022 Exam 2 Sample B

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  $y\,=\,\ln {\frac {(x+1)^{4}}{(2x-5)(x+4)}}.$

 Problem 2 

Sketch the graph of $y=\left({\frac {1}{2}}\right)^{x+1}-4$.

 Problem 3 

Find the derivative: $f(x)\,=\,2x^{3}e^{3x+5}$.

 Problem 4 

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

What is the present value of $3000, paid 8 years from now, in an investment that pays 6%interest,

(a) compounded quarterly?

(b) compounded continuously?

 Problem 5 

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

 Problem 6 

Find the area under the curve of  $y=6x^{2}+2x$ between the $y$-axis and $x=2$.

 Problem 7 

Find the antiderivatives:

(a) $\int xe^{3x^{2}+1}\,dx.$

(b) $\int _{2}^{5}4x-5\,dx.$

 Problem 8 

Find the quantity that produces maximum profit, given demand function $p=70-3x$ and cost function  $C=120-30x+2x^{2}.$

 Problem 9 

Find all relative extrema and points of inflection for the function $g(x)=x^{3}-3x$. 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 min and which is the max (i.e., which test did you use?).

 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).