Consider the vector-valued function 2 P(1) = ( cos(1) + 20 VI05 sin(t), - Cos(t) + V105 sin(), - sin(t)) (a) Compute the first derivative and the magnitude of the first derivative. (Hint: Make sure to simplify your answer for the magnitude as much as possible! There will be magic cancellations. (b) Are r(t) and r00(t) perpendicular for all t? (Hint: You do not need to compute the second derivative. Use the formula (IOP)' = (*"(t) - (1)= 2"(1) - P'(t) %3D and your answer from part a.) (c) Use your answer from part b. to compute the principal normal vector in terms of the second derivative r00(t). (Hint: Your answer should involve the symbol r°o). Do not compute the second derivative).
Consider the vector-valued function 2 P(1) = ( cos(1) + 20 VI05 sin(t), - Cos(t) + V105 sin(), - sin(t)) (a) Compute the first derivative and the magnitude of the first derivative. (Hint: Make sure to simplify your answer for the magnitude as much as possible! There will be magic cancellations. (b) Are r(t) and r00(t) perpendicular for all t? (Hint: You do not need to compute the second derivative. Use the formula (IOP)' = (*"(t) - (1)= 2"(1) - P'(t) %3D and your answer from part a.) (c) Use your answer from part b. to compute the principal normal vector in terms of the second derivative r00(t). (Hint: Your answer should involve the symbol r°o). Do not compute the second derivative).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Text
Consider the vector-valued function
= (
4
cos(t) +
2
sin(t), -cos(t) +
V5
4
sin(t),
V105
f(t)%3D
20
V105
VI05 sin(e)
(a) Compute the first derivative and the nmagnitude of the first derivative.
(Hint: Make sure to simplify your answer for the magnitude as much as possible! There will be magic cancellations.)
(b) Are ^r(t) and ^r0°(t) perpendicular for all t? (Hint: You do not need to compute the second derivative. Use the
formula
(1*OF)' = (*"(t) · (1) = 26"(1) · i'(1)
and your answer from part a.)
(c) Use your answer from part b. to compute the principal normal vector in termns of the second derivative r00(E).
(Hint: Your answer should involve the symbol r 0t). Do not compute the second derivative).
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