# 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:

• 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?