Math 132/1 calculus quiz

Name: Student ID: Problem 1. Compute ∫_{1}^{2} \frac{dx}{1 − 2x} and simplify. Problem 2. If u(x, y) = e^{y^{2}x}, what is ∂u/∂x and ∂u/∂y? Problem 3. Find a function v(x, y) such that ∂v/∂x = x + y and ∂v/∂y = x − y. Problem 4. What kind of shape (square, line, disk, circle, etc.) is described by the set {(2 + cos θ, sin θ) : 0 ≤ θ ≤ π}? Problem 5. Find all values of x for which the series ∑_{n=0}^{∞}(x−2)^{n} converges? Problem 6. The arc-tangent function tan^{−1}(x) is an anti-derivative of \frac{1}{1+x^{2}} . Based on this fact, or otherwise, compute ∫_{0}^{4} \frac{dy}{1 + 4y^{2}} . Problem 7. Estimate ∫_{0}^{1} x^{2} + \frac{sin(x^{2})}{300} dx to two decimal places. Explain your reasoning.