Key to the Practice Final Exam

Problem 1
x = 3a

Problem 2
y = 2 e^0.03t ,    23.1 hours

Problem 3
Through (1,4) reflect across the vertical axis.  The new asymptote is y = 3.

Problem 4
0.078125 meters

Problem 5
Vertices:  (2,-2 + ), (-2,-2 + ), (2,-2 - ), (-2,-2 - ), 
Foci:  (2- ,-2),  (2 + , -2)
e = /2

Problem 6
(3,5,1)

Problem 7

1. 4	2 - 3x
   -2	-4

   
   

2. x/(x - 2)	-2/(x - 2)
   -1/(x - 2)	1/(x - 2)

   Undefined for x - 2
   

3. 6 - 6x

Problem 8

13

Problem 9
For n = 1, 1 = 1 o.k.
Now assume the theorem is true for n = k - 1, then
        S_i=1^k-1i = (k - 1)(k)/2
Goal:  S_i=1^k i = (k)(k + 1)/2
we have
        S_i=1^k i = S_i=1^k-1 i + k
         = (k - 1)(k)/2 + k
         = (k - 1)(k)/2 + 2k/2
        = [(k - 1)(k) + 2k]/2
        = [k² - k + 2k]/2
        = [k² + k]/2
        = (k)(k + 1)/2

by mathematical induction the theorem is true.

Problem 10

1. False, they could be negative 
2. False, if x = 1/e, then ln(1/e) = -1 is less than 0.
3. False, the bottom focus lies below the vertex and can dip below the x-axis.
4. False, let A be the zero matrix

Problem 11
107,616,795

Problem 12
35