Q: For the curve $$\vec{r}(t) = (1 + t^2$)\vec{i} + (2t - 3)\vec{j} + 6t\vec{k}$$, write the acceleration in the form $$a_T \vec{T} + a_N \vec{N}.$$

$$a_T = \frac{d}{dt} |\vec{v}|$$, \ a_N = \frac{|\vec{v} \times \vec{a}|}{|\vec{v}|}$

Question 1

Which of the following best describes the set of points satisfying $$x^2$$ + $$y^2 = 48$ in the plane z = 1$?

Accepted Answer

A circle of radius $$\sqrt{48} in $$the plane $$z = 1$$

Question 2

Given the equation $$x^2$$ + $$y^2$$ + $$z^2 = 9$, which of the following is a correct geometric description?

Accepted Answer

A sphere of radius $$3$$ centered at the origin

Question 3

Find the distance between the points $$P_1 = (8$$, 5, -1)$$ and P_2$ = (9, 6, -2). $$Which formula should you use?

Accepted Answer

$$\sqrt{(x_2$ - x_1$)^2$$ + ($$y_2$$ - $$y_1)^2$ + (z_2$ - z_1$)^2$$}$

Question 4

Given vectors $$\vec{v} = 5\vec{i} - 8\vec{j} + 7\vec{k}$$ and $$\vec{u} = -5\vec{i} + 4\vec{j} + 10\vec{k}$$, what is $$\vec{v} + \vec{u}$$?

Accepted Answer

$$0\vec{i} - 4\vec{j} + 17\vec{k}$$

Question 5

Express the vector $$\vec{a} = 2\vec{i} + 8\vec{j} - 3\vec{k} as a $$product of its length and direction.

Accepted Answer

$$\sqrt{77} \left( \frac{2}{\sqrt{77}}\vec{i} + \frac{8}{\sqrt{77}}\vec{j} - \frac{3}{\sqrt{77}}\vec{k} \right)$$

Question 6

Find the projection of $$\vec{v} = 7\vec{i} - \vec{j} + 3\vec{k}$$ onto $$\vec{u} = 3\vec{i} - 9\vec{j}.$$

Accepted Answer

Question 7

Given the parametric equations $$x = -2 + 2t$$, $$y = -5 + 6t$$, $$z = 0 + t$$, what is the direction vector of the line?

Accepted Answer

$$(2, 6, 1)$$

Question 8

Find the equation of the plane through the point $$(7, -6, 2)$$ and parallel to the plane $$3x - 5y + 4z = 6.$$

Accepted Answer

$$3x - 5y + 4z = 59$$

Question 9

Evaluate the integral $$\$$int_0$$^1 \left( 2e$$^{2t}$$ - 3t \right) dt.$$

Accepted Answer

Question 10

A projectile is fired at a speed of $$720 m/sec at an $$angle of $$45^\circ. $$How long will it take to fall $$1.5 km $$downward? (Assume $$g = 9.8 m/s$^2)$$

Accepted Answer

$$445 s$$

Question 11

Find the arc length of the curve $$\vec{r}(t) = 6t\vec{i} + 9t^2$\vec{j} + 2t^3$\vec{k}$$ from $$t = 0 to$$ $$t = 1.$$

Accepted Answer

$$\$$int_0$$^1 \sqrt{36 + 324t^2$ + 36t^4$} dt$$

Question 12

Find the unit tangent vector $$\vec{T}$$ for the curve $$\vec{r}(t) = (10t)\vec{i} + (2t)\vec{j} + (3t)\vec{k}.$$

Accepted Answer

$$\frac{1}{\sqrt{113}} (10\vec{i} + 2\vec{j} + 3\vec{k})$$

Question 13

For the curve $$\vec{r}(t) = (1 + t^2$)\vec{i} + (2t - 3)\vec{j} + 6t\vec{k}$$, write the acceleration in the form $$a_T \vec{T} + a_N \vec{N}.$$

Accepted Answer

$$a_T = \frac{d}{dt} |\vec{v}|$$, \ a_N = \frac{|\vec{v} 	imes \vec{a}|}{|\vec{v}|}$

Question 14

Find the principal unit normal vector $$\vec{N}$$ for $$\vec{r}(t) = (\sin t)\vec{i} + (\cos t)\vec{j}.$$

Accepted Answer

$$-\cos t\vec{i} + \sin t\vec{j}$$

Question 15

Find the curvature $$\kappa of $$the space curve $$\vec{r}(t) = t\vec{i} + t^2$\vec{j} at$$ $$t = 1.$$

Accepted Answer

$$\kappa = \frac{1}{2}$$

Question 16

Given the velocity vector $$\vec{v}(t) = -8t\vec{j} + 20t\vec{k}$$, what is the position vector $$\vec{r}(t) if$$ $$\vec{r}(0) = \vec{0}$$?

Accepted Answer

$$\vec{r}(t) = -4t^2$\vec{j} + 10t^2$\vec{k}$$

Question 17

Find the intersection of the planes $$3x + 4y - 10z = 2$$ and $$2x - 2y + 2z = 9.$$

Accepted Answer

$$x = -2 + 2t, y = -5 + 6t, z = 0 + t$$

Question 18

Evaluate the triple scalar product $$\vec{a} \cdot (\vec{b} 	imes \vec{c})$$ for $$\vec{a} = 5\vec{i} + 2\vec{j} - 3\vec{k}$$, $$\vec{b} = 6\vec{i} - 7\vec{j} + 5\vec{k}$$, $$\vec{c} = 7\vec{i} - 5\vec{j} + 9\vec{k}.$$

Accepted Answer

$$73$$

Question 19

A particle moves along $$\vec{r}(t) = (2t)\vec{i} + (t^2$)\vec{j}. $$What is the speed at $$t = 2$$?

Accepted Answer

$$\sqrt{(2)^2$ + (4)^2$} = \sqrt{20}$$

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