UVA Mathematics Department
Advisory Calculus Placement Exam

Exam B:

1. Find the derivative of \(f(x)  =  \frac{ \sin^2 x}{\sqrt{1+x^2}}\). Do not simplify your answer.

2. If \(f(9) = 1, f'(9) = 3, g(1) = 0,  g'(1) = -2\), and \(h(x) = x^{3/2} e^{g(f(x))}\), then what is \(h'(9)\)?  Simplify your answer.

3. Below is a portion of the graph of a function \(f\).

Plot of f

For the following, give all values of \(a\) in the interval \((-5,4)\) satisfying the given condition. If there are none, write "none". No work is required.

(a) all \(a\) not in the domain of \(f\): 

(b) all \(a\) such that \(f\) is not continuous at \(a\):

(c) all \(a\) such that  \( \displaystyle \lim_{x \rightarrow a}f(x)\) does not exist: 

(d)  all \(a\) at which \(f\) is not differentiable:  

4. Find and classify the critical numbers of \(f(x) =x^{4/5}(x-4)^2\), indicating for each critical number whether it yields a local maximum value of \(f\), a local minimum value, or neither.

5.  Find the area of the region bounded by the curves \(y = x^2\) and \(y =x + 2\).

6.  A canister is dropped from a helicopter 500 m above the ground. Its parachute does not open, but the canister has been designed to withstand an impact speed of 120 m/s. Will it burst? Assume that after it is dropped, the canister experiences (only) the acceleration due to gravity, approximately 10 m/s\(^2\).

7.  Find \(\displaystyle{ \int  \frac{e^x}{1+e^x} \, dx}\).

8.   Evaluate   \(\displaystyle{\int_0^{\pi/12} \tan^3(3x)\sec^2(3x)\, dx}\).

9. Over the time interval \(1 \le t \le 100\) the temperature of a freezer compartment is given by

\(f(t) = \frac{t - 4}{t^2}\),

where \(t\) is measured in hours and \(f(t)\) is measured in degrees Celsius.
What's the maximum temperature of the freezer over this time interval?

10. Both Rugby Road and Madison Lane meet University Avenue at right angles. (See the diagram below.)  Dolly is biking on University Ave at 9 ft/sec and has passed Madison Ln heading toward Rugby. James is biking away from University Ave along Rugby traveling at 8 ft/sec.  At what rate is  the distance between Dolly and James changing at the instant when James is 40 ft from and Dolly is 30 ft from the intersection of Rugby Rd and University Ave?   Is the distance between them increasing or decreasing at this instant?  

Roads image |_| , with | = Rugby _ =
            Univeristy, | = Madison