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Math 320 Third Test ______________________ Calculus III Name (10 points)
This exciting 50-minute test covers sections 13.1-6 of Calculus by Finney and Thomas. Relax and do well. Each of the ten problems is worth nine points. Check your work -- especially on the easy problems.
- On the left is a drawing of several level curves of a function f( x , y ). Draw ∇f as a vector at the three indicated points (show their relative lengths).
- Evaluate the following limits.
a.
lim P→(1,0,−1)
e x + z z 2 + cos xy
b.
lim P→(1,1,0)
x^2 z − 2 xyz + y^2 z xz − yz
- By considering different paths of approach, show that
x + y x^2 + y^2
has no limit as ( x,y )
approaches (0,0).
- Let f(x,y,z) = e^3 x +4 y cos 5 z. Find the appropriate second order partials and the
determine if f satisfies the Laplace equation:
∂2f ∂x
∂2f ∂y
∂2f ∂z = 0. Place your results
below.
∂2f ∂x
∂2f ∂y
∂2f ∂z
Satisfies Laplace? yes^ no
- In the following two problems, find
∂ w ∂ t in terms of t at the given point.
a. w = x 2
- y 2 , x = sin t 2 , y = t 5 ; when t = π.
b. w = xy + ln z , x = ( t + s )^2 , y = ln ts , z =
t s ; when^ t^ =1,^ s =1.
- The derivative of f( x,y ) at (1,2) in the direction of the vector i + j is 2 2 and in the
direction of − 2 j is −3. What is the derivative f(x,y) in the direction of the − i − 2 j?
- Let f(x,y) = xy + xz + yz and P0 = (1,2,2).
a. Find ∇f (at the point Po).
b. Find the equation for the tangent plane at Po.
c. Find the parametric equations for the normal line at Po.
- (Use the gradient or linearization to find) by about how much f( x,y,z ) = xy + xz + yz will change as the point P( x,y,z ) moves from the origin at a distance of ds = 0. units in the direction of 2 i + 2 j − k?
