Question 1

Let $$f(x) = x^2$ e$$^{-x}$$. Which of the following is the correct expression for f$$'(x)$$?

Accepted Answer

$$f'(x) = 2x e$$^{-x}$$ - x^2$ e$$^{-x}$$

Question 2

Evaluate the definite integral $$\$$int_0$$^1 x^2$ e$$^{-x}$$ \, dx$$ using integration by parts.

Accepted Answer

$$2 - 3e$$^{-1}$$

Question 3

Which of the following is the correct formula for integration by parts?

Accepted Answer

$$\int u \, dv = uv - \int v \, du$$

Question 4

Given the vector field $$\vec{F}(x, y) = (y, x)$$, compute $$
abla \cdot \vec{F}.$$

Accepted Answer

$$2$$

Question 5

Let $$\vec{F}(x, y) = (y, x). $$Compute $$
abla 	imes \vec{F} in $$two dimensions.

Accepted Answer

$$0$$

Question 6

Which of the following best describes the geometric meaning of the gradient $$
abla f at a $$point?

Accepted Answer

It points in the direction of the greatest rate of increase of $$f.$$

Question 7

Suppose $$f(x, y) = x^2$ + y^2$. What is the value of $$
abla f$$ at the point (1$$, 2)$$?

Accepted Answer

$$(2, 4)$$

Question 8

Evaluate the line integral $$\int_C (y \, dx + x \, dy)$$ where $$C is $$the straight line from $$(0, 0) to$$ $$(1, 1).$$

Accepted Answer

$$1$$

Question 9

Which of the following is the correct statement of Green's Theorem?

Accepted Answer

$$\oint_C (P \, dx + Q \, dy) = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA$$

Question 10

Let $$\vec{F}(x, y) = (y, x). $$Use Green's Theorem to compute $$\oint_C (y \, dx + x \, dy)$$ where $$C is $$the unit circle centered at the origin.

Accepted Answer

$$\pi$$

Question 11

Suppose $$\vec{F}(x, y) = (y, x)$$ and $$C is $$the boundary of the rectangle with vertices $$(0,0)$$, $$(a,0)$$, $$(a,b)$$, $$(0,b). $$What is $$\oint_C (y \, dx + x \, dy)$$?

Accepted Answer

$$ab$$

Question 12

Which of the following is a necessary condition for a vector field $$\vec{F} to be $$conservative in a simply connected domain?

Accepted Answer

$$
abla 	imes \vec{F} = 0$$

Question 13

Let $$f(x, y) = x^2$ y. $$Compute $$\frac{\partial f}{\partial x} at$$ $$(1, 2).$$

Accepted Answer

$$4$$

Question 14

Suppose $$\vec{F}(x, y) = (2x, 3y). $$Compute $$
abla \cdot \vec{F}.$$

Accepted Answer

$$2 + 3 = 5$$

Question 15

Evaluate $$\$$int_0$$^1 x e$$^{-x}$$ \, dx$$ using integration by parts.

Accepted Answer

$$1 - e$$^{-1}$$

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