Consider the unit sphere x2 +y2 +z2 = 1 and the cone (z+√2)2 =
x2 +y2. Show that these surfaces are tangent where they intersect,
that is, for a point on the intersection, these surfaces have the
same tangent plane
If a circle C with radius 1 rolls along the outside of
the circle x2 +
y2 = 36, a fixed point P on
C traces out a curve called an epicycloid, with
parametric equations x = 7
cos(t) − cos(7t),
y = 7 sin(t) −
sin(7t). Graph the epicycloid.
Find the area it encloses.
Multivariable Calculus
[A] Consider the region R in the first quadrant that is outside
the circle r = 1 and inside the four-leaved rose r = 2 sin 2θ).
(A.1) Draw a sketch of the circle and the four-leaved rose
(include the entire graph) and shade the region R. Feel free to use
your graphing calculator.
(A.2) Write the following double integral as an iterated
integral in polar coordinates. Do not evaluate the integral in this
part. Be sure to...
Solve the following equations in non-negative integers.
1. x2 -
y2 = 221
2. a + b = ab
3. gcd(a,b)lcm(a,b) = b +
9
4. x4 +
2x3 - y2(1+2x) +
x2(1-y2) = 2299
Show that the set ℝ2R2, equipped with operations
(?1,?1)+˜(?2,?2)=(?1+?2+1,?1+?2−1)(x1,y1)+~(x2,y2)=(x1+x2+1,y1+y2−1)
? ⋅˜ (?,?)=(??+?−1,??−?+1)
(1)defines a vector space over ℝR.
(2)Show that the vector space ?V defined in question 1 is
isomorphic to ℝ2R2 equipped with its usual vector space operations.
This means you need to define an invertible linear map
?:?→ℝ2T:V→R2.
Let F= (x2 +
y + 2 + z2) i + (exp(
x2 ) + y2)
j + (3 + x) k . Let a
> 0 and let S be part of the spherical
surface x2 + y2 +
z2 = 2az + 15a2
that is above the x-y plane and the disk formed in the
x-y plane by the circular intersection between the sphere
and the plane. Find the flux of F outward across
S.
a)
Find the volume of the solid obtained by revolving the region in
the first quadrant bounded by the curves y= x^(1/2) & y= x^5
about the x-axis
b) Find the volume of the solid obtained by revolving the
region between the curve
f(x)= x^(1/3) , the line y=2, and the line x=8 about the
y-axis