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About LinkBacksSuppose thatand that
for all of t. Must
for all of t??
Great. I had the right answer, but i am not sure that i had the right thought process... i had said something along the lines of: IfOriginally Posted by VonNemo19
Absolutely not. There are an infinte number of functions that pass through the pointhaving slope 5.
, then
. when x=0, y=-7, but when
so that negates the question. Is my thought process skewed? Or is that a valid approach?
Yes:Suppose that
.
. Therefore
.
Correct. But not having a slope 5 for all values of t ...Absolutely not. There are an infinte number of functions that pass through the point
Thanks, I was reviewing the question yesterday and realised that entire thing before was Wrong! Because we are not looking at the inverse, but rather, the derivative! I was about to make a note of that here. Didn't see you had responded.Yes:
Correct. But not having a slope 5 for all values of t ...
Thanks!
This is if we are looking at the inverse function but we are not. The question asked about the derivative! We had the totally wrong idea here... I hope this post doesn't mislead anyone...Absolutely not. There are an infinte number of functions that pass through the point
Yes. By the mean value theorem, two functions that have the same derivative must differ by a constant:Suppose that
Let h(x)= f(x)- g(x). f'(t)= g'(t), then h'(x)= 0 for all t. By the mean value theorem, h(b)- h(a)/(b- a)= 0 so h(b)- h(a)= 0 and h(b)- h(a)= 0. That is, h is a constant and so f and g differ by a constant.
Since f(t)= 5t has f'(x)= 5 for all t,any function with derivative always equal to 5 must be of the form g(x)= 5x+ C. In order that g(0)= -7, C must be -7.
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