Calculus II, Spring 2014

Midterm 2

May 08, 2014

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1. Evaluate

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

2. True or false? Prove it or give a counter example.

Assume f(x, y), f_{x}(x, y) and f_{y}(x, y) are all continuous in R^{2}. Let C = {(x, y), f(x, y) = f(0, 0)} and T be a tangent vector of C at (0, 0). Then ∇f(0, 0) · T = 0.

3. Use Lagrangian multipliers (and only Lagrangian multipliers) to find extreme values of f(x, y, z) = xy + 2z^{2} on

\{ \begin{matrix}x^{2} + y^{2} + z^{2} = 9 \\ x − y = 0 \end{matrix}

4. Find the absolute maximum and minimum of f(x, y) = 2 + 2x + 2y − x^{2} − y^{2} in the region bounded by x = 0, y = 0 and x + y = 6.

5. Let f(x, y) = x^{3} + y^{3} and g(r, θ) = f(r cos θ, r sin θ). Evaluate ∂_{r}^{2}g + (∂_{r}g)/r + (∂_{θ}^{2}g)/r^{2}

6. Derive the Taylor expansion of f(x, y, z) around (0, 0, 0) up to quadratic terms of x, y and z and an expression of the remainder term, R_{2}. You may assume that f and all its first and second derivatives are all continuous in R^{3}.

7. Evaluate (\frac{∂U}{∂P})_{V} and (\frac{∂U}{∂T})_{V} at (P, V, T) = (1, 2, 2) where U(P, V, T) = T exp(−P/V) with the constraint P V = T.

8. Evaluate ∫_{0}^{2} ∫_{y}^{2} \frac{sin x}{x}dxdy.

9. Let f(x, y) = \frac{x^{3} + y^{3}}{x^{2} + y^{2}} , for (x, y) ≠ (0, 0) and f(0, 0) = 0. P = (0, 0) and u^{θ} = (cos θ, sin θ),

θ ∈ [0, 2π].

(a) Is f continous at (0, 0)? Explain.

(b) For fixed θ, write down the definition of the directional derivative (\frac{df}{ds})_{u^{θ},P} and evaluate

it.

(c) Does f have a linear approximation at (0, 0)? Explain.