Cram University
 
Username: Password:

Sign up (it's free!)   | |   forgot password?

 

Calc 3 11.8

 

<== Calculus 3
Number Question Answer
See the work
1. For each of the following functions, find the maximum and mimimum values of the function on the circular disk: x^{2} + y^{2} \leq 1. Do this by looking at the level curves and gradiants.

(A) f(x, y) = x + y + 1:
maximum value =
minimum value =

(B) f(x, y) = 1\!x^{2} + 2\!y^{2}:
maximum value =
minimum value =

(C) f(x, y) = 1\!x^{2} - 2\!y^{2}:
maximum value =
minimum value =

A)
maximum value = 2.414
minimum value = -0.414

B)
maximum value = 2
minimum value = 0

C)
maximum value = 1
minimum value = -2
The work
2. Find the maximum and minimum values of f(x, y) = 7 x + y on the ellipse x^{2} + 25 y^{2} = 1
maximum value:
minimum value:
Max: 7.0029
Min: -7.0028
The work
3. Find the maximum and minimum values of f(x, y, z) = 2\!x + 1\!y + 3\!z on the sphere x^{2} + y^{2} + z^{2} = 1.
maximum value =
minimum value =
Max: 3.742
Min: -3.742
The work
4. Find the maximum and minimum volumes of a rectangular box whose surface area equals 4000 square cm and whose edge length (sum of lengths of all edges) is 320 cm. Hint: It can be deduced that the box is not a cube, so if x, y, and z are the lengths of the sides, you may want to let x represent a side with x ≠ y and x ≠ z. Maximum value is ,
occuring at (, ,). Minimum value is ,
occuring at (, ,).
Max: 16000
at ( 40, 20, 20)

Min: 14814.81
at ( 40/3, 100/3, 100/3)
The work
5. The plane x + y + 2z = 20 intersects the paraboloid z = x^2 +
y^2 in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin.
Point farthest away occurs at
(, ,).
Point nearest occurs at
(, ,).
Farthest: ( -2.5, -2.5, 12.5)
Nearest: ( 2, 2, 8)
The work
6. question Answer The work
7. question Answer The work
8. question Answer The work
9. question Answer The work
10. question Answer The work
11. question Answer The work
12. question Answer The work