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Solved: Walking on a surface Consider the following
Chapter 11, Problem 49E(choose chapter or problem)
Walking on a surface Consider the following surfaces specified in the form z = f(x, y) and the curve C in the xy-plane given parametrically in the form x = g(t), y = h(t).
a. In each case, find z'(t).
b. Imagine that you are walking on the surface directly above the curve C in the direction of increasing t. Find the values of 1 for which you are walking uphill (that is, z is increasing).
\(z=x^{2}+4 y^{2}+1, C: x=\cos t, y=\sin t ; 0 \leq t \leq 2 \pi\)
Questions & Answers
QUESTION:
Walking on a surface Consider the following surfaces specified in the form z = f(x, y) and the curve C in the xy-plane given parametrically in the form x = g(t), y = h(t).
a. In each case, find z'(t).
b. Imagine that you are walking on the surface directly above the curve C in the direction of increasing t. Find the values of 1 for which you are walking uphill (that is, z is increasing).
\(z=x^{2}+4 y^{2}+1, C: x=\cos t, y=\sin t ; 0 \leq t \leq 2 \pi\)
ANSWER:
Solution 49E
Step 1 of 4:
In this problem we need to find the z'(t).
Given:
a. In each case, find z'(t).
We have
The chain rule for one independent variable states that,
We have
We have