Spring 1997

Final Exam

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1.

Here is a table of values of a function f(x,y):

X

Y

-1

0

1

1.0

1.2

1.4

-0.5	0.1	0.7
3.6	4.3	4.8
7.7	8.3	8.9

Is f linear? Justify your answer.

2.

If $\vec{u}$ and $\vec{v}$ are perpendicular vectors, then $\vec{u} \cdot \vec{v} = \vert\vert\vec{u}\vert\vert \vert\vert\vec{v}\vert\vert$. True or False? Justify your answer.

3.

If f is differentiable at (a,b), f_x(a,b) = 0 and f_y(a,b) = 0, then the directional derivative of f at (a,b) is zero in any direction. True or False? Justify your answer.

4.

Which of the following diagrams represents the parametric curve $x = \sin t, y = \cos t, 0 \le t \le \pi$?

Justify your answer.

5.

Which of the following is a contour diagram for $f(x,y) = \sin x$?

Justify your answer.

6.

The figure below shows a vector field $\vec{F}$ and an oriented curve C. Is $\int_C \vec{F} \cdot d\vec{r}$ positive, zero, or negative? Justify your answer.

7.

Is $\vec{F}(x,y) = -0.5y \vec{i} + 0.5 x \vec{j}$ a conservative vector field? Justify your answer.

8.

The two planes 3z - 2x + y = 5 and -6z + 4x - 2y = 0 are parallel. True or False? Justify your answer.

9.

The ideal gas law says that

PV = RT

for a certain fixed amount of gas (called a mole of gas), where P is the pressure (in atmospheres), V is the volume (in cubic meters), T is the temperature (in degrees Kelvin), and R is a positive constant.

(a)

Find $\frac{\partial P}{\partial T}$and $\frac{\partial P}{\partial V}$.

(b)

A mole of a certain gas is at a temperature of 298 degrees Kelvin, a pressure of 1 atmosphere, and a volume of 0.0245 m³. Calculate $\frac{\partial P}{\partial V}$ for this gas. Give the units of your answer and explian what it means in practical terms.

10.

(a)

Express $\int_R f(x,y)dA$ as an iterated integral, where R is the region in the xy-plane shown below.

(b)

Evaluate $\int_R e^{-x^2 - y^2} dA$ where R is the circle of radius 2 centered at the origin.

11.

(a)

Write a brief paragraph about Stokes' Theorem. You should include a statement of the theorem, defining any symbols used, and you should explain in words what the theorem says.

(b)

Suppose that $\mbox{curl} \vec{F}$ is parallel to the x-axis and points in the direction of the positive x-axis at every point in three-space. Suppose that C is a circle in the yz-plane, oriented clockwise when viewed from the positive x-axis. Is the circulation of $\vec{F}$ around C positive, zero, or negative? Explain your answer.

12.

Let $f(x,y) = x + y + \frac{1}{x} + \frac{4}{y}$.

(a)

Find the critical points of f.

(b)

Classify the critical points as local maxima, minima, or saddle points.

13.

Consider the vector field

$$\vec{F}(x,y,z) = \frac{x}{(x^2 + y^2 + z^2)^{3/2}}\vec{i} + ... ...ac{z}{(x^2 + y^2 + z^2)^{3/2}}\vec{k}, (x,y,z) \neq (0,0,0).$$

(a)

Calculate the divergence of $\vec{F}$. Simplify your answer.

(b)

Use your answer in part (a) to calculate the flux of $\vec{F}$ out through the sphere of radius 1 centered at the point (0,0,2).

14.

Consider the function f(x,y) = 3x e^x2y.

(a)

Find $\mbox{grad} f(-1,0)$.

(b)

In what direction from the point (-1,0) is f increasing most rapidly? In what direction is the rate of change of f equal to zero?