Consider the surfaces x^2 + y^2 + z^2 = 1 and (z +√2)2 = x^2 +
y^2, and let (x0, y0, z0) be a point
in their intersection. Show that the surfaces are tangent at this
point, that is, show that the
have a common tangent plane at (x0, y0, z0).
3. Consider the following pdf:
Q(x)
= {3x^2 / 2 -1? x ?1
{0 otherwise
a. Set up an acceptance-rejection algorithm for this
distribution. Use the majorizing function g(x) = 3/2.
b. Assuming a linear congruential generator with parameters a:
21, m: 100, c: 13, and x0: 7, generate two random variates from the
distribution q(x).
c. Can you make any comments about the potential period of the
LCG in part
b?
The surface z = 3x^(2) + (1/6)x^(3) - (1/8)x^(4) - 4y^(2) is
intersected by the plane 2x - y = 1. The resulting intersection is
a curve on the surface. Find a set of parametric equations for the
line tangent to this curve at the point (1,1,-23/24).
Consider the following planes. 5x − 3y + z = 2, 3x + y − 5z = 4
(a) Find parametric equations for the line of intersection of the
planes. (Use the parameter t.) (x(t), y(t), z(t)) = (b) Find the
angle between the planes. (Round your answer to one decimal
place.)
1) Let P(x) = 3x(x − 1)3 (3x + 4)2 . List
the zeros of P and their corresponding multiplicities.
2) Let f(x) = −18(x + 3)2 (x − 2)3 (x +
71)5 . Describe the end behavior of f by filling in the
blank below.
As x → −∞, f(x) → .
As x → ∞, f(x) → .
3) The polynomial of degree 4, P(x) has a root of multiplicity 2
at x = 3 and roots of...
Find the flux of the vector field F = (3x + 1, 2xe^z , 3y^2 z +
z^3 ) across the outward oriented faces of a cube without the front
face at x = 2 and with vertices at (0,0,0), (2,0,0), (0,2,0) and
(0,0,2).
Consider the definite integral ∫05 ((3x −
1)/(x + 2)) dx
a. How large an n do we need to use to approximate the value of
the integral to within 0.001 using the Midpoint Rule?
b. How large an n do we need to use to approximate the value of
the integral to within 0.001 using Simpson’s Rule?
1.
f(x)= 3x / x^2 + 1
- Vertical Asymtote
As x → −, f(x) →
As x → +, f(x)
→
- Any hole in graph
- Horizontal asymtote
As x → −, f(x) →
As x → +, f(x)
→