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MATH 251 (4,6,7 ), Dr. Z. , Exam 1, Thurs., Oct. 16, 2017, SEC 118
FRAME YOUR FINAL ANSWER(S) TO EACH PROBLEM
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Types: Number, Function of variable(s), 2D vector of numbers, 3D vector of numbers, 2D vector of functions, 3D vector of functions, equation of a plane, parametric equation of a line, equation of a line, equation of a surface, equation of a line, DNE (does not exist).
- (10 pts.) Find ∂z∂x and ∂z∂y at the point (1, 1 , 1) if z(x, y) is given implictly by the equation x^3 + y^3 + z^3 − 2 xyz = 1.
The type of the answers is:
ans. (^) ∂x∂z (1, 1) = ∂z∂y (1, 1) =
- (10 points) Find the directional derivative of f (x, y, z) = x^3 y^4 z^5 at P = (1, − 1 , 1) in the direction pointing from the point P to the point Q = (1, 2 , 2). (Hint: first find the vector PQ.)
The type of the answer is:
ans.
- (10 points) Find an equation of the tangent plane to the following surface at the given point xy + 2yz + 3xz = 6 , (1, 1 , 1).
The type of the answer is:
ans.
- (10 points) Use the linearization of f (x, y, z) =
2 x + 3y + 4xz to approximate f (1. 01 , 0. 99 , 1 .02).
The type of the answer is:
ans.
- (10 points, altogether) Do the following limits exist? If they do, find them. Explain!
a. (3 points)
lim (x,y,z)→(1, 1 ,1)
ln(x^2 + y^2 + z^2 ) x + y + z
b. (3 points)
lim (x,y,z)→(0, 0 ,0)
x + y + z x^3 + y^3 + z^3
c. (4 points)
lim (x,y,z)→(0, 0 ,0)
(x + y + z) sin(
x + y + z
- (10 points) A certain particle has acceleration
a(t) = 〈 et, − sin t, −4 cos 2t 〉 ,
and at t = 0 its velocity is 〈 1 , 1 , 0 〉 and its position vector is 〈 1 , 0 , 1 〉, find its velocity and position vector at time t = π 2.
The type of the answer(s) is:
ans.
velocity vector at t = π/2:
position vector at t = π/2:
- (10 points) Find an equation to the plane that passes through the points (6, 0 , 0), (0, 4 , 0), (0, 0 , 3).
The type of the answer is:
ans.
