UVA Mathematics Department Advisory Calculus Placement Exam Exam C:

1. Evaluate  $$\displaystyle{\int \frac{1}{\sqrt{x^2+2}} \, \mathrm{d}x}$$.

2.  Evaluate $$\displaystyle \int \frac{16}{x^3-4x} \, \mathrm{d}x$$.

3. The region under the graph of $$y = \sin x$$, $$0 \le x \le \pi$$, is rotated 360 degrees about the $$y$$-axis forming a solid of revolution $$S$$.  Find the volume of $$S$$.

4. Solve the differential equation $$\frac{dy}{dx} = y^2x^2 + y^2$$, obtaining an explicit solution.

5.  Show that the series $$\displaystyle \sum_{n=2}^\infty \frac{1}{n(\ln n)^2}$$ converges.

6.  Find the Maclaurin Series for $$\displaystyle f(x) = \frac{x}{3+x^2}$$.  What is the radius of convergence?

7.  Determine the radius and interval of convergence of the power series $$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n 2^n} (x - 1)^{n}$$.

8.   Suppose that $$(x,y)$$ has distance $$r\ge 1$$ from the origin.  Use polar coordinates to show that

$$(|x| + |y|)\ln(x^2 + y^2) \le 4r\ln(r).$$

9.  Find an equation of the line tangent to the curve $$x = 1 + \ln t, y = t^2 + 2$$ at the point $$(1, 3)$$.

10.  Find the length of the curve $$x = e^t -t, y = 4e^{t/2}$$, $$0 \le t \le 2$$.