The derivative as the slope of the tangent line

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The derivative as the slope of the tangent line
(at a point)
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What is a derivative?

• A function
• the rate of change of a function
• the slope of the line tangent to the curve

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The tangent line
single point of intersection
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slope of a secant line
f(a) - f(x)
a - x
f(x)
f(a)
x
a
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slope of a (closer) secant line
f(a) - f(x)
a - x
f(x)
f(a)
a
x
x
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closer and closer
a

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watch the slope...

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watch what x does...
x
a

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The slope of the secant line gets closer and
closer to the slope of the tangent line...

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As the values of x get closer and closer to a!
x
a

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The slope of the secant lines gets closer to the
slope of the tangent line...
...as the values of x get closer to a
Translates to.
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f(x) - f(a)
lim
x - a
a
x
as x goes to a
Equation for the slope
Which gives us the the exact slope of the line
tangent to the curve at a!
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similarly...
f(xh) - f(x)
(xh) - x
f(xh) - f(x)
h
f(ah)
h
f(a)
ah
a
(For this particular curve, h is a negative value)
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thus...
lim f(ah) - f(a)
h 0
h
AND
lim f(x) - f(a)
x a
x - a
Give us a way to calculate the slope of the line
tangent at a!
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Which one should I use?
(doesnt really matter)
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A VERY simple example...
want the slope where a2
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as x a2
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As h 0
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back to our example...
When a2, the slope is 4
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in conclusion...

• The derivative is the the slope of the line
  tangent to the curve (evaluated at a point)
• it is a limit (2 ways to define it)
• once you learn the rules of derivatives, you WILL
  forget these limit definitions
• cool site to go to for additional
  explanationshttp//archives.math.utk.edu/visual.c
  alculus/2/