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Chapter 3 | Derivatives
Example 3.1 Finding a Tangent Line
Find the equation of the line tangent to the graph of f ( x ) = x 2 at x =3.
Solution First find the slope of the tangent line. In this example, use Equation 3.3 . m tan = lim x →3 f ( x )− f (3) x −3
Apply the definition.
x 2 −9 x −3
f ( x ) = x 2 and f (3) =9.
= lim
Substitute
x →3
( x −3)( x +3)
= lim ( x +3) = 6 Factor the numerator to evaluate the limit. Next, find a point on the tangent line. Since the line is tangent to the graph of f ( x ) at x =3, it passes through the point ⎛ ⎝ 3, f (3) ⎞ ⎠ . We have f (3) =9, so the tangent line passes through the point (3, 9). x →3 x −3 = lim x →3 Using the point-slope equation of the line with the slope m =6 and the point (3, 9), we obtain the line y −9=6( x −3). Simplifying, we have y =6 x −9. The graph of f ( x ) = x 2 and its tangent line at 3 are shown in Figure 3.6 .
Figure 3.6 The tangent line to f ( x ) at x =3.
Example 3.2 The Slope of a Tangent Line Revisited
Use Equation 3.4 to find the slope of the line tangent to the graph of f ( x ) = x 2 at x =3.
Solution The steps are very similar to Example 3.1 . See Equation 3.4 for the definition.
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