Name:

Lab Section:

MATH 215 – Fall 2004

FINAL EXAM

Show your work in this booklet.

Do NOT submit loose sheets of paper–They won’t be graded

Problem Points Score

1 15

2 10

3 25

4 10

5 15

6 15

7 10

TOTAL 100

Some useful trigonometric identities:

sin2θ+ cos2θ= 1 cos 2θ= cos2θ−sin2θsin 2θ= 2 sin θcos θ

sin2θ=1−cos 2θ

2cos2θ=1 + cos 2θ

2

Spherical coordinates:

x=ρcos(θ) sin(φ)y=ρsin(θ) sin(φ)z=ρcos(φ)

1

Problem 1. (15 points) This problem is about the function

f(x, y, z) = 3zy + 4xcos(z).

(a) What is the rate of change of the function of fat (1,1,0) in the direction from this point to

the origin?

(b) Give an approximate value of f(0.9,1.2,0.11).

CONTINUED ON THE NEXT PAGE

2

(c) Recall that f(x, y, z) = 3zy + 4xcos(z).

The equation f(x, y, z) = 4 implicitly deﬁnes zas a function of (x, y), if we agree that z= 0 if

(x, y) = (1,1).

Find the numerical values of the derivatives

∂z

∂x(1,1) and ∂z

∂y (1,1).

3

Problem 2. (10 points)

(a) Find and classify the critical points of the function f(x, y) = −2x2+ 8xy −9y2+ 4y−4.

(b) Find the equation of the tangent plane to the surface z+ 2x2−8xy + 9y2−4y= 0 at the

point (2,0,−8).

4

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