Multivariable Calculus Final Exam Fall 2008, Exams of Calculus

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MATH 2110Q Multivariable Calculus Fall 2008

Final Exam

Show your work for full credit. Good luck!

Name :

P roblem P oints Score

T otal 100

1. The acceleration function of a particle is given by a(t) =< 3 t, 9 , − 3 t 2 >,

where 0 ≤ t ≤ 4. The initial velocity of the particle is v(0) =< 1 , 0 , 0 >

and its initial position is r(0) =< 2 , 0 , 4 >.

(a) Compute the velocity of the particle at time t.

(b) Compute the position of the particle at time t.

2. Find the absolute maximum and minimum values of the function

f (x, y) = 3x

2 − 12 x +

3 y

2

on the closed triangular region D with vertices (0, 0), (2, 0), (0, 2). Show

your work, and explain your reasoning.

3. Find the volume of the solid bounded by the paraboloid z = x 2 + y 2 and

the plane z = 4.

5. Determine whether or not F is a conservative vector field on R 2 . If it

is, find a function f such that F = ∇f ; otherwise explain why it is not

conservative.

(a) F(x, y) =< x

2 − sin(xy),

y + cos(xy) >

(b) F(x, y) =< e x

• 6xy, 3 x 2 − 2 cos y >

6. Let f (x, y) = sin(πx 2 ) + x 2 y 3 , and let C be the curve r(t) =<

t, 1 − t 2

,

with 0 ≤ t ≤ 9. Use the Fundamental Theorem for line integrals to

compute

C

∇f · dr