Transcript Math 205: Midterm #2 Jeopardy!

Math 205: Midterm
#2
Jeopardy!
April 16, 2008
Jeopardy!
Definitions
Partial
Partial
Derivatives 1 Derivatives 2
Integration
100
100
100
100
200
200
200
200
300
300
300
300
400
400
400
400
Definitions: 100
State Clairaut’s theorem. What part
of the second derivative test uses
this theorem?
Back
Definitions: 200
State Fubini’s Theorem and how it
applies to the following integral:
Back
Definitions: 300
Suppose z = f(x,y) is a function in R3. If
(a,b) is a point in the domain,
describe the geometric meaning
behind fx(a,b).
Back
Definitions: 400
Define what it means for a function
to be continuous at a point (a,b)
and give a function in R3 that is
discontinuous.
Back
Partial Derivatives 1: 100
Compute the directional derivative
of z = 2x2 - y3 at the point (0,1) in the
direction of the vector u = 2i - j.
Back
Partial Derivatives 1: 200
Compute ∂z/∂t of z = 3xcosy,
where x = 3st - t2 and y = s - 2sint.
Back
Partial Derivatives 1: 300
Suppose you are at the point (0,π) a hill
given by the function:
z = 15 - x2 + cos(xy) - y2.
If the positive y-axis represents north,
and the positive x-axis represents east,
what is your rate of ascent if you head
northwest?
Back
Partial Derivatives 1: 400
Find ∂z/∂y of the equation given by:
zexz = y2 - yz.
Back
Partial Derivatives 2: 100
Compute f of the function
z = f(x,y) = 3x3/2y1/2.
Back
Partial Derivatives 2: 200
Determine the tangent plane at the
point (0, π/2) for the function
z = 2cos(xy) - x2.
Back
Partial Derivatives 2: 300
Find and classify all critical points of
the function z = 1- 3x2 - y2 + 2xy.
Back
Partial Derivatives 2: 400
Find and classify all critical points of
the function z = 2xy-1 subject to the
constraint x2 + y2 = 1.
Back
Integration: 100
Reverse the order of integration in the
following integral:
Back
Integration: 200
Use polar coordinates to set up (but
not evaluate) an integral to
determine the volume under the
sphere x2 + y2 + z2 = 4 within the first
octant.
Back
Integration: 300
Set up an integral to describe the
area within the curve r = 2cos.
Back
Integration: 400
Find the volume of the surface that
lies under the cone z = 4 - √(x2+y2)
and between the planes z = 1 and
z = 2.
Back