Dots and Crosses
u = <3, -1, 4> and v = <-2, 5, 1>.
Calculate the dot product u⋅v.
-7
What shape do the xy, yz, and xz traces look like for the surface 4x2 + y2 + 9z2 = 9.
Circles
Acceleration vector of any projectile motion problem
a(t) = <0,-g>
Find the limit as (x,y) ->(2,-1) of the function, or prove DNE:
f(x,y) = (3x2y + 2x)/(y2+5)
-4/3
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
a = i + 2j - k and b = 3i - j + 2k.
Calculate the cross product a x b.
<3, -5, -7>
What type of quadric surface is z2 = 4x2 + 9y2?
Cone
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>
Find the limit as (x,y) -> (3,3) of the function, or prove DNE:
f(x,y) = (x2-y2) / (x-y)
6
Find the vector projection of b onto a given a = <6, -5, 2> and b = <9, 0, 1>.
<336/65, -280/65, 112/65>
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
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
35.3 meters
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)
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)>
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
Gimme a contour plot of f(x,y) = sqrt(16 - x2 - y2).
Circles getting smaller
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
Find the limit as (x,y) -> (0,0) of the function, or prove DNE:
f(x,y) = (x3y) / (x2+y2)
0 (prove with squeeze)
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
Let |u| = 3, |v| = 5, and u ⋅ v = 9. Find the value of |u x v|.
12
Draw the 3 traces for the surface z = y2/4 - x2 (for each trace, let k=-1,0,1).
Ch12 Notes
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
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)
Parameterize the curve r(t) = <etcos(t), etsin(t), et> 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)>