UVA Mathematics Department
Advisory Calculus Placement Exam

Exam A:

1. Find the derivative of \(\displaystyle g(x)  =  \frac{x e^{x^ 2 + 1}}{3x + 2}\). Do not simplify your answer.

2.  If \(f(9) = 1, f'(9) = 3, g(1) = 4\), \(g'(1) = -2\), and \(h(x) = x^{3/2} g(f(x))\), then what is \(h'(9)\)?  Simplify your answer.

3.  Suppose \(s(t)  = t^3 - 9t^2 +15t\) is the position of a particle traveling along a coordinate line, where \(s\) is measured in meters and \(t\), in seconds. At what times will the particle be at rest? 

4. Find and classify the critical numbers of \(f(x) = x^5(x-11)^6\), indicating for each critical number whether it yields a relative maximum value of  \(f\), a relative minimum value, or neither.

5. Find \(f(x)\) given that \(f '(x) = 3x^2 + 4x - 1\) and \(f(2) = 9\).

6.  Find the area of the region bounded by the curves \(y = 4-x^2\) and \(y = 0\).

7.  The number of items produced by a manufacturer is given by \( p= 100 xy^3 \),
where \(x\) is the amount of capital and \(y\) is the amount of labor, amounts that change over time. At a particular point in time:

Determine the rate of change in the number of items produced at this point in time.

8.  Find \(\displaystyle{ \int \left(e^x + 1\right)^2 \, dx}\).

9. Evaluate   \( \displaystyle \int_{e}^{e^4} \frac{1}{x\sqrt{\ln x}} \,dx \).

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

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

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?