Q: Given $$\vec{r}(t) = f(t)\vec{i} + g(t)\vec{j} + h(t)\vec{k}$$, which of the following statements is true about the limits of $$\vec{r}(t) as$$ $$t \to a$$?

$$\lim_{t \to a} \vec{r}(t) = \lim_{t \to a} f(t)\vec{i} + \lim_{t \to a} g(t)\vec{j} + \lim_{t \to a} h(t)\vec{k}$$

Question 1

Which of the following best describes a vector function $$\vec{r}(t) in $$the context of space curves?

Accepted Answer

A function whose domain is a set of real numbers and whose range is a set of vectors.

Question 2

Given $$\vec{r}(t) = f(t)\vec{i} + g(t)\vec{j} + h(t)\vec{k}$$, which of the following statements is true about the limits of $$\vec{r}(t) as$$ $$t 	o a$$?

Accepted Answer

$$\lim_{t 	o a} \vec{r}(t) = \lim_{t 	o a} f(t)\vec{i} + \lim_{t 	o a} g(t)\vec{j} + \lim_{t 	o a} h(t)\vec{k}$$

Question 3

Suppose $$\vec{r}(t) = (1 + t^2$)\vec{i} + e$$^{t^2}$$\vec{j} + \sin t\vec{k}. $$What is $$\lim_{t 	o 0} \vec{r}(t)$$?

Accepted Answer

$$1\vec{i} + 1\vec{j} + 0\vec{k}$$

Question 4

Which of the following is the correct vector equation for a space curve given by $$x = f(t)$$, $$y = g(t)$$, $$z = h(t)$$?

Accepted Answer

$$\vec{r}(t) = f(t)\vec{i} + g(t)\vec{j} + h(t)\vec{k}$$

Question 5

Given the vector function $$\vec{r}(t) = \cos t\vec{i} + \sin t\vec{j} + t\vec{k}$$, what type of curve does this represent in space?

Accepted Answer

A helix winding around the $$z-$$axis.

Question 6

If $$\vec{r}(t) = t\vec{i} + t^2$\vec{j} + t^3$\vec{k}$$, which of the following points lies on the curve when $$t = 2$$?

Accepted Answer

$$(2, 4, 8)$$

Question 7

Which of the following is a necessary condition for the limit $$\lim_{t 	o a} \vec{r}(t) to $$exist?

Accepted Answer

The limits of all component functions $$f(t)$$, $$g(t)$$, and $$h(t)$$ must exist as $$t 	o a.$$

Question 8

A curve in space is defined by the intersection of the cylinder $$x^2$$ + $$y^2 = 1$ and the plane y + z = 2$. Which vector function represents this curve?

Accepted Answer

$$\vec{r}(t) = \cos t\vec{i} + \sin t\vec{j} + (2 - \sin t)\vec{k}$$

Question 9

Suppose $$\vec{r}(t) = f(t)\vec{i} + g(t)\vec{j} + h(t)\vec{k}$$, where $$f$$, $$g$$, and $$h$$ are continuous on $$[a, b]. $$What does the set of all points $$\vec{r}(t)$$ for $$t in$$ $$[a, b]$$ represent?

Accepted Answer

A space curve.

Question 10

Which of the following is NOT a correct way to write the vector function $$\vec{r}(t)$$ for a space curve?

Accepted Answer

$$\vec{r}(t) = f(t) + g(t) + h(t)$$

Question 11

If $$\vec{r}(t) = \cos t\vec{i} + \sin t\vec{j} + t\vec{k}$$, what is the projection of the curve onto the $$xy-$$plane?

Accepted Answer

A circle of radius 1 centered at the origin.

Question 12

Which of the following real-world phenomena can be modeled by a vector function in three dimensions?

Accepted Answer

The path of a drone flying through the air.

Question 13

Given $$\vec{r}(t) = t\vec{i} + \sin t\vec{j} + \cos t\vec{k}$$, which of the following describes the motion of a particle following this path?

Accepted Answer

The particle oscillates in the $$y$$ and $$z$$ directions while moving linearly in $$x.$$

Question 14

If $$\vec{r}(t) = t\vec{i} + t^2$\vec{j} + t^3$\vec{k}$$, what is the $$z-$$coordinate of the point on the curve when $$t = -1$$?

Accepted Answer

$$-1$$

Question 15

Which of the following is a correct interpretation of the vector function $$\vec{r}(t) = f(t)\vec{i} + g(t)\vec{j} + h(t)\vec{k}$$?

Accepted Answer

It gives the position of a particle in space as a function of time.

Question 16

Suppose $$\vec{r}(t) = \cos t\vec{i} + \sin t\vec{j} + t\vec{k}. $$What is the value of $$\vec{r}(\pi)$$?

Accepted Answer

$$(-1, 0, \pi)$$

Question 17

Which of the following is a plausible application of vector functions in engineering?

Accepted Answer

Modeling the trajectory of a satellite in orbit.

Question 18

If $$\vec{r}(t) = \cos t\vec{i} + \sin t\vec{j} + t\vec{k}$$, what is the $$y-$$coordinate when $$t = \frac{\pi}{2}$$?

Accepted Answer

$$1$$

Section 3.1 - Vector Functions and Space Curves

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