rule $\frac{d}{d x} (e^{x}) = e^{x}$ , trigonometric rule $\frac{d}{d x} (\sin x) = \cos x$ and power rule $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ .