Question 1

Let $$f(x, y) = x^2$ e$$^{y^2}$$. What is the partial derivative of f$ with respect to y$?

Accepted Answer

$$x^2 2y$$ $$e^{y^2}$$

Question 2

Let $$F(x, y, z) be a $$differentiable function such that the gradient of $$F at $$the point $$(1, -1, 2) is$$ $$
abla F(1, -1, 2) = (3, -2, 4). $$Which of the following is the directional derivative of $$F at$$ $$(1, -1, 2) in $$the direction of the vector $$\mathbf{v} = (2, 1, 2)$$?

Accepted Answer

$$\frac{16}{3}$$

Question 3

Consider the function $$f(x, y) = \frac{1}{x^2$ + y^2$}. $$What is the domain of $$f(x, y)$$?

Accepted Answer

All points in $$\mathbb{R}^2$$ except $$(0, 0)$$

Question 4

Let $$R be $$the region in the $$xy-$$plane bounded by $$y = x^2$ and y = 4$. Which of the following double integrals gives the area of R$?

Accepted Answer

$$\int_{-2}^{2} \int_{x^2$}^{4} dy\,dx$$

Question 5

Let $$\vec{r}(t) = (\cos t, 2\sin t, t^2$)$$ for $$t \in [0, 2\pi]. $$Find $$\frac{d\vec{r}}{dt}$$, the derivative of $$\vec{r}.$$

Accepted Answer

$$(-\sin t, 2\cos t, 2t)$$

Question 6

Which of the following is a critical point for the function $$f(x, y) = 2x^2$ + y^2$ - 4x - 2y$$?

Accepted Answer

$$(2, 1)$$

Question 7

Consider a vector-valued function $$\vec{r}(t)$$ where the unit tangent vector $$\vec{T}$$ and the principal unit normal vector $$\vec{N} at$$ $$t = 1$$ are $$\vec{T}(1) = \left(\frac{3}{5}, \frac{4}{5}\right)$$ and $$\vec{N}(1) = \left(-\frac{4}{5}, \frac{3}{5}\right). $$What is the binormal vector $$\vec{B}(1)$$?

Accepted Answer

$$(0, 0, 1)$$

Question 8

Consider the curve in the $$xy-$$plane given by the equation $$x^2$$ + \$$ln(y^2) = 4$. What is the slope of the line tangent to the curve at the point (0$$, $$e^2$$)$$?

Accepted Answer

$$0.25$$

Question 9

Consider the function $$f(x, y) = \frac{10x^2$ + 2y^2$}{(2x^2$ + y^2$)^2 + 4$$}$$. Evaluate the limit through the path y = 0$ as x$$ 	o 0$.

Accepted Answer

$$0$$

Question 10

Let $$f(x, y) = x^2$ - 4y^2$ + 4y + 4. $$Find all critical points and classify them as local minima, maxima, or saddle points.

Accepted Answer

Saddle point at $$(0, 0)$$; local minimum at $$(1, 1)$$ and $$(-1, -1)$$

Question 11

Evaluate the following double integral: $$\iint_{0 \leq x \leq 1,\ 0 \leq y \leq 2} x^2$ e$$^{xy}$$ dy\,dx.$$

Accepted Answer

$$\frac{e^2$ - 1}{4}$$

Question 12

Let $$R be $$the triangular region in the $$xy-$$plane with vertices $$(0, 0)$$, $$(1, 0)$$, and $$(1, 1). $$Find the absolute maximum and minimum values of $$f(x, y) = 4x - 2y on$$ $$R.$$

Accepted Answer

Maximum: $$4 at$$ $$(1, 0)$$; Minimum: $$-2 at$$ $$(0, 1)$$

Question 13

Find the equation of the plane tangent to the surface $$x^2 - 2y - 2z = 4$ at the point (-2$$, -1, 1)$$.

Accepted Answer

$$-2x - 2y - 4z = -2$$

Question 14

Suppose $$f(x, y) = x^2$ + y^2$ - 4x - 2y + 4. $$What are the coordinates of the critical point, and what type is it?

Accepted Answer

$$(2, 1)$$, local minimum

Question 15

Let $$f(x, y) = x^2$ + y^2$. What is the value of the directional derivative of f$ at (1$$, 2)$$ in the direction of the vector $$\mathbf{u} = (3, 4)$$?

Accepted Answer

$$10/\sqrt{25}$$

Question 16

Which of the following best describes the gradient vector $$
abla f(x, y) at a $$point $$(x_0$, y_0$)$$?

Accepted Answer

It is always perpendicular to the level curve of $$f at$$ $$(x_0$, y_0$)$$

Question 17

Evaluate $$\frac{\partial}{\partial x} (x^2$ y^3$ + e$$^{xy}$$) at $$the point $$(1, 0). $$

Accepted Answer

$$2$$

Question 18

Suppose $$f(x, y) = x^2$ - y^2$. Which of the following statements is true about the critical point at (0$$, 0)$$?

Accepted Answer

$$(0, 0) is a $$saddle point

Study Guide: Multivariable Calculus Exam Topics (Critical Points, Gradients, Multiple Integrals, and Tangent Planes)

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