Math 205: Midterm #2

Jeopardy!

Math 205: Midterm #2

Jeopardy!April 16, 2008April 16, 2008

Jeopardy!Jeopardy!DefinitionsDefinitions Partial Partial

DerivativeDerivatives 1s 1

Partial Partial DerivativeDerivative

s 2s 2

IntegratioIntegrationn

100 100 100 100

200 200200 200200 200200

300 300300 300300 300300

400 400400 400400 400400

Definitions: 100Definitions: 100

State Clairaut’s theorem. What part of the second derivative test uses this theorem?

State Clairaut’s theorem. What part of the second derivative test uses this theorem?

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Definitions: 200Definitions: 200

State Fubini’s Theorem and how it applies to the following integral:

State Fubini’s Theorem and how it applies to the following integral:

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Definitions: 300Definitions: 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).

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).

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Definitions: 400Definitions: 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.

Define what it means for a function to be continuous at a point (a,b) and give a function in R3 that is discontinuous.

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Partial Derivatives 1: 100Partial Derivatives 1: 100Compute the directional derivative

of z = 2x2 - y3 at the point (0,1) in the direction of the vector u = 2i - j.

Compute the directional derivative of z = 2x2 - y3 at the point (0,1) in the direction of the vector u = 2i - j.

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Partial Derivatives 1: 200Partial Derivatives 1: 200Compute ∂z/∂t of z = 3xcosy,

where x = 3st - t2 and y = s - 2sint.

Compute ∂z/∂t of z = 3xcosy, where x = 3st - t2 and y = s - 2sint.

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Partial Derivatives 1: 300Partial Derivatives 1: 300Suppose 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?

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?

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Partial Derivatives 1: 400Partial Derivatives 1: 400Find ∂z/∂y of the equation given by:

zexz = y2 - yz. Find ∂z/∂y of the equation given by:

zexz = y2 - yz.

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Partial Derivatives 2: 100Partial Derivatives 2: 100Compute f of the function

z = f(x,y) = 3x3/2y1/2.Compute f of the function

z = f(x,y) = 3x3/2y1/2.

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Partial Derivatives 2: 200Partial Derivatives 2: 200Determine the tangent plane at the

point (0, π/2) for the function z = 2cos(xy) - x2.

Determine the tangent plane at the point (0, π/2) for the function z = 2cos(xy) - x2.

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Partial Derivatives 2: 300Partial Derivatives 2: 300Find and classify all critical points of

the function z = 1- 3x2 - y2 + 2xy. Find and classify all critical points of

the function z = 1- 3x2 - y2 + 2xy.

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Partial Derivatives 2: 400Partial Derivatives 2: 400Find and classify all critical points of

the function z = 2xy-1 subject to the constraint x2 + y2 = 1.

Find and classify all critical points of the function z = 2xy-1 subject to the constraint x2 + y2 = 1.

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Integration: 100Integration: 100

Reverse the order of integration in the following integral:

Reverse the order of integration in the following integral:

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Integration: 200Integration: 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.

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.

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Integration: 300Integration: 300

Set up an integral to describe the area within the curve r = 2cos.

Set up an integral to describe the area within the curve r = 2cos.

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Integration: 400Integration: 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.

Find the volume of the surface that lies under the cone z = 4 - √(x2+y2) and between the planes z = 1 and z = 2.

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