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:

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