BE f (t) = (eat, e-at) G(t)= (cos(at), sin(at)), WITH a > 0. CHECK IFa) F"(t) = a²F(t)ANDG" (t) = -a²G(t).b) F"(t) · F(t) > OND G" (t) - G(t) < 0.

Answered: BE f(t) = (et, e¬at) G(t)= (cos(at),…

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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Question

BE f (t) = (eat, e-at) G(t)= (cos(at), sin(at)), WITH a > 0. CHECK IF
a) F"(t) = a²F(t)ANDG" (t) = -a²G(t).
b) F"(t) · F(t) > OND G" (t) - G(t) < 0.

Expert Solution

Step 1

(a)

If $F (t) = (e^{a t}, e^{- a t}) a n d G (t) = (\cos (a t), \sin (a t))$ , then after differentiating

$\begin{array}{rcl} F' (t) & = & (\frac{d}{d t} (e^{a t}), \frac{d}{d t} (e^{- a t})) \\ = & (a e^{a t}, - a e^{- a t}) \\ G' (t) & = & (\frac{d}{d t} (\cos (a t), \frac{d}{d t} (\sin (a t))) \\ = & (- a \sin (a t), a \cos (a t)) \end{array}$

After again differentiating,

$\begin{array}{rcl} F'' (t) & = & (\frac{d}{d t} (a e^{a t}), \frac{d}{d t} (- a e^{- a t})) \\ = & (a^{2} e^{a t}, a^{2} e^{- a t}) \\ = & a^{2} (a^{2} e^{a t}, a^{2} e^{- a t}) \\ = & a^{2} F (t) \\ G' (t) & = & (\frac{d}{d t} (- a \sin (a t), \frac{d}{d t} (a \cos (a t))) \\ = & (- a^{2} \cos (a t), - a^{2} \sin (a t)) \\ = & - a^{2} (\cos (a t), \sin (a t)) \\ = & - a^{2} G (t) \end{array}$