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The derivative as the slope of the tangent linePowerPoint Presentation

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)

What is a derivative?

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

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

The slope of the secant lines slope of the tangent line...

gets closer

to the slope of the tangent line...

...as the values of x

get closer to a

Translates to….

f(x) - f(a) slope of the tangent line...

lim

x - a

x

a

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!

similarly... slope of the tangent line...

f(x+h) - f(x)

(x+h) - x

= f(x+h) - f(x)

h

f(a+h)

h

f(a)

a+h

a

(For this particular curve, h is a negative value)

thus... slope of the tangent line...

lim f(a+h) - 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!

Which one should I use? slope of the tangent line...

(doesn’t really matter)

as x a=2 slope of the tangent line...

As h 0 slope of the tangent line...

in slope of the tangent line...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 explanations:http://archives.math.utk.edu/visual.calculus/2/

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