Musa beyah
10/1/16
53103353
(AP Statistics MA14)
289B02SP
Pages 360361 Exercises 7.1, 7.5, 7.7
7.1
One broken egg
a. Discrete b. Continuous c. Discrete d. Discrete e. Continuous
7.5
y is continuous va
Chain Rule
1. The temperature on a hot surface is given by
T = 100 e
(x2 +y 2 )
.
A bug follows the trajectory r(t) = ht cos(2t), t sin(2t)i.
a) What is the rate that temperature is changing as the bu
Problems for 18.112 Final Examination
(Open Book)
Dec. 20, 2006
1. (20) Let a, b, c be complex numbers satisfying
ba
ac
=
.
ca
bc
Considering the triangle with vertices a, b, c. Prove
b a = c a =
Solution for 18.112 Final Examination
Problem 1.
Method 1 (Geometric way): Let A, B, C be points on the complex plane corresponding to complex numbers a, b, c. Then
AB = b a, BC = c b, CA = a c
and
A
Derivatives  vector functions.
of
Recall that if x is a point of
a scalar function of
x,

R
"
and if
then the derivative of
f
f ( 5 ) is
(if it
exists) is the vector
For some purposes, it will be co
Notes  double integrals.
on
(Read 11.111.5 of Apostol.)
Just as for the case of a single integral, we have the
following condition for the existence of a double integral:
 defined is
on
f  integra
Solution for 18.112 ps 1
1(Prob1 on P11).
Solution:
a < 1, b < 1 = (1 aa)(1 b < 1
b)
= 1 aa b + a < 1
b
abb
ab ab > a + b ab ab
= 1 + a b
ab
a
b
= (1 a ab) > (a b)(
b)(1
a b)
ab
=
< 1.
1 ab
Solution for 18.112 ps 3
1(Prob 2 on P83).
Solution: First we will prove the following two lemmas.
Lemma 1. Reection carries circles to circles.
Proof: Reection is the composition of two maps: conjuga
® Let C be the ellipse x2/4 + y2/9 = 1 traversed once in
the counterclockwise direction, and define
:2 r + 2
G(z) =  d{ for any 2 inside C
C I 2
Then find G(1) , G' (i) and G"(i)
00 x HINT:
G) E
Rough Solutions for PSet 12
1. (12.10:7) We rst parameterize the sphere of radius a by r(, ) = (a cos sin , a sin sin , a cos )
where 0 2 and 0 . (We write rst so that the normal points
outward.) Then
Solution for 18.112 ps 2
1(Prob 7 on P58).
Solution: Let A be a discrete set in separable metric space (X, d), then for any
a A, there is ra > 0 such that
A Nra (a) = cfw_a.
It follows that for any a
Solutions for PSet 11
1. (11.28:14) We substitute u = x y and v = x + y. the Jacobian of the
transformation (x, y) (u, v) is
1 1
1 1
This has determinant 2 and the vertices of the parallelogram S corr
Worksheet #3: Unions and Intersections
Answer Key
Name_
1. If P(X) = 0.23 and P(X and Y) = 0.12 and P(X or Y) = .34, find P(Y ).
c
X
0.11
Y
0.12
0.11
2. Let
P A B
3
6
P Ac B
1
6
P A Bc
1
6
(a) Draw