022 Exam 2 Sample 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 the derivative of  $y\,=\,\ln {\frac {(x+5)(x-1)}{x}}.$

 Problem 2 

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

 Problem 3 

Find the antiderivative of 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 \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  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 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 $x=4$.

 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 $C\,=\,200-30x+x^{2}$.

 Problem 8 

Use differentials to approximate the change in profit given $x=10$  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 $P(x)=-4x^{2}+90x-128$.

 Problem 9 

Find all relative extrema and points of inflection for the function $g(x)={\frac {2}{3}}x^{3}+x^{2}-12x$. 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.

Contributions to this page were made by John Simanyi