The Tangent Line Problem

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The Tangent Line Problem

• Part of section 1.1

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Tangent Line Problem
One of the classic problems in calculus is the
tangent line problem. You are probably very good
at finding the slope of a line. Since the slope
of a line (and line always implies straight in
the world of math) is the same everywhere on the
line, you could pick two points on the line and
calculate the slope. With a curve, however, the
slope is different depending where you are on the
curve. You aren't able to just pick any old two
points and calculate the slope. Think about the
parabola,
Whose graph looks like
Pretend you are a little bug and this graph is a
mountain. The slope of the mountain is different
at each point on the mountain. Watch how
easy/hard it looks for the bug to climb. Just
look at the bugs angle.
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Without calculus, we could estimate the slope at
a particular point by choosing an additional
point close to our point in question and then
drawing a line between the points and finding the
slope.
P
So if I was interested in finding out the slope
at point P, I could estimate the slope by using a
point Q, drawing a secant line (which crosses the
graph in two places), and then finding the slope
of that secant line. This is continued on the
next slide.
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Q
P
So, we get an approximate slope and sometimes
just an estimate is fine. To get an even better
approximation, we can move Q closer to P and that
secant line begins to look more and more like a
tangent line to the graph.
Ill draw a tangent line at P in orange. The
tangent line and the graph share the same slope
at point P.
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Q
P
If we could get the secant line to look more like
the tangent line, the slope of the secant line
would be more like the slope of the tangent line.
You might be asking yourself, why is this
helpful?
Well, if you could zoom in on point P, youd see
that the tangent line and the graph look so
similar, you can hardly tell them apart. So, by
finding an approximate slope of the tangent line,
youd be finding an approximate slope of the
graph at the point P.
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With calculus, we are able to find the actual
slope of the tangent line.
Without it, we can only estimate.