Q: A vector function $$\vec{r}(t) is $$continuous at $$t = t_0$ if:

$$\lim_{t \to t_0$} \vec{r}(t) = \vec{r}(t_0$)$$

Q: What is the maximum height $$y_{\text{max}}$$ reached by a projectile launched from the origin with initial speed $$v_0 at an $$angle $$\alpha$$?

$$y_{\text{max}} = \frac{v_0$^2 \sin^2$ \alpha}{2g}$$

Question 1

Which of the following best describes the position vector $$\vec{r}(t) of a $$particle moving in space?

Accepted Answer

$$\vec{r}(t) is a $$vector function that gives the position of the particle as a function of time.

Question 2

Given $$\vec{r}(t) = f(t)\vec{i} + g(t)\vec{j} + h(t)\vec{k}$$, which of the following is the correct formula for the derivative $$\vec{r}'(t)$$?

Accepted Answer

$$\vec{r}'(t) = f'(t)\vec{i} + g'(t)\vec{j} + h'(t)\vec{k}$$

Question 3

A vector function $$\vec{r}(t) is $$continuous at $$t = t_0$ if:

Accepted Answer

$$\lim_{t 	o t_0$} \vec{r}(t) = \vec{r}(t_0$)$$

Question 4

Which of the following statements about the velocity vector $$\vec{v}(t) is $$correct?

Accepted Answer

$$\vec{v}(t) is $$the derivative of the position vector $$\vec{r}(t)$$ with respect to time.

Question 5

If a particle moves on a sphere such that its position vector $$\vec{r}(t) is $$always of constant length, which of the following must be true?

Accepted Answer

$$\vec{r}(t) \cdot \vec{r}'(t) = 0$$

Question 6

Which rule correctly describes the derivative of the dot product of two differentiable vector functions $$\vec{u}(t)$$ and $$\vec{v}(t)$$?

Accepted Answer

$$\frac{d}{dt}[\vec{u}(t) \cdot \vec{v}(t)] = \vec{u}'(t) \cdot \vec{v}(t) + \vec{u}(t) \cdot \vec{v}'(t)$$

Question 7

A projectile is launched from the origin with initial speed $$v_0 at an $$angle $$\alpha$$ above the horizontal. What is the formula for its position vector $$\vec{r}(t)$$ under ideal projectile motion (neglecting air resistance)?

Accepted Answer

$$\vec{r}(t) = (v_0$ \cos \alpha) t \vec{i} + \left[(v_0$ \sin \alpha) t - \frac{1}{2}gt^2$\right] \vec{j}$$

Question 8

What is the maximum height $$y_{	ext{max}}$$ reached by a projectile launched from the origin with initial speed $$v_0 at an $$angle $$\alpha$$?

Accepted Answer

$$y_{	ext{max}} = \frac{v_0$^2 \sin^2$ \alpha}{2g}$$

Question 9

Which of the following integrals gives the arc length $$L of a $$smooth curve $$\vec{r}(t) = x(t)\vec{i} + y(t)\vec{j} + z(t)\vec{k}$$ for $$t in$$ $$[a, b]$$?

Accepted Answer

$$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\$$right)^2$$ + \left(\frac{dy}{dt}\$$right)^2$$ + \left(\frac{dz}{dt}\$$right)^2$$} dt$$

Question 10

Suppose $$\vec{r}(t) = (t^2$)\vec{i} + (2t)\vec{j} + (3)\vec{k}. $$What is the velocity vector $$\vec{v}(t)$$?

Accepted Answer

$$\vec{v}(t) = (2t)\vec{i} + (2)\vec{j} + (0)\vec{k}$$

Question 11

A particle moves along a curve with position vector $$\vec{r}(t). $$Its speed at time $$t is $$given by:

Accepted Answer

$$|\vec{r}'(t)|$$

Question 12

Which of the following is the correct formula for the definite integral of a vector function $$\vec{r}(t) = f(t)\vec{i} + g(t)\vec{j} + h(t)\vec{k}$$ from $$a to$$ $$b$$?

Accepted Answer

$$\int_a^b \vec{r}(t) dt = \left( \int_a^b f(t) dt \right)\vec{i} + \left( \int_a^b g(t) dt \right)\vec{j} + \left( \int_a^b h(t) dt \right)\vec{k}$$

Question 13

A ball is thrown from the ground with an initial speed of $$20 m/s at an $$angle of $$30^\circ$$ above the horizontal. What is the total flight time? (Assume $$g = 9.8 m/s$^2)$$

Accepted Answer

$$t = \frac{2 	imes 20 	imes \sin 30^\circ}{9.8}$$

Question 14

Which of the following best describes the chain rule for differentiating a vector function $$\vec{r}(f(t))$$?

Accepted Answer

$$\frac{d}{dt} \vec{r}(f(t)) = \vec{r}'(f(t)) \cdot f'(t)$$

Question 15

If $$\vec{r}(t) is a $$differentiable vector function and $$C is a $$constant vector, what is $$\frac{d}{dt}(C)$$?

Accepted Answer

$$0$$

Question 16

A particle moves along a curve with velocity $$\vec{v}(t). $$The arc length from $$t = a to$$ $$t = b is $$given by:

Accepted Answer

$$\int_a^b |\vec{v}(t)| dt$$

Vector-Valued Functions and Motion in Space: Curves, Tangents, Integrals, and Arc Length

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