MATH261 EXAM II FALL 2013
NAME:
SI:
Problem Points
1
18
2
17
3
19
4
15
5
17
6
14
Total
100
SECTION NUMBER:
You may NOT use calculators or
any references. Show work to receive full credit.
GOOD LUCK !!!
Score
1. Consider the function f (x, y, z) = xy + eyz , and suppose that x = uv 2 , y = 3u, and
z = ln v.
(a) (6pts) Write down the general multivariable Chain Rule that computes ∂f /∂v in
this case.
(b) (12pts) Use your formula in (a) to compute ∂f /∂v when u = 3, v = 1.
2. Consider the function F (x, y, z) = xz 2 sec(y).
(a) (9pts) Find the directional derivative at P (1, π/3, 1) in the direction v = h3, 0, 1i.
(b) (6pts) Is F increasing or decreasing in the direction of v? Give a reason for your
answer.
(c) (2pts) What is the maximum rate of change in F at P ?
3. Consider the surfaces
1
F (x, y, z) = x2 y cos(z) − xey−1 = 0
4
G(x, y, z) = x ln y − yz = 0
(a) (4pts) Verify that each surface contains the point P (4, 1, 0).
(b) (12pts) Find the equations of the tangent planes to the surfaces at P . The answer
may be in the form A(x − x0 ) + B(y − y0 ) + C(z − z0 ) = 0.
(c) (3pts) Find a vector equation of the line tangent to both surfaces at P .
x3 + 8xy − y 2 Classify each of the
4. (15pts) Consider the function f (x, y) = −4x4 + 16
3
critical points given below as a local max, local min, saddle point or a point where the
second derivative test fails. Fill in the table.
Critical Point Max Min Saddle Fails
(0,0)
(2,8)
(-1,-4)
5. (a) (9pts) Find the linearization L(x, y) of f (x, y) = ln(x + 1) + 21 xy at the point
(0, 2). The final form of the linearizaion must be L(x, y) = A + Bx + Cy.
(b) (8pts) Find an upper bound on the magnitude of the error in the approximation
f (x, y) ≈ L(x, y) for the rectangular prism |x| ≤ 0.2 and |y − 2| ≤ 0.1.
√
6. (14pts) Consider the finite area R in the xy-plane bounded by y = 0, x = y, and
x = 1. Using a double integral, find the volume of the solid bounded above by the
3
surface z = ex and below by the area in the xy plane, R. Include a graph of the
region.