Ultimate Multivariable Calculus Jeopardy

Just Plane Math

450

Find the equation of the plane perpendicular to the line

L(t)=<2-4t,2+3t,t-1> 

containing the point (7,8,-9)

-4(x-7) + 3(y-8) + (z+9) = 0

500

Find the equation of the plane through

(1,-1,0), (3,0,0), and (3,1,4) 

N=(2,-4,1), e.g.

2(x-1) - 4(y+1) + z = 0

550

Find the equation of the plane containing the line

L= <3+t,1,-5>

which is parallel to the line

L=<1,-3+t,6>

z+5=0 or z=-5

600

Find the tangent hyperplane to the surface in R⁴ of the function

w=f(x,y,z) = x²+y²-z²

at the point (1,-2,3,-4)

w=-4+2(x-1) + -4(y+2) + 8(z+4)

Vector Vibes

450

Find the angle between

v=<1,-4,-8> and w=<2,6,-3>

you can give your answer as an inverse trig function.

arccos(2/63)

500

Find the area of the triangle containing the points 

(0,0,0), (1,-3,-1), (2,0,1)

7/2

550

Find the distance between the point (-9,2,-12) and the plane

4x-y+8z = 28

18

600

Find the volume of the 4-dimensional tetrahedron with vertices

(0,0,0,0), (1,0,0,0), (0,-2,0,0), (0,0,3,0), (0,0,0,-4)

Recall that there are 4!=24 such tetrahedra in a 4-dimensional parallelopiped.

1

Conic-tions

450

See Card (C) - Hyperbolic Paraboloid

C, lambda, e, 4

500

See Card (2) - z²=x²-y²

2 l G iota

550

See Card (i) - Level set

H i 3 beta

600

DAILY DOUBLE

Find the surface of intersection in 5-dimensional space of the cone

x²+y²+v²+w²=z²  (z >= 0)

and the hyperboloid

-x²-y²+2z²-v²-w²=1

Parametrickery

450

Find the arc length of the curve

s(t) = <e^{t^2}, sin(e^{t^2}), cos(e^{t^2}) >

over the interval [0,  sqrt( ln(7) ) ]

6 sqrt(2)

500

Find the curvature of the curve

s(t) = <e^t, e^t, 1 > 

zero: it is a line

550

Find the unit normal vector to the curve

s(t) =< (1/2)t², (1/2)cos(t²), (1/2)sin(t²) > 

N=<0, - sin(t²), -cos(t²) >

600

Find the curvature of the curve of intersection of the cylinder

x²+z²=25

and the hyperboloid

x²-y²+z² = 16

1/5

Surface Tension

450

Find the 4th order mixed partial f_xyxy for the function

f(x,y) = x² sin(y) - y² cos(x)

f_xyxy(x,y) = -2 sin(y) + 2 cos(x)

500

Let f(x,y) = y²-x²

x=r sin(t)

y=r cos(t)

Compute (df/dr) at (x,y) = (2,0)

-4

550

DAILY DOUBLE

Compute the limit as (x,y) goes to (0,0) of

[(e^{sqrt(x^2+y^2)}-1)(sin(1 / sqrt(x²+y²)))]

                                                         

           sqrt(x²+y²)

600

Tiny pyramids are to be manufactured with a square base of side length s=6 cm and height h=8 cm. The error in the side length is up to 0.15 cm and the error in the height is up to 0.25 cm. What is the approximate error in volume? Use the formula

V=(1/3)s²h

dV is about 7.8 cm³