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:

- the manufacturer has 2 units of capital;
- capital is increasing at a rate of 1 unit per month;
- the manufacturer has 3 units of labor; and
- labor is decreasing at a rate of 1/3 unit per
month.

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?