Answered: Let y be the curve with vector form…

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Topics In Analytic Geometry

Section6.2: Introduction To Conics: parabolas

Problem 4ECP: Find an equation of the tangent line to the parabola y=3x2 at the point 1,3.

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Let y be the curve with vector form r(t) = (1/(t +2), sin(sin(sin t)), e-t).
(a) Determine whether the curve y passes through the point (1,0, 1).
(b) Compute lim r(t), or show that this limit does not exist.
t一→0
(c) Compute r(t) dt.
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Expert Solution

Step 1

Given, $γ$ is a curve with vector form $r (t) = (\frac{1}{t + 2}, \sin (\sin (\sin t)), e^{- t}) .$

(a)

Claim: $γ$ does not pass through (1,0,1).

Suppose that the curve $γ$ passes through the point $(1, 0, 1) .$

Then there exists some $t \in ℝ$ such that $\frac{1}{t + 2} = 1, \sin (\sin (\sin t)) = 0, e^{- t} = 1;$ all three are satisfied.

Solving them,

$\frac{1}{t + 2} = 1 \Rightarrow t = - 1 .$

$\sin (\sin (\sin t)) = 0 \Rightarrow t = 0 .$

$e^{- t} = 1 \Rightarrow t = 0 .$

But t can't be 0 as well as -1 at the same time; which is a contradiction.

Hence, $γ$ does not pass through (1,0,1).

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Let y be the curve with vector form r(t) = (1/(t +2), sin(sin(sin t)), e-t). (a) Determine whether the curve y passes through the point (1,0, 1). (b) Compute lim r(t), or show that this limit does not exist. t一→0 (c) Compute r(t) dt. -