Difference between revisions of "022 Sample Final A"

Revision as of 11:51, 30 May 2015

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).

@@ Line 3: / Line 3: @@
 == [[022_Exam_2_Sample_B,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]] ==
-<span class="exam">Find the derivative of &thinsp;<math style="vertical-align: -60%">y\,=\,\ln \frac{(x+1)^4}{(2x - 5)(x + 4)}.</math>
+<span class="exam">Find all first and second partial derivatives of the following function, and demostrate that the mixed second partials are equal: <math>\qquad f(x, y) = \frac{2xy}{x-y}</math>
 == [[022_Exam_2_Sample_B,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
-<span class="exam"> Sketch the graph of <math style="vertical-align: -52%">y = \left(\frac{1}{2}\right)^{x + 1} - 4</math>.
+<span class="exam"> 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?
 == [[022_Exam_2_Sample_B,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
-<span class="exam"> Find the derivative of <math style="vertical-align: -18%">f(x) \,=\, 2x^3e^{3x+5}</math>.
+<span class="exam"> Find the antiderivative: <math>\int \frac{6}{x^2 - x - 12}</math>
 == [[022_Exam_2_Sample_B,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
-<span class="exam"> '''Set up the equation to solve. You only need to plug in the numbers-not solve for the particular values!'''
+<span class="exam"> Use implicit differentiation to find <math>\frac{dy}{dx}: \qquad x+y = x^3y^3</math>
-<span class="exam">What is the present value of $3000, paid 8 years from now, in an investment that pays 6%interest,
+== [[022_Exam_2_Sample_B,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
+<span class="exam"> Find producer and consumer surpluses for the following situation:
-::<span class="exam">(a) compounded quarterly?
-::<span class="exam">(b) compounded continuously?
-== [[022_Exam_2_Sample_B,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
+<math>\text{Supply curve: }\qquad p = 18 + 3x^2 \qquad \text{Demand curve: }\qquad p = 150 - 4x</math>
-<span class="exam"> Find the antiderivative of <math>\int \frac{2e^{2x}}{e^2x + 1}\, dx.</math>
 == [[022_Exam_2_Sample_B,_Problem_6|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 6&nbsp;</span>]] ==
-<span class="exam">Find the area under the curve of&thinsp; <math style="vertical-align: -13%">y = 6x^2 + 2x</math> between the <math style="vertical-align: -15%">y</math>-axis and <math style="vertical-align: -1%">x = 2</math>.
+<span class="exam"> Sketch the curve, including all relative exterma and points of inflection. <math>y = 3x^4 - 4x^3</math>
 == [[022_Exam_2_Sample_B,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==
-<span class="exam">Find the antiderivatives:
+<span class="exam">Find the present value of the income stream <math>Y = 20 + 30x</math> from now until 5 years from now, given an interest rate <math>r = 10%.</math> (note that once you plug in the limits of integration, you are finished- you do not need to simplify our answer beyond that step).
-::<span class="exam">(a) <math> \int xe^{3x^2+1}\,dx.</math>
-<br>
-::<span class="exam">(b) <math>\int_2^54x - 5\,dx.</math>
 == [[022_Exam_2_Sample_B,_Problem_8|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 8&nbsp;</span>]] ==
 <span class="exam">
-Find the quantity that produces maximum profit, given demand function <math style="vertical-align: -15%">p = 70 - 3x</math> and cost function&thinsp; <math style="vertical-align: -8%">C = 120 - 30x + 2x^2.</math>
+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)
+<math>f(k, l) = 200k^{0.6}l^{0.4}</math>
 == [[022_Exam_2_Sample_B,_Problem_9|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 9&nbsp;</span>]] ==
-<span class="exam">
+<span class="exam"> Given demamd <math>p = 116 - 3x</math> &nbsp;, and cost <math>C = x^2 + 20x + 64</math>&nbsp;, find:
-Find all relative extrema and points of inflection for the function <math style="vertical-align: -16%">g(x) = x^3 - 3x</math>. Be sure to give coordinate pairs for each point. You do not need to draw the graph. Explain how you know which point is the local minimum and which is the local maximum (i.e., which test did you use?).
+<span class="exam">::a) Marginal revenue when x = 7 units.
+<span class="exam">::b) The quantity(x-value) that produces minimum average cost.
+<span class="exam">::c) Maximum profit (find the x-value and the profit itself)
 == [[022_Exam_2_Sample_B,_Problem_10|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 10&nbsp;</span>]] ==
 <span class="exam">'''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<sup>th</sup> side costs $6 per foot. Find the most economical dimensions of the region (that is, minimize the cost).