21-259  Calculus in 3-D  Spring 2003

Answers to Review #1

1. (a) \sqrt{11}

(b) −1

(c) θ = cos^{−1} (1/\sqrt{11} \sqrt{35})

(d) \frac{10}{11} ⟨1, 3, −1⟩

(e) cos α = 5/\sqrt{35} cos β = −1/\sqrt{35}, cos γ = 3/\sqrt{35} (f) 64

2. (a) 7(x − 3) − 2(y + 6) − 3(z − 8) = 0

(b) \frac{x − 1}{7} = \frac{y − 3}{−2} = \frac{z + 5}{−3};

\{ \begin{matrix}x = 1 + 7t \\ y = 3 − 2t \\ z = −5 − 3t \end{matrix}

3. \frac{x − 1}{6} = \frac{y − 3}{−3} = \frac{z}{−3}

4. d = −2 − 3\sqrt{6} or d = 3\sqrt{6} − 2

5. \sqrt{10}/\sqrt{3}

6. b⃗_{1} = ⟨\frac{4}{5}, 0, \frac{−2}{5}⟩ , b⃗_{2} = ⟨\frac{1}{5}, 2, \frac{2}{5}⟩

7. (\frac{−1}{2}, \frac{7}{4}, \frac{\sqrt{3}}{4}) , (\frac{1}{2}, \frac{1}{4}, \frac{−\sqrt{3}}{4})

8. x − 2y + z = 0

9. \frac{1}{2}

10. \frac{−1}{3} ⟨4, 1, −5⟩.

11. \sqrt{\frac{1}{2}(A^{2} + B^{2} + C^{2})}.

12. (a) T⃗(t) = \frac{1}{\sqrt{2}} ⟨cos 2t, sin 2t, 1⟩ .

N⃗(t) = ⟨− sin 2t, cos 2t, 0⟩.

B⃗(t) = \frac{1}{\sqrt{2}} ⟨− cos 2t, − sin 2t, 1⟩ , κ(t) = e^{−t}

(b) 1

13. 2(8 − 2^{3/2}).

14. (a) 1 (b) DNE

15.

16.

f_{x} = cos(x − +3y) + 2xy^{3}
f_{y} = 3 cos(x + 3y) + 3x^{2}y^{2} + 4y
f_{xx} = − sin(x + 3y) + 2y^{3}
f_{xy} = −3 sin(x + 3y) + 6xy^{2}
f_{yy} = −9 sin(x + 3y) + 6x^{2}y + 4