2'7 Tangents, Velocity, and Other Rates of Change

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2.7 Tangents, Velocity, and Other Rates of
Change
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Definition Secant and Tangent Lines
Secant Line A line passing thorough two points
on a graph of a function.
Tangent Line A line that touches the graph at a
point. The tangent line may cross the graph at
other points depending on the graph.
P
f(x1)
x1
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Slope of the Tangent Line
The slope of the tangent line can be found from
the slope of the secant line.
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Slope of the Tangent Line
Now let x1 a and x2 x. This changes the
formula to
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Slope of the Tangent Line
To determine the slope of the tangent line, let a
approach x (as in our animation). In terms of
limits, we are finding the slope of the secant
line as x ? a.
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Slope of the Tangent Line
Now let x1 a and x2 ah, where h is the
distance from x1 to x2. This changes the formula
to
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Slope of the Tangent Line
To determine the slope of the tangent line using
this form, let a h approach a. This is
equivalent to allowing h ? 0.
?
means h ? 0
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The Derivative
The derivative of a function f at a number a,
denoted by f '(a), is
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Interpretations of the Derivative
The tangent line to y f (x) at (a, f (a)) is
the line through (a, f (a)) whose slope is equal
to f '(a), the derivative of f at a.
The derivative f ' (a) is the instantaneous rage
of change of y f(x) with respect to x when x
a.