If your solution to stated problem does not match the given solution, you should self-critique per instructions at

   http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm

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Your solution, attempt at solution.  If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it.  This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

At the end of this document, after the qa problems (which provide you with questions and solutions), there is a series of Questions, Problems and Exercises. 

Question:  `q001.  Let f(x,y,z) = x^2*y*e^3x + (x - y + z)^2. Find the following expressions.

• f(0,0,0)
• f(1,-1,1)
• f(-1,1,-1)
• d/dx(f(x,x,x))
• d/dy(f(1,y,1))
• d/dz(f(1,1,z^2))

Your solution: 

Confidence rating:
 

Given Solution: 

So (d/dx) f(x, x, x) is the x derivative of this expression, equal to

(x^3) ' * e^(3x) + x^3 * e^(3x) ' + (x^2) ' = 3 x^2 e^(3x) + 3 x^3 e^(3x) + 2 x = 3 (x^2 + x^3) e^(3x) + 2x.

f(1, y, 1) = 1 * y^2 * e^(3 * 1) + (1 - y + 1)^2 = y^2 * e^3 + (2-y)^2. 

The derivative with respect to y of this expression is 2 y e^3 - 2 ( 2 - y), which simplifies to 2 y ( e^3 + 1) - 4.

There are at least two ways to find the derivative of f(1, 1, z^2) with respect to z.

One is to substutute and take the derivative. We obtain

f(1, 1, z^2) = e^3 + z^4, and easily find the derivative to be 4 z^3.

Another more generally applicable way (more work that it's worth if all you want to know is the derivative of f(1, 1, z) ) is to find the derivative of f(x, y, z^2) with respect to z. This will involve the chain rule, applied to the 'innermost' function z^2 (the derivative of which is 2 z), and the result is

2 z * 2 (x - y + z)^2.

For x = y = 1, this reduces to the former result 4 z^3.

 

Self-critique (if necessary):

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Question:  `q002.  Find the domain and range of the function f(u,v) = sqrt(u cos v).

Your solution: 

Confidence rating:
 

Given Solution: 

This occurs when u >= 0 and cos(v) >= 0, or when u <= 0 and cos(v) <= 0.

u >= 0 on the right-hand half of the u-v plane. 

cos(v) >= 0 when -pi/2 <= v <= pi/2, or more generally when -pi/2 + 2 n pi <= v <= pi/2 + 2 n pi, for n = ..., -2, -1, 0, 1, 2, ... .  The corresponding regions of the u-v plane are alternating infinite horizontal strips of width pi.

The domain corresponding to u >= 0 and cos(v) >= 0 are therefore alternating horizontal strips in the right half-plane.

The domain corresponding to u <= 0 and cos(v) <= 0 are alternating the horizontal strips in the left half-plane corresponding to pi/2 + 2 n pi <= v <= 3 pi/2 + 2 n pi.

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Question:  `q003.  Sketch and describe the level surface f(x,y,z) = 1 when f(x,y,z) = 2x^2 + 2z^2 - y.

Your solution: 

Confidence rating:
 

Given Solution: 

This corresponds to the equation 2 x^2 + 2 z^2 - y = 1.  This is a quadric surface, an elliptic paraboloid.  Its intersection with any plane parallel to the x-y plane, and also with any plane parallel to the y-z plane, is a parabola.  Its intersection with any surface parallel to the x-z plane is either an ellipse (for y < -1), the point (0, 0, -1) for the plane y = -1, and empty for y > -1.

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Question:  `q004. 

According to the ideal gas law, PV = kT where P is pressure, V is volume, T is temperature, and k is some constant. Suppose a tank contains 3500in^3 of some gas at a pressure of 24lb/in^2 when the temperature is 270K.

• Determine k for this gas.
• Express T as a function of P and V using the k found in the previous step and describe the isotherms.

Your solution: 

Confidence rating:
 

Given Solution: 

k = P V / T = 24 lb/in^3 * 3500 in^3 / (270 K) = 320 lb / K, approx..

So T = P V / k = P V / (320 lb/K) = .003 K / lb * P V.

An isotherm occurs when T is constant, in which case

P V = constant

and

P = constant / V.

This is a hyperbola in the P V plane, asymptotic to the x and y axes, with the line P = V as the axis of symmetry.  (very similar to the graph of y = 1 / x).

The constant is ( .003 K / lb ) / T.  The greater the value of T, the greater the constant and the further the hyperbola's closest approach to the origin.

Self-critique (if necessary):

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