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# The derivative as the slope of the tangent line - PowerPoint PPT Presentation

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 tangent line. single point of intersection. slope of a secant line. f(a) - f(x). a - x. f(x). f(a). x.

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## PowerPoint Slideshow about 'The derivative as the slope of the tangent line' - zohar

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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

single point

of intersection

f(a) - f(x)

a - x

f(x)

f(a)

x

a

f(a) - f(x)

a - x

f(x)

f(a)

x

a

x

watch what x does...

x

a

As the values of slope of the tangent line...x get closer and closer to a!

x

a

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)

A VERY simple example... slope of the tangent line...

want the slope

where a=2

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

As h 0 slope of the tangent line...

back to our example... slope of the tangent line...

When a=2,

the slope is 4

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/