# UVA Mathematics Department 1320/2310 Diagnostic

This diagnostic is designed to provide feedback to a) help you decide the best course to start in and b) how we can best support your learning. Some suggestions:

• The diagnostic is self-paced. You will want to set aside enough time to work through each of the 10 questions in one sitting.
• Please refrain from using calculators as you will not be able to do so when taking exams in our courses.
• This diagnostic will not confer academic credit, so relax and do your best!

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 $$x = e^t -t, y = 4e^{t/2}$$, $$0 \le t \le 2$$.