Difference between revisions of "022 Exam 2 Sample A"

Latest revision as of 21:07, 13 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 the derivative of $y\,=\,\ln {\frac {(x+5)(x-1)}{x}}.$

Problem 2

Find the antiderivative of $y\,=\,3x^{2}-12x+8.$

Problem 3

Find the antiderivative of $\int {\frac {1}{3x+2}}\,dx.$

Problem 4

Find the antiderivative of $\int (3x+2)^{4}\,dx.$

Problem 5

Set up the equation to solve. You only need to plug in the numbers - not solve for particular values!

How much money would I have after 6 years if I invested $3000 in a bank account that paid 4.5% interest,

(a) compounded monthly?

(b) compounded continuously?

Problem 6

Find the area under the curve of $y\,=\,{\frac {8}{\sqrt {x}}}$ between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=4} .

Problem 7

Find the quantity that produces maximum profit, given the demand function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\,=\,90-3x} and cost function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\,=\,200-30x+x^2} .

Problem 8

Use differentials to approximate the change in profit given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 10} units and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dx = 0.2} units, where profit is given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x) = -4x^2 + 90x - 128} .

Problem 9

Find all relative extrema and points of inflection for the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) = \frac{2}{3}x^3 + x^2 - 12x} . Be sure to give coordinate pairs for each point. You do not need to draw the graph.

Problem 10

Use calculus to set up and solve the word problem: Find the length and width of a rectangle that has a perimeter of 48 meters and maximum area.

@@ Line 3: / Line 3: @@
 == [[022_Exam_2_Sample_A,_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: -40%">y\,=\,\ln \frac{(x+5)(x-1)}{x}.</math>
+<span class="exam">Find the derivative of &thinsp;<math style="vertical-align: -42%">y\,=\,\ln \frac{(x+5)(x-1)}{x}.</math>
 == [[022_Exam_2_Sample_A,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
@@ Line 21: / Line 21: @@
 == [[022_Exam_2_Sample_A,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==
-<span class="exam">Find the quantity that produces maximum profit, given the demand function <math>p\,=\,90-3x</math> and cost function <math>C\,=\,200-30x+x^2</math>.
+<span class="exam">Find the quantity that produces maximum profit, given the demand function <math style="vertical-align: -15%">p\,=\,90-3x</math> and cost function <math style="vertical-align: -5%">C\,=\,200-30x+x^2</math>.
 == [[022_Exam_2_Sample_A,_Problem_8|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 8&nbsp;</span>]] ==
 <span class="exam">
-Use differentials to approximate the change in profit given <math>x = 10</math> units and <math>dx = 0.2</math> units. <math>P = -4x^2 + 90x - 128</math>
+Use differentials to approximate the change in profit given <math style="vertical-align: -5%">x = 10</math> units and <math style="vertical-align: 0%">dx = 0.2</math> units, where profit is given by  <math style="vertical-align: -15%">P(x) = -4x^2 + 90x - 128</math>.
 == [[022_Exam_2_Sample_A,_Problem_9|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 9&nbsp;</span>]] ==
 <span class="exam">
-Find all relative extrema and points of inflection for the following function. Be sure to give coordinate pairs for each point. You do not need to draw the graph. <math>g(x) = \frac{2}{3}x^3 + x^2 - 12x</math>
+Find all relative extrema and points of inflection for the function <math style="vertical-align: -45%">g(x) = \frac{2}{3}x^3 + x^2 - 12x</math>. Be sure to give coordinate pairs for each point. You do not need to draw the graph.
 == [[022_Exam_2_Sample_A,_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:''' Find the length and width of a rectangle that has perimeter 48 meters and maximum area.
+'''Use calculus to set up and solve the word problem:''' Find the length and width of a rectangle that has a perimeter of 48 meters and maximum area.