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nydimaria [60]
3 years ago
12

Consider the surface z = x 2+4y 2+1. Suppose you are walking on this surface directly above a curve C in the xy-plane, where the

parameterized curve C is given by C : x(t) = cost, y(t) = sin t. Find the values of t for which you are walking uphill (increasing z).
Mathematics
1 answer:
Bess [88]3 years ago
5 0

With x(t)=\cos t and y(t)=\sin t, we have

z(t)=x(t)^2+4y(t)^2+1

z(t)=\cos^2t+4\sin^2t+1

z(t)=3\sin^2t+2

Then z(t) has critical points where

\dfrac{\mathrm dz}{\mathrm dt}=6\sin t\cos t=3\sin2t=0

\implies\sin2t=0\implies2t=n\pi\implies t=\dfrac{n\pi}2

where n is any integer.

z(t) is increasing wherever \sin2t>0, which happens for

\dfrac{2n\pi}2

\boxed{n\pi

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