## anonymous one year ago Pretty lost on this problem! Help needed! Q: Suppose that 3 less than or equal to f^(x) is less than or equal to 5 for all values of x? Show that 18 is less than or equal to f(8)- f(2) is less than or equal to 30.

1. anonymous

I dont really know where to start..

2. dan815

Show that 18 is less than or equal to f(8)- f(2) is less than or equal to 30. what is thsi part saying

3. anonymous

Here lemme right it out: Suppose that \[3\le f'x \le 5 \]

4. dan815

oh xD thats a derivative haha

5. anonymous

and show that \[18 \le f(8) - f(12) \le 30 \]

6. dan815

okk

7. anonymous

lol yea

8. dan815

basically, lets consider the max slope and min slope

9. dan815

f'(x) has a min of 3 and max of 5 slope, what can we say about the distance change from f(8) to f(12) then

10. dan815

i see you wrote f(2) up in your question then f(12) rewritten which one is it exactly?

11. anonymous

12. dan815

okay so

13. dan815

do you have any ideas so far?

14. anonymous

how would you solve for what f(8) and f(2) is if they dont even give you what the orignal function is?

15. dan815

we dont know f(8) and f(2) but we can see the max and min difference

16. dan815

|dw:1437625240205:dw|

17. dan815

lets the slope function is some arbitrary function like this that si contained in slope 3 to 5

18. dan815

you can say the max change will be if its slope 5 everywhere and hte min change will be if the slope is 3 everywhere

19. anonymous

yea thats given from the first info

20. dan815

so you see f(8)-f(2) if f'(X) WAS 5 EVERYWHERE and then f(8)-f(2) if f'(x) was 3 everywhere

21. dan815

does this make sense?

22. anonymous

yea, thanks for capitalizing it

23. dan815

lol

24. dan815

does it really make sense?

25. anonymous

NO. I dont even get what this question is asking honestly... im super confused.

26. dan815

ok ok lets start over

27. anonymous

I need to go back to square 1 or something

28. dan815

dont worry

29. dan815

okay f'(x) the first derivatve of x means its giving u the slope of your function everywhere

30. anonymous

yes, I get that

31. dan815

okay now, they tell you the slope isalways somewhere in between 3 and 5

32. anonymous

Yes, so we assumed 3 is the min slope and 5 is the max slope

33. dan815

now our job is to find the most change possible and the least change possible for our function f(x) given this info

34. dan815

we can say the most change happens when the slope is always at 5 and the least change when the slope is always at 3

35. dan815

now a constnat slope means the function is a line so to find the max and min changes we have to look at lines here

36. dan815

|dw:1437625618806:dw|

37. dan815

i dont know what f(8) and f(2) are but since the slope is constant i do know the changge in height

38. anonymous

love the drawing, its makes more sense to me that way

39. dan815

|dw:1437625668687:dw|

40. dan815

|dw:1437625704207:dw|

41. anonymous

wait, how were you able to assume that you could draw the right triangles like that in the diagram?

42. dan815

that just me finding the change of height for a line with slope 5

43. dan815

since its f(8)-f(2) the change in x was 8-2 and to find the change in y i know the slope of the line is 5, and slope = rise/run so 5 slope means for every run there is 5 rise so 8-2=6 rrun so for 6 run there is 6*5 rise

44. dan815

and the reason its right angle triangle is because run is horizontal and rise is vertical horizontal and vertical lines have a 90 degree angle

45. anonymous

OMG that was so easy!!!!

46. dan815

lol xD

47. dan815

im glad u find htis easy now

48. anonymous

. Oops

49. anonymous

Meant to say thanks! Appreciate the help! :)

50. dan815

no its good,. its supposed tofeel easy once your understand it

51. dan815

you already thanked me by saying omg its so easy