1 2.1 - the tangent and velocity problem © john seims (2008)
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2.1 - The Tangent and Velocity Problem
© John Seims (2008)
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Definition: Secant Line
f(x2)f(x1)
x1 x2
PQ
Secant Line – A line passing thorough two points on a graph of a function. The slope of the secant line is called the average rate of change.
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( ) ( )f x f xm
x x
Average Rate of Change
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Definition: Tangent Line
f(x1)
x1
P
Tangent Line – A line touching the graph of a function at a point P. The line may touch the graph elsewhere and still be considered a tangent line at x = x1.
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Instantaneous Rate of Change
f(x1)
x1
P
The slope of the tangent line to the graph of a function at a point is called the instantaneous rate of change of the function at x1.
How can we determine the slope of the tangent line using a secant line?
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The Slope of a Tangent Line
http://www.scottsarra.org/applets/calculus/SecantTangent.html To estimate the slope of a tangent line at a point x1, choose a point _____ that is very ________ to x1 and determine the ___________ of the line between the coordinates of these two points.