Math 2A Vector Calculus Chapter 11 Test Fall ’07 Name__________________________ Show your work. Don’t use a calculator. Write responses on separate paper.
1.
The temperature-humidity index I (or humidex, for short) is the
perceived air temperature when the actual temperature is T and the relative humidity is h, so we can write .
The following table of values of I is
an excerpt from a table compiled by the National Oceanic and Atmospheric
Administration.
a. For what value of h is ? b.
Approximate
3. Find the limit if it exists. If it doesn’t exist, explain why. a.
b.
4. Suppose a. Find b. Find c.
Find 5.
Find an equation for the tangent plane to the surface
with parametric equations at the point . 6.
Given that ,
x = u v, y
= 2uv, and z = u + v, use the Chain Rule to find and when u
= 4
and v = 5.
7. The temperature at a point is given by where T is measured in degrees Centigrade and x, y, z are in meters. a. Find the rate of change of temperature at the point P(0,2,1) in the direction toward the point (0,5,5). Give your answer in °C/m. b.
In which direction does the temperature increase
fastest at P? Write your answer as an ordered triple in
the form . 8. Consider a. Find the critical points. b. Find all maxima, minima and saddle points in the first octant.
9. Find the extreme values of for .
Math 2A Vector Calculus Chapter 11 Test Solutions Fall ’07
1. The temperature-humidity index I (or humidex, for short) is the perceived air temperature when the actual temperature is T and the relative humidity is h, so we can write . The following table of values of I is an excerpt from a table compiled by the National Oceanic and Atmospheric Administration.
a. For what value of h is ? SOLN: b.
Using the formula with the gradient, 3. Find the limit if it exists. If it doesn’t exist, explain why. a.
shows that as we approach the origin along
different lines y = mx, we arrive at a z value that changes…thus the limit does
not exist. b.
Substitute and . Then (x,y) approaching zero along any path is
equivalent to r shrinking to zero
and we can show the limit exists: 4. Suppose a. Find . SOLN: b. = c.
Find . SOLN: by Foucault’s theorem, this is the
same as above. 5.
Find an equation for the tangent plane to the surface
with parametric equations at the point . 6.
Given ,
x = u v, y
= 2uv, and z = u + v, find and when u
= 4
and v = 5.
7. The temperature at a point is given by where T is measured in degrees Centigrade and x, y, z are in meters. a.
Find the rate of change of temperature at the point P(0,2,1) in the direction toward the
point (0,5,5). Give your answer in °C/m. b.
In which direction does the temperature increase
fastest at P? Write your answer as an ordered triple in
the form . 8. Consider a.
Find the critical points. SOLN:
b.
Find extreme values. 9.
Find the extreme values of for .
This means and, equating components,
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