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 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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What is a derivative l.jpg
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 l.jpg
The tangent line

single point

of intersection


Slope of a secant line l.jpg
slope of a secant line

f(a) - f(x)

a - x

f(x)

f(a)

x

a


Slope of a closer secant line l.jpg
slope of a (closer) secant line

f(a) - f(x)

a - x

f(x)

f(a)

x

a

x




Watch what x does l.jpg
watch what x does...

x

a



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

x

a


Slide11 l.jpg

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….


Slide12 l.jpg

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 l.jpg
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)


Slide14 l.jpg
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 l.jpg
Which one should I use? slope of the tangent line...

(doesn’t really matter)


A very simple example l.jpg
A VERY simple example... slope of the tangent line...

want the slope

where a=2


Slide17 l.jpg

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


Slide18 l.jpg

As h 0 slope of the tangent line...


Back to our example l.jpg
back to our example... slope of the tangent line...

When a=2,

the slope is 4


In conclusion l.jpg
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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