DQ 3

Question 1

For this question, treat 'z' as a function of the independent variable, 'y'. That is z = f(y) = z(y)

$$z(y) = \frac{ab}{y} \ \ \ \ \ \ \ \ \ \\ \ \ \ \ a,b \ = \ constants \newline \ \nonumber \newline$$

$$\int_{y_1}^{y_2} \ z(y) \ dy \ = \ ? \nonumber$$

A.

$$ab \left [ln\left(\frac{y_2}{y_1} \right) \right] \nonumber$$

B.

$$ab \left [ln ( y_2 - y_1) \right] \nonumber$$

C.

$$ab \left [ln(y_2) - ln(y_1) \right] \nonumber$$

D.

$$ab \left [ ln(y_1 y_2) \right] \nonumber$$

E.

$$-ab \left [ln\left(\frac{y_1}{y_2} \right) \right ] \nonumber$$

F.

$$\ none \ of \ the \ above \ \nonumber$$

Question 2

Consider a fluid for which the relation between its pressure (P), volume(V), and temperature(T) is given by:

$$PV \ = \ nRT \nonumber$$

What is the area under the curve representing a volume change from V1 to V2 (V1 less than V2) while T is constant. R and n are constants.

A.

$$nRT \left [ ln(V_2) - ln(V_1) \right] \nonumber$$

B.

$$nRT \left [ ln ( V_2 - V_1) \right ] \nonumber$$

C.

$$nRT \left [ ln \left ( \frac{V_2}{V_1} \right) \right ] \nonumber$$

D.

$$ln(V_1) \nonumber$$

E.

$$\ none \ of \ the \ above \ \nonumber$$

Question 3

Consider a function of 2 independent variables, g(x,y) where x and y are the independent variables. A partial derivative of g(x,y) is sometimes necessary to evaluate. Abstractly, this would meet a need to assess a change in g(x,y) when x changes while y is constant. The case of y changing with x constant is also a possibility. A partial derivative is written as the following for the differential change in g(x,y) with a differential change in x with y constant:

$$\left (\frac{\partial g(x,y)}{\partial x} \right)_y \ \nonumber$$

The partial derivative amounts to taking the derivative of the function g(x,y) and treating one of the independent variables as a constant while taking the derivative with respect to the other independent variable. The total differential of g(x,y), d(g(x,y)), is given as:

$$d(g(x,y)) \ = \ \left (\frac{\partial g(x,y)}{\partial x} \... ... (\frac{\partial g(x,y)}{\partial y} \right)_x dy \ \nonumber$$

For the following function, g(x,y), determine the total differential.

$$g(x,y) \ = \ x^2 + xy + y^2 \ \nonumber$$

A.

$$d(g(x,y)) \ = \ (2x)dx+(2y)dy \nonumber$$

B.

$$d(g(x,y)) \ = \ (2x+y)dx+(x+2y)dy \nonumber$$

C.

$$d(g(x,y)) \ = \ (2x+y)dz+(x+2y)dy \nonumber$$

D.

$$d(g(x,y)) \ = \ (2x+y)dg+(x+2y)df \nonumber$$

E.

$$\ none \ of \ the \ above \ \nonumber$$

Question 4

Find the total differential for the function:

$$f(x,y, z) \ = \ ln(x) \ + \ yz+x^2y^2z^2 \nonumber$$

A.

$$d(f(x,y,z)) \ = \ xdx \ + \ (z + 2x^2yz^2)dy + (y + 2x^2y^2z)dz \nonumber$$

B.

$$d(f(x,y,z)) \ = \ ( \frac{1}{2x^2 + zy + 3} + 2xy^2z^2)dx \ + \ (z + 2x^2yz^2)dy + (y + 2x^2y^2z)dz \nonumber$$

C.

$$d(f(x,y,z)) \ = \ ( \frac{1}{x} + 2xy^2z^2)dx \ + \ (z + 2x^2yz^2)dy + (y + 2x^2y^2z)dz \nonumber$$

D.

$$d(f(x,y,z)) \ = \ ( \frac{1}{x} + 2xy^2z^2)dz \ + \ (z + 2x^2yz^2)dx + (y + 2x^2y^2z)dy \nonumber$$

E.

$$\ none \ of \ the \ above \ \nonumber$$

Question 5

What is the area under the curve represented by f(x) between the values of x1 and x2 (x1 less than x2)?

A.

$$\int_{x_3}^{y_2} f(x) \ dx \ \nonumber$$

B.

$$(-1) \ \int_{x_2}^{x_1} f(x) \ dx \ \nonumber$$

C.

$$\int_{x_1}^{x_2} f(x) \ dx \ \nonumber$$

D.

$$\int_{x_2}^{x_1} -f(x) \ dx \ \nonumber$$

E.

$$ln(Y) \nonumber$$

F.

$$\ none \ of \ the \ above \ \nonumber$$