If your solution to stated problem does not match the given solution, you should self-critique per instructions at http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.
Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
011. `query 11
Question:
`q1.6.16 (was 1.6.14 intervals of cont for (x-3)/(x^2-9)
What are the intervals of continuity for the given function?
Your solution:
Confidence Rating:
Given Solution:
`a The function is undefined where x^2 - 9 = 0, since
division by zero is undefined.
x^2 - 9 = 0 when x^2 = 9, i.e., when x = +-3.
So the function is continuous on the intervals (-infinity,
-3), (-3, 3) and (3, infinity).
The expression (x - 3) / (x^2 - 9) can be simplified. Factoring the denominator we get
(x - 3) / [ (x - 3) ( x + 3) ] = 1 / (x + 3).
This 'removes' the discontinuity at x = +3. However in the given fom (x-3) / (x^2 + 9)
there is a discontinuity at x = -3. **
Self-critique (if necessary):
Self-critique Rating:
Question:
`q1.6.24 (was 1.6.22 intervals of cont for |x-2|+3, x<0;
x+5, x>=0
What are the intervals of continuity for the given function?
Your solution:
Confidence Rating:
Given Solution:
`a The graph of y =
|x-2|+3 is translated 2 units in the x direction and 3 in the y direction from
the graph of y = |x|. It forms a V with
vertex at (2, 3).
The given function follows this graph up to x = 0. It has slope -1 and y-intercept at y = | 0 -
2 | + 3 = 5.
The graph then follows y = x + 5 for all positive x. y = x + 5 has y-intercept at y = 5. From that point the graph increases along a
straight line with slope 1.
So the graph of the given function also forms a V with
vertex at (0, 5).
Both functions are continuous up to that point, and both
continuously approach that point. So the
function is everywhere continuous. **
Self-critique (if necessary):
Self-critique Rating:
Question:
`q1.6.66 (was 1.6.54 lin model of revenue for franchise
Is your model continuous? Is actual revenue continuous?
Your solution:
Confidence Rating:
Given Solution:
`a revenue comes in 'chunks'; everytime someone pays. So the
actual revenue 'jumps' with every payment and isn't continuous. However for a
franchise the jumps are small compared to the total revenue and occur often so
that a continuous model isn't inappropriate for most purposes. **
GOOD STUDENT EXAMPLE
The model would not be continuous. The VT vs. Boston College football game I watched actually inspired me on this one. A football franchise makes most, if not all, of its revenue on one day out of the week and that’s game day. Attendance and concessions will differ from each game depending on the anticipation beforehand so you may make a fortune on a game against a rival team and you may make less than average on a game that everyone knows the outcome of.
Add comments on any surprises or insights you experienced as
a result of this assignment.
Self-critique (if necessary):
Self-critique Rating:
Student question:
What are the intervals of continuity for the function f(x) = [[x-2]] + x?
Instructor Response:
[[x-2]] indicates 'the greatest integer in x - 2'.
For example, if x = 3.9 we have [ [ x - 2 ] ] = [ [3.9 - 2]] = [ [ 1.9] ] =
1.
[[ 1.9 ]] is the greatest integer less than or equal to 1.9, which is 1.
if x = 3.99 we have [ [ x - 2 ] ] = [ [3.99 - 2]] = [ [ 1.99] ] = 1
[[1.99]] is still 1.
if x = 3.99999 we have [ [ x - 2 ] ] = [ [3.99999 - 2]] = [ [ 1.99999] ] = 1
[[1.99999]] is still 1.
However as soon a x = 4 we get [[4-2]] = [[2]] = 2.
[[ 2 ]] is the greatest integer less than or equal to 2, which since 2 is an
integer is just 2.
Now if x = 3.9 we have [[x-2]] + x = [[3.9 - 2]] + 3.9 = 1 + 3.9 = 4.9.
If x = 4 we get [[x-2]] + x = 6.
What happens for values of x between 3.9 and 4, especially for values of x that
approach 4 as a limit?
How can you use these examples to understand this function and determine its intervals of continuity?