#$&*
course MTH 173
12/14 2:00
Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
029. `query 29
*********************************************
Question: `qQuery 3.10.23 (previously 4.7.24) (was problem 7 p 290 ) prove if g' < h' on (a,b} and g(b) = h(b) then h < g on (a,b)--g,h both cont on [a,b] diff on (a,b)Explain why you expect, that for the given conditions, the function h will be strictly less than the function g on the interval.
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
Let f(x) = h(x) - g(x) on the interval (a,b]
Given that h(b) = g(b)
Thus f(b) = h(b) - g(b) = 0
Given that h(x) and g(x) is continuous on [a,b] and differentiable on (a,b)
The derivative of f(x) function will be
f’(x) = h’(x) - g’(x)
Given that h’(x) > g’(x) throughout the interval (a,b]
f’(x) will thus have a positive value for all x values on the interval (a,b]
Since the end value of f(x) interval is 0 that is f(b) = 0 and f(x) has a positive slope throughout (a,b],
It indicates that f(x) is throughout increasing and finally reaches a value equal to 0
Since it reaches a value equal to 0 the value for x values less than x = b should be negative
Thus f(x) < 0 for b a < x < b
Thus for the interval (a,b) h(x) - g(x) < 0
Thus we would expect h(x) to be strictly less than g(x) for the interval (a,b)
confidence rating #$&*: 3
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution:
`aSince f ' (x) < 0 on the interval the function is decreasing on the interval, hence since f(b) = 0 it follows that f(x) > 0 on the interval.
From this it follows that g(x) - h(x) > 0 on the interval and g(x) > h(x). **
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary): OK
------------------------------------------------
Self-critique Rating: OK
*********************************************
Question: `qQuery Add comments on any surprises or insights you experienced as a result of this assignment.
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution:
`aI was surprised (but not disappointed) that the query was only on one question. I did gain insight in that after I first typed in my original answer, I realized that it was wrong. I had proved (quite successfully, I thought) that the Racetrack principle was wrong! I'm hoping that my revised answer is more correct.
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary): OK
------------------------------------------------
Self-critique Rating: OK
""
"
Self-critique (if necessary):
------------------------------------------------
Self-critique rating:
""
"
Self-critique (if necessary):
------------------------------------------------
Self-critique rating:
#*&!
&#Very good responses. Let me know if you have questions. &#