1
Quiz 3 MATH 251, Section 505
Due, February, 24th, 2015
last name : . . . . . . . . . . . . . . . . . .
first name : . . . . . . . . . . . . . . . . . .
”An Aggie does not lie, cheat or steal, or tolerate those who do”
signature : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Write up your result, detail your calculations if necessary and BOX your final answer.
Your final work have to be neat, so use pencil first if you want.
1. [10pts] Find parametric and symmetric equations of the line passing through A(1, 2, 3) and B(−2, 4, 2).
2. [10pts] Determine whether the planes P1 : x + y = 1 + z and P2 : 2x + 4z = 3y + 5 are parallel,
orthogonal, or neither. Find cosine of the angle between the planes.
3. [10pts] Find the domain of z = f (x, y) =
z ≥ 0?
p
3 − x2 − y 2 and sketch it. What is the range of f when
2
4. [5pts] What is the domain of f (x, y) =
√
x−
√5 .
y
5. Let C be the curve with equations r(t) = h2 − t2 , 2t − 1, ln ti.
(a) [4pts] What is the point where C intersects the xz-plane ?
(b) [8pts] Find an equation of the tangent line to C at (1, 1, 0).
6. [10pts] Identify the level curves of the surface given by z = e−(x
2 +y 2 )
.
7. [15pts] Find the second partial derivatives of f (x, y) = y 3 + 5y 2 e4x − cos(x2 ).
3
8. Given the surface xy 2 y 3 = 12 and the point P (3, 2, 1).
(a) [5pts] Find an equation of the tangent plane to the given surface at P .
(b) [8pts] Find the equation of the normal line at P to the given surface.
9. Let f (x, y) = 3x2 y + y 3 − 3x2 − 3y 2 .
(a) [10pts] Find all critical points of f .
(b) [10pts] Classify the critical points (local maximum, local minimum or saddle points).
4
LAST NAME :
FIRST NAME :