The derivative as the slope of the tangent line

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Presentation transcript:

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 a

slope of a (closer) secant line f(a) - f(x) a - x f(x) f(a) x x a

closer and closer… a

watch the slope...

watch what x does... x a

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

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

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….

f(x) - f(a) 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... f(x+h) - f(x) (x+h) - x = f(x+h) - f(x) h f(a+h) h f(a) (For this particular curve, h is a negative value)

thus... lim f(a+h) - f(a) h lim f(x) - f(a) AND x - a Give us a way to calculate the slope of the line tangent at a!

Which one should I use? (doesn’t really matter)

A VERY simple example... want the slope where a=2

as x a=2

As h 0

back to our example... When a=2, the slope is 4

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 explanations:http://archives.math.utk.edu/visual.calculus/2/