$$\kappa = \frac{|v \times a|}{|v|^3}$$

Question 1

Given the vector function $$r(t) = \langle 3\cos(t), 3\sin(t), 4t \rangle$$, what is the curvature $$\kappa(t) at $$any point $$t$$?

Accepted Answer

$$\frac{3}{25}$$

Question 2

Which formula correctly expresses the arc length $$L of a $$vector function $$r(t) = \langle f(t), g(t), h(t) \rangle$$ from $$t = a to$$ $$t = b$$?

Accepted Answer

$$L = \int_a^b \sqrt{f'(t)^2$ + g'(t)^2$ + h'(t)^2$} \, dt$$

Question 3

For the function $$f(x, y) = \ln(x + y)$$, what is the domain?

Accepted Answer

All $$(x, y)$$ such that $$x + y \u003e 0$$

Question 4

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

Accepted Answer

All $$(x, y) on $$and outside the circle of radius 1 centered at the origin

Question 5

The graph of $$f(x, y) = \sqrt{x^2$ + y^2$} is $$best described as:

Accepted Answer

The top half of a cone

Question 6

Which of the following is the correct formula for curvature $$\kappa$$ using velocity $$v$$ and acceleration $$a$$?

Accepted Answer

$$\kappa = \frac{|v 	imes a|}{|v|^3}$$

Question 7

What kind of curves are the level curves of $$f(x, y) = \sqrt{y - x^2$}$$ for $$c \geq 0$$?

Accepted Answer

Parabolas

Question 8

For the function $$F(x, y, z) = x^2$ + y^2$ + z^2$, what are the level surfaces F(x$$, y, z) = c$$ for c$$ \u003e 0$$?

Accepted Answer

Spheres centered at the origin

Question 9

What is the shape of the level curves of $$f(x, y) = 5 + e$$^{2x - 3y}$$?

Accepted Answer

Lines

Question 10

Suppose $$f(x, y) = x^2$ + y^2$. Which of the following best describes its level curves?

Accepted Answer

Circles centered at the origin

Question 11

A topographical map is an example of which mathematical concept?

Accepted Answer

Level curves

Question 12

If $$r(t) is a $$smooth parameterized curve, which of the following is the unit tangent vector $$T(t)$$?

Accepted Answer

$$T(t) = \frac{r'(t)}{|r'(t)|}$$

Question 13

Which of the following best describes the range of a function $$f(x, y)$$?

Accepted Answer

The set of real numbers $$z$$ that $$f(x, y)$$ assumes as $$(x, y)$$ varies over the domain

Question 14

Which of the following is NOT a possible domain for a function of two variables?

Accepted Answer

The set of all points $$(x, y)$$ such that $$x^2$$ + $$y^2$$ \u003c 0$$

Question 15

Which of the following best describes the graph of $$f(x, y) = x + y$$?

Accepted Answer

A plane in $$\mathbb{R}^3$$

Question 16

If $$r(t) = \langle t, t^2$, t^3$ \rangle$$, what is $$|r'(t)|$$?

Accepted Answer

$$\sqrt{1 + 4t^2$ + 9t^4$}$$

Calc 9: Arc Length, Curvature, and Functions of Several Variables

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