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I dont really know where to start..
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
Here lemme right it out: Suppose that \[3\le f'x \le 5 \]
oh xD thats a derivative haha
and show that \[18 \le f(8) - f(12) \le 30 \]
basically, lets consider the max slope and min slope
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
i see you wrote f(2) up in your question then f(12) rewritten which one is it exactly?
its f(2), sorry about that
do you have any ideas so far?
how would you solve for what f(8) and f(2) is if they dont even give you what the orignal function is?
we dont know f(8) and f(2) but we can see the max and min difference
lets the slope function is some arbitrary function like this that si contained in slope 3 to 5
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
yea thats given from the first info
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
does this make sense?
yea, thanks for capitalizing it
does it really make sense?
NO. I dont even get what this question is asking honestly... im super confused.
ok ok lets start over
I need to go back to square 1 or something
okay f'(x) the first derivatve of x means its giving u the slope of your function everywhere
yes, I get that
okay now, they tell you the slope isalways somewhere in between 3 and 5
Yes, so we assumed 3 is the min slope and 5 is the max slope
now our job is to find the most change possible and the least change possible for our function f(x) given this info
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
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
i dont know what f(8) and f(2) are but since the slope is constant i do know the changge in height
love the drawing, its makes more sense to me that way
wait, how were you able to assume that you could draw the right triangles like that in the diagram?
that just me finding the change of height for a line with slope 5
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
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
OMG that was so easy!!!!
im glad u find htis easy now
Meant to say thanks! Appreciate the help! :)
no its good,. its supposed tofeel easy once your understand it
you already thanked me by saying omg its so easy