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Chapter 2 | Limits
Definition The secant to the function f ( x ) through the points ⎛
⎝ a , f ( a ) ⎞
⎛ ⎝ x , f ( x ) ⎞
⎠ and
⎠ is the line passing through these points. Its
slope is given by
f ( x )− f ( a ) x − a .
(2.1)
m
sec =
The accuracy of approximating the rate of change of the function with a secant line depends on how close x is to a . Aswe see in Figure 2.5 , if x is closer to a , the slope of the secant line is a better measure of the rate of change of f ( x ) at a .
Figure 2.5 As x gets closer to a , the slope of the secant line becomes a better approximation to the rate of change of the function f ( x ) at a .
The secant lines themselves approach a line that is called the tangent to the function f ( x ) at a ( Figure 2.6 ). The slope of the tangent line to the graph at a measures the rate of change of the function at a . This value also represents the derivative of the function f ( x ) at a , or the rate of change of the function at a . This derivative is denoted by f ′( a ). Differential calculus is the field of calculus concerned with the study of derivatives and their applications. For an interactive demonstration of the slope of a secant line that you can manipulate yourself, visit this applet ( Note: this site requires a Java browser plugin): Math Insight (http://www.openstax.org/l/20_mathinsight) .
Figure 2.6 Solving the Tangent Problem: As x approaches a , the secant lines approach the tangent line.
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