jeopardy 203. formulas 100 lines 100 planes 100 surfaces 100 curves 100 formulas 101 lines 200...
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Jeopardy 203
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Formulas 100
Lines 100
Planes 100
Surfaces 100
Curves 100
Formulas 101
Lines 200
Planes 200
Surfaces 200
Curves 200
Formulas 102
Lines 300
Planes 300
Surfaces 300
Curves 300
Formulas 103
Lines 400
Curves 400
Curves 500
Curves 600
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The angle between vectors u and v.
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Arccos(u·v/|u||v|)
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|u x v|=
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|u||v|sin(θ),where θ
is the angle between the vectors
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Volume of parallelopiped determined by vectors u, v, and w.
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|u·(v x w)|
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Distance from point (x1, y1, z1) to the plane ax+by+cz+d=0
.
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|ax1+by1+cz1+d| / √(a²+b²+c²)
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Symmetric form of line that goes through points (3,5,7) and (1,8,4)
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Any of the following:
1 8 4
2 3 3
x y z
3 5 7
2 3 3
x y z
1 8 4
2 3 3
x y z
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Equation of line, in parametric form, that goes through point (3,2,1) and is normal to the plane 3x+2z=4y-5
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x=3+3t, y=2-4t, z=1+2t
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From t=1 to t=3, a particle is moving along the curve
r(t)=<3t^2, 6/t, 5>. At t=3, its velocity becomes constant. What is its position at t=6?
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The particle’s position will be: (81,0,5)
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What is the line of intersection of the planes 3x+2y+6z = 11 and 4x-2y-3z = 3?
Answer in parametric form.
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X=2+6t, y=5/2+33t, z = -14t
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Find the equation of the plane that goes through the points (6,4,2), (9,7,5), and (11,16,11).
Answer in most reduced form.
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3x+4y-7z=20
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Find the equation of the plane that contains the lines
Answer in standard form
5 4 4
3 3 3
x y z
5 4 4
5 3 6
x y z
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9x+y-8z=81
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What are the equations of the planes that are parallel to and 3 units away from
4x-4y+2z=9?
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4x-4y+2z=-94x-4y+2z=27
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Find the domain of the function
f(x,y)=ln(x²-y²)
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The right and left quadrants of the plane.
I.e. the right and left regions determined by the lines y=
±x
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Describe (name the type) of the x=k, y=k, and z=k traces of
-x²/25 +y²/36 – z²/4 = 100
And name the type of this surface
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X=k traces are hyperbolas centered on y-axis
Y=k traces are ellipsesZ=k traces are hyperbolas centered
on y-axis
The surface is a hyperboloid of two sheets
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Give the equation, in standard form, of the surface whose projection to the x-y plane are the lines y=
±3x/2, and whose projection to the x-z plane is z=9x². What is the name of this surface?
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z/36 = x²/4-y²/9
Hyperbolic paraboloid
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Give the vector equation of the standard spiral (centered on the z-axis) and with radius 1.
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r(t)=<cos t, sin t, t>
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What is the unit tangent vector to the curve r(t)=<t^2,t-t^2, t^2+3t-2>
at t=2?
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1/√(74) <4,-3,7>
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Write an integral for the length of the curve <t²,2t-1,cosπ t>
From (0,-1,1) to (4,3,1)
(no need to evaluate the integral)
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2 2 2 2
04 4 sint t dt
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Find the length of the curve
from x=1 to x=4.
2 ln( )
2 4
x xf x
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7.5 +1/4 ln(4)
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Find the curvature of the ellipse <3cos t, sin t> at (3,0)
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3
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Find the normal vector N(t) for the curve
<4cos t, 4 sin t, t²>
at t=0.
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N(0) = 2/√5 <-1,0,1/2>