21-259  Calculus in 3-D  Spring 2003

Answers to Review Problems for Test 2

1. (a) ∇f(x, y, z) = ⟨y, x + 2yz, y^{2}⟩

(b) \frac{−37}{\sqrt{29}}

2. ∇f(0, 0, 0) = \frac{−5}{\sqrt{41}} ⟨6, −1, 2⟩

3. Tangent Plane: −2(x + 1) + 16(y − 2) − (z − 17) = 0

Normal Line: \frac{x + 1}{−2} = \frac{y − 2}{16} = \frac{z − 17}{−1}

4. ∇f(2, 1) = ⟨−6, 10⟩

5. (2, 1, −2) and (−2, −1, 2)

6. 3(rs)^{2}(r^{2} − s^{2})^{3}r − 6s [(rs)^{3} + 2(r^{2} − s^{2})^{3}] (r^{2} − s^{2})^{2} − 12(2r + s)^{2}

9. 4u\frac{d^{2}w}{du^{2}} + 4\frac{dw}{du}

10. (a) relative min at (−1, 1)

(b) relative min at (0, 0); saddle points at (\frac{1}{2}, −1) and (2, 2)

11. Absolute min occurs at (2,1). Absolute max occurs at (0,4). f(2, 1) = −4 and f(0, 4) = 9

12. (a) Possible maxima and minima at (0, 1, −3), (0, −1, 3) [corresponding to λ = ±\frac{1}{2}]

and at (λ, \frac{1}{2λ}, \frac{−3}{2λ}) with λ = ±\sqrt{\frac{10 + \sqrt{90}}{2}} and λ = ±\sqrt{\frac{10 − \sqrt{90}}{2}}.

(b) Possible maxima and minima at (\frac{\sqrt{2}}{3}, \sqrt{2}) , (\frac{−\sqrt{2}}{3}, −\sqrt{2}) , (−\frac{\sqrt{2}}{3}, \sqrt{2}) ,

(\frac{\sqrt{2}}{3}, −\sqrt{2}),

13. (a) ∫_{0}^{1}∫_{x^{2}}^{x}f(x, y)dydx

(b) ∫_{1}^{e^{4}} ∫_{\frac{1}{2} lny}^{2}f(x, y)dxdy

(c) ∫_{0}^{4} ∫_{0}^{y/2}f(x, y)dxdy

14. (a) \frac{2}{9}
(b) \frac{(2^{3/2} − 1)}{4}