Calculus II, Spring 2016

Midterm 2

May 10, 2016 Show all details.

1. True or False?

If f(x, y) is differentiable at (0, 0), then f(x, y) is continuous at (0, 0).

2. Evaluate

\frac{d}{dy} ∫_{1}^{2+y^{2}} \frac{cos(xy)}{x}dx.

3. Find the equation of plane normal to the following curve at (1, −1, 1)

\{ \begin{matrix}x^{2} + 2y^{2} + 3z^{2} = 6 \\ x + y + z = 1 \end{matrix}

4. Show that, for all a ∈ ℝ, the point (x_{0}, y_{0}) = (0, 0) is a critical point of the function

f_{a}(x, y) = (a − 1)(x + y)^{2} + (a + 1)(x − 2y)^{2}.

For what values of a is the point (0, 0) a local minimum, a local maximum and a saddle point, respectively?

5. Find absolute maxima and minima of f(x, y) = x^{2} +xy +y^{2} −6x+2 on the rectangular 0 ≤ x ≤ 5, −3 ≤ y ≤ 3.

6. Give an example of a constraint optimization problem that, upon applying the method of Lagrangian multipliers, results in a system of 5 equations with 5 unknowns (and write down the equations). Need not solve it.

7. Use Taylor’s formula to find the quadratic approximation of f(x, y, z) = \frac{1}{1 − x − y + z} near the origin.

8. Evaluate (\frac{∂u}{∂x})_{y} at (x, y, z, w) = (1, 1, 1, 1) where u(x, y, z, w) = x^{2} + y^{2} + z^{2} + w^{2} with the constraint x + y + z + w = 4 and x − y + z − w = 0.

9. Evaluate

∫_{0}^{2} ∫_{y}^{2} x^{2} cos(xy) dxdy

10. Change

∫_{√\overline{2}}^{2} ∫_{√\overline{4−y^{2}}}^{y} dxdy

into an equivalent polar integral and evaluate the polar integral.

11. Is ∫_{1}^{∞} sin \frac{1}{x^{2}} convergent? Explain.

12. Evaluate ∫_{n=0}^{∞} \frac{x^{n}}{n + 2} on |x| < 1 using computational rules of power series.