16. Consider f (x, y) = ex² + y² .(a) Draw the level curve of f (x, y) for k = e.(b) Find and sketch a normal vector to the level curve in part a.) at the point22

Answered: 6. Consider f (x, y) = ex² + y² . (a)…

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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10. Find f and then f: f"(x) = 2-2, f (1) = 0, ƒ(1) = 2
Consider the function z = 4x²+4y²-4 z = f(x, y). Find equations for the xy-trace, xz-trace, and yz-trace for the graph of Find equations for the level curves f(x, y) = c for c= -3, 0, 1, 4. Then sketch and luoel these curves in the xy-plane (on the same set of axes).
4. Let f (x) = x³ -x² + 5. a) Find the y-intercept of f. y-intercept: b) Find f' and f", and determine where each are 0 and/or do not exist (DNE). If none, write "none". f' = 0: f' DNE: f" = 0: f" DNE: c) E Do a sign analysis on f' and f". d) Find the intervals on which f is increasing and decreasing. Increasing: Decreasing: e) Find the intervals on which f is concave up and concave down. Concave up: Concave down: f) answers as (x, y) points. Find all local maxima, local minima, and inflection points of f. Be sure to write your Local max: Local min: Inflection point(s): -4 -3 -1 g) Sketch the graph of f.
3. Suppose that the temperature on a metal plate is given by the function T with T(x, y) = 100 - (x² +4y²) where the temperature is measured in degrees Fahrenheit and a and y are each measured in feet. Now suppose that an ant is walking on the metal plate in such a way that it walks in a straight line from the point (1, 4) to the point (5, 6). (a) Find parametric equations for the ant's a and y coordinates as it walks the line from (1,4) to (5,6) in one minute at a constant speed. (b) What can you say about dr dt and dy dt for every value of t? (c) Determine the instantaneous rate of change in temperature with respect to t that the ant is experi- encing at the moment it is halfway from (1, 4) to (5,6). Include units in your answer.

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6. Consider f (x, y) = ex² + y² . (a) Draw the level curve of f (x, y) for k =