Ayyy yuh uh huh on g - GM (good mornin)

Dots and Crosses

100

u = <3, -1, 4> and v = <-2, 5, 1>.

Calculate the dot product u⋅v.

-7

200

a = i + 2j - k and b = 3i - j + 2k.

Calculate the cross product a x b.

<3, -5, -7>

300

a = <4,-8,5>, b = <-3,2,3>, P(-4,6,3)

Write a formula for a plane (in general form) made by the two vectors a,b and contains the point P.

34x + 27y + 16z = 74

400

Find the area of the triangle with vertices P(1,0,-1), Q(3,1,2), and R(2,3,0).

(3*sqrt(10))/2

500

Let |u| = 3, |v| = 5, and u ⋅ v = 9. Find the value of |u x v|.

12

Name that Shape

100

What shape do the xy, yz, and xz traces look like for the surface 4x² + y² + 9z² = 9.

Circles

200

What type of quadric surface is z² = 4x² + 9y²?

Cone

300

With a hyperboloid of 2 sheets in the form -x² - y² + z² = 1, what shapes are the 3 traces? 

xy - Circles

yz and xz - Hyperboloids

400

Gimme a contour plot of f(x,y) = sqrt(16 - x² - y²).

Circles getting smaller

500

Draw the 3 traces for the surface z = y²/4 - x² (for each trace, let k=-1,0,1).

Ch12 Notes

Bombs Away

100

Acceleration vector of any projectile motion problem

a(t) = <0,-g>

200

Velocity and position vectors of any given acceleration problem (use |V0| for speed and (x0, y0) for initial position)

v(t) = <|V0|cos(θ), |V0|sin(θ) - gt>

r(t) = <|V0|cos(θ)t + x0, |V0|sin(θ)t - .5gt^2 + y0>

300

A cannonball is launched from the ground at an angle of 30 degrees at 20 m/s. How far does the ball go before it hits the ground? (Assume g= 9.8)

35.3 meters

400

A cannonball is launched at an angle of 60 degrees at a speed of 10 m/s. What is the maximum height of the ball? (Assume g=9.8)

3.83m

500

An archer stands on a ledge 10m above the ground and wants to hit a target 50m away that is 15m above the ground. If they know that they shoot at 25 m/s, what launch angle would allow them to hit the target? (Assume g=9.8)

Hint: 1/cos(θ)^2 = 1 + tan(θ)^2

Hint: Use quadratic formula with tan(θ) instead of x

33.65 or 62.06

I've Reached my Limit

100

Find the limit as (x,y) ->(2,-1) of the function, or prove DNE:

f(x,y) = (3x²y + 2x)/(y²+5)

-4/3

200

Find the limit as (x,y) -> (3,3) of the function, or prove DNE:

f(x,y) = (x²-y²) / (x-y)

6

300

Find the limit as (x,y) -> (0,0) of the function, or prove DNE:

f(x,y) = x² * sin(1/(x²+y²))

0 (Use squeeze thm)

400

Find the limit as (x,y) -> (0,0) of the function, or prove DNE:

f(x,y) = (x³y) / (x²+y²) 

0 (prove with squeeze)

500

Find the limit as (x,y) -> (0,0) of the function, or prove DNE:

f(x,y) = xy²cos(y) / (x² + 4y⁴)

DNE (x=y² and x=2y²)

Vectorrrr

100

Give me a symmetric equation that represents a line between the points A(4,1,-7) and B(2,-6,3)

(x-4)/2 = (y-1)/-7 = (z+7)/10

200

Find the vector projection of b onto a given a = <6, -5, 2> and b = <9, 0, 1>.

<336/65, -280/65, 112/65>

300

Find a vector-valued function r(t) that paramaterizes the curve given by the intersection of the following surfaces:

x² + y² = 4

x + 2y + z = 4

<2cos(t), 2sin(t), 4 - 2cos(t) - 4sin(t)>

<t, sqrt(4 - t²), 4 - t - 2sqrt(4 - t²)>

400

Find the equation of a plane written in general form that passes through the points P(2,1,-1), Q(3,0,2), R(1,-1,1).

4x - 5y - 3z = 6

500

Parameterize the curve r(t) = <e^tcos(t), e^tsin(t), e^t> in terms of arc length from 0 <= t <= ln(2).

r(s) = <(1 + s/sqrt(3))cos(ln(1 + s/sqrt(3))), (1 + s/sqrt(3))sin(ln(1 + s/sqrt(3))), 1 + s/sqrt(3)>