Math 324 C&F Vector Calculus Autumn 2004
Assignment $\char93 1$ - Review Problems
Due Monday, October 4

1.

Let ${\bf A} = \langle 3, -2, 0 \rangle$ and ${\bf B} = \langle 0, 5, 1 \rangle$. Compute $\vert{\bf A}\vert$, $\vert{\bf B}\vert$, ${\bf A} \cdot {\bf B}$, and ${\bf A} \times {\bf B}$.

2.

Give parametric equations for the following curves.

(a)

The circle of radius 2 centered at (-1,3).

(b)

The line segment joining (3,-1,0) to (-4, 2, 2).

(c)

The ellipse 12x² + y² = 4.

3.

Draw pictures (in ${\bf R}^3$) of the following.

(a)

The graph of the function z = f(x,y) = 4 - x² - y².

(b)

Let F(x,y,z) = x² - y. Draw the level surface F = 1.

(c)

The intersection of the surfaces given by x² + y² = 4 and y² + z² = 4.

4.

Put the following equations into the form (x - h)² + (y - k)² = r² by completing the square.

(a)

x² + 2x + y² = 0

(b)

y² - 4 = x - x² - y

5.

What's the difference between a point and a vector? In what ways are they related?

6.

Find the equations of the planes described below.

(a)

The plane containing the points (1,1,1), (4, -7, 2), and (0,0,0).

(b)

The plane with normal vector ${\bf n} = \langle 3,2,-1 \rangle$passing through the point (6,-4,1).

7.

Compute the determinant of each of the following matrices.

$$A = \left( \begin{array}{rr} 2 & 1 \ -4 & 2 \end{array} \... ... 6 & 4 & 8 \ 0 & 3 & 0 \ -1 & 1 & 2 \end{array} \right).$$

8.

Here are some integral problems to think about.

(a)

Let $F(x) = \cos(x^2)$. Compute

$$\begin{eqnarray*}\int_0^{\sqrt{\pi}/2} F'(x) dx. \end{eqnarray*}$$

(b)

Compute

$$\begin{eqnarray*}\frac{d}{dx} \int_{\sqrt{x}}^{x^2} \sqrt{t} \sin{t} \; dt. \end{eqnarray*}$$

9.

Chap. 10.3, #18, #19

10.

Chap. 10.3, #24, #25