1 2.1 - the tangent and velocity problem © john seims (2008)

5
1 2.1 - The Tangent and Velocity Problem © John Seims (2008)

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Page 1: 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)

Page 2: 1 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.

2 1sec

2 1

( ) ( )f x f xm

x x

Average Rate of Change

Page 3: 1 2.1 - The Tangent and Velocity Problem © John Seims (2008)

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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.

Page 4: 1 2.1 - The Tangent and Velocity Problem © John Seims (2008)

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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?

Page 5: 1 2.1 - The Tangent and Velocity Problem © John Seims (2008)

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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.