022 Sample Final A

This is a sample, and is meant to represent the material usually covered in Math 22 for the second exam. An actual test may or may not be similar. Click on the  boxed problem numbers  to go to a solution.

 Problem 1 

Find all first and second partial derivatives of the following function, and demostrate that the mixed second partials are equal: $\qquad f(x,y)={\frac {2xy}{x-y}}$

 Problem 2 

A dairy farmer plans to enclose a rectangular pasture adjacent to a river. To provide enough grass for the her, the pasure must contain 200 square meters of grass. No fencing is required along the river. What dimensions will use the smallest amount of fencing?

 Problem 3 

Find the antiderivative: $\int {\frac {6}{x^{2}-x-12}}$

 Problem 4 

Use implicit differentiation to find ${\frac {dy}{dx}}:\qquad x+y=x^{3}y^{3}$

 Problem 5 

Find producer and consumer surpluses for the following situation:

${\text{Supply curve: }}\qquad p=18+3x^{2}\qquad {\text{Demand curve: }}\qquad p=150-4x$

 Problem 6 

Sketch the curve, including all relative exterma and points of inflection. $y=3x^{4}-4x^{3}$

 Problem 7 

Find the present value of the income stream $Y=20+30x$ from now until 5 years from now, given an interest rate $r=10\%.$ (note that once you plug in the limits of integration, you are finished- you do not need to simplify our answer beyond that step).

 Problem 8 

Find ther marginial productivity of labor and marginal productivity of capital for the following Cobb-Douglas productio function: (note: you must simplify so your solution does not contain negative exponents)

$f(k,l)=200k^{0.6}l^{0.4}$

 Problem 9 

Given demamd $p=116-3x$  , and cost $C=x^{2}+20x+64$ , find:

::a) Marginal revenue when x = 7 units. ::b) The quantity(x-value) that produces minimum average cost. ::c) Maximum profit (find the x-value and the profit itself)

 Problem 10 

Use calculus to set up and solve the word problem: A fence is to be built to enclose a rectangular region of 480 square feet. The fencing material along three sides cost $2 per foot. The fencing material along the 4^th side costs $6 per foot. Find the most economical dimensions of the region (that is, minimize the cost).