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Physics 272: Homework Set #10

Read 14.7,14.9 of Stein and Barcellos.

1. Around Town

(a) At the locations listed below, clearly draw the gradient of the elevation on the topographical map (next page). You can either print out the map to later scan or draw on it electronically. Use the same scale for all vectors: that is, draw a large vector when the magnitude of the gradient is large and a proportionately smaller vector when the magnitude of the gradient is small.

i. Carr Hall.

ii. Several points along the path of the wrecking ball (which in 2007 rolled down North Main Street, starting at Pelletier library and stopping in the trunk of a car waiting for a red light at the corner of North Main and Randolph).

iii. The apartment complex your instructor lived in when he was a toddler (corner of Limber and Jefferson, now the Allegheny Commons student residence complex).

iv. The house your instructor grew up in (near the corner of Limber and Devore).

v. Your instructor’s elementary school (First District, near bottom of map).

vi. The place where your instructor got married (Ford Chapel).

vii. A couple other places of your choosing.

(b) Mark the map with an X at a local maximum.

2. Gradients and Directional Derivatives

For each of the following functions, (i) find the gradient of f, (ii) evaluate the gradient at the given point P, (iii) determine the direction that yields the maximum directional derivative at that point, (iv) find the magnitude of the gradient at that point, and (v) find the directional derivative at the given point in the indicated direction v. (Note that the given v below are not necessarily unit vectors, although , , and are unit vectors.) (vi) Is your answer to part (iv) or part (v) larger? Why is this not surprising?

3. Critical Points and Extrema

Find the critical points and, with the help of the second derivative test, any relative extrema of the given functions.

4. Keeping Your Iguana Happy on a Budget

You plan to build a reptile enclosure with some desired volume V for your pet iguana. The rectangular base will be slate and the vertical sides will be glass. The top is open to the air. The slate costs three times as much per square foot as the glass. Find the dimensions of the enclosure that keeps the cost of the materials as low as possible. As a numerical value for V hasn’t been specified here, your answer for each dimension will involve V.