Q: Which of the following is the correct definition of the limit of a vector-valued function $$\vec{r}(t) as$$ $$t$$ approaches $$a$$?

$$\lim_{t \to a} \vec{r}(t) = \vec{L} if$$ $$\lim_{t \to a} |\vec{r}(t) - \vec{L}| = 0$$

Q: Which of the following is a correct way to compute the limit $$\lim_{t \to a} \vec{r}(t)$$ for $$\vec{r}(t) = (x(t), y(t), z(t))$$?

Compute $$\lim_{t \to a} x(t)$$, $$\lim_{t \to a} y(t)$$, and $$\lim_{t \to a} z(t)$$ separately.

Question 1

Given the equation $$x^2 - 8x$$ + $$y^2 + 6y$$ = $$z^2 - 10z$, what type of surface does this equation represent?

Accepted Answer

A cone

Question 2

Which of the following is a correct parameterization for the line segment between points $$Q$$ and $$R in $$space?

Accepted Answer

$$\vec{r}(t) = Q + t\vec{v}$$ for $$0 \leq t \leq 1$$

Question 3

Which of the following vector-valued functions traces out a circle of radius 1 centered at the origin in the $$xy-$$plane?

Accepted Answer

$$\vec{r}(t) = (\cos(t), \sin(t))$$

Question 4

Which of the following parameterizations traces out a spiral in three dimensions?

Accepted Answer

$$\vec{s}(t) = (\cos(t), \sin(t), t)$$

Question 5

Given $$\vec{r}(t) = (t^2$, t^2$ + 3, t^2$ - 1)$$, what type of curve does this parameterize?

Accepted Answer

A ray (half-line)

Question 6

Which of the following is the correct definition of the limit of a vector-valued function $$\vec{r}(t) as$$ $$t$$ approaches $$a$$?

Accepted Answer

$$\lim_{t 	o a} \vec{r}(t) = \vec{L} if$$ $$\lim_{t 	o a} |\vec{r}(t) - \vec{L}| = 0$$

Question 7

Evaluate $$\lim_{t 	o 2} (t^2$ - 4t + 4, t + 1, 7). $$What is the result?

Accepted Answer

$$(0, 3, 7)$$

Question 8

Which of the following vector-valued functions is contained in the line $$x = y = z$$?

Accepted Answer

$$\vec{r}(t) = (t^2$, t^2$, t^2$)$$

Question 9

A spiral is parameterized by $$\vec{r}(t) = 4\cos t\,\mathbf{i} + \sin t\,\mathbf{j} + \frac{t}{2\pi}\,\mathbf{k}. On $$what surface does this spiral lie?

Accepted Answer

$$\frac{x^2$}{16} + y^2$ = 1$$

Question 10

Which of the following parameterizations traces out an ellipse in the $$xy-$$plane?

Accepted Answer

$$\vec{c}(t) = (a\cos t, b\sin t)$$

Question 11

Suppose $$\vec{r}(t) = (t+3, 3t-1, 7). $$What type of curve does this parameterize?

Accepted Answer

A line

Question 12

Which of the following statements is true about the orientation of the curve $$\vec{r}(t) = (\cos t, \sin t)$$ for $$0 \leq t \leq 2\pi$$?

Accepted Answer

It traces the unit circle in a counterclockwise direction.

Question 13

Given the vector-valued function $$\vec{r}(t) = (\cos(e^t), \sin(e^t), e^t)$$, what is the general shape of the curve?

Accepted Answer

A spiral

Question 14

Which of the following is a correct way to compute the limit $$\lim_{t 	o a} \vec{r}(t)$$ for $$\vec{r}(t) = (x(t), y(t), z(t))$$?

Accepted Answer

Compute $$\lim_{t 	o a} x(t)$$, $$\lim_{t 	o a} y(t)$$, and $$\lim_{t 	o a} z(t)$$ separately.

Question 15

A slinky curve is given by $$\vec{r}(t) = (A(t)\cos t, A(t)\sin t, 15t)$$ where $$A(t) = 3 + \cos 15t. $$What is the projection of this curve onto the $$xy-$$plane?

Accepted Answer

$$\vec{p}(t) = (A(t)\cos t, A(t)\sin t)$$

Question 16

Which of the following is NOT a valid parameterization for a line passing through point $$P$$ and parallel to vector $$\vec{v}$$?

Accepted Answer

$$\vec{r}(t) = P + t\vec{u}$$ where $$\vec{u} is $$not parallel to $$\vec{v}$$

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