022 Sample Final A

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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 for the function ${\displaystyle f(x,y)={\frac {2xy}{x-y}}.}$

Problem 2

A dairy farmer plans to enclose a rectangular pasture adjacent to a river. To provide enough grass for his cows, the fenced pasture must contain 200 square meters of grass. If no fencing is required along the river, what dimensions will use the smallest amount of fencing?

Problem 3

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

Problem 4

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

Problem 5

Find producer and consumer surpluses if the supply curve is given by ${\displaystyle p=18+3x^{2}}$, and the demand curve is given by ${\displaystyle p=150-4x}$.

Problem 6

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

Problem 7

Find the present value of the income stream ${\displaystyle Y=20+30x}$ from now until 5 years from now, given an interest rate ${\displaystyle r=10\%.}$
(Note: 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 marginal productivity of labor and marginal productivity of capital for the following Cobb-Douglas production function:

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

(Note: You must simplify so your solution does not contain negative exponents.)

Problem 9

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

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

Problem 10

Set up the formula to find the amount of money one would have at the end of 8 years if she invests \$2100 in an account paying 6% annual interest, compounded quarterly.

Problem 11

Find the derivative: ${\displaystyle g(x)={\frac {\ln(x^{3}+7)}{(x^{4}+2x^{2})}}}$ .

(Note: You do not need to simplify the derivative after finding it.)

Problem 12

Find the antiderivative: ${\displaystyle \int x^{2}e^{3x^{3}}dx.}$

Problem 13

Use differentials to find ${\displaystyle dy}$ given ${\displaystyle y=x^{2}-6x,~x=4,~dx=-0.5.}$

Problem 14

Find the following limit: ${\displaystyle \qquad \lim _{x\rightarrow \,-3}{\frac {x^{2}+7x+12}{x^{2}-2x-15}}}$.