rate_of_change
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Hello wonderful mathematics people, this is Anna Cox from Kellogg Community College
the average speed is the distance covered divided by the time elapsed. The average rate of change of a secant line: a secant line is just
a line that connects two points on any curve, so if we have some given curve
x one f(x one) and x two f(x two), the average rate of change is really just the slope or the change of y over the change of x
so f(x two) minus f(x one) over x two minus x one
Know if we thought about that change of thatx two minus x one equaling some new variable let's call it each h
if h is x two minus x one, then we know that x two has to equal
h plus x one, where h isn't equal to zero
so we could think of that slope in a new format or an alternative form of f, instead of x two
x one plus h minus f(x one) all divided by h
for an instantaneous rate of change, what we're trying to do is we're trying to figure out what happens
at a single point. So we wanna make that h get smaller and smaller and smaller
so if we started with
the x one and the x two further out
We want the h to get smaller. So now we're gonna take the x one
with an x two that's a little closer
and then we're gonna do that again. The x one with an x two that still closer
and eventually if we make that h go closer and closer and closer to zero
we're going to have a tangent line at a specific point on the curve. So the instantaneous rate of change and tangent line
is really that instantaneous point when that h is really, really, really small
of the slope. So we're getting certain points closer and closer and closer to the x one to figure out the instantaneous rate of change
thank you and have a wonderful day