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Unformatted text preview: Example
Let f (x) = 1
2x + 1 . Use the deﬁnition of the derivative to ﬁnd f (1) = lim f (1 + h) − f (1)
h h→0 , (1) and then write the equation of the tangent line to the curve y = f (x) at the point
(1, f (1)).
Solution: Making the substitution f (1 + h) = 1/[2(1 + h) + 1] into (1), it follows that f (1) = lim h→0 = lim h→0 = lim h→0 = lim h→0 = lim h→0 1
1
−
2(1 + h) + 1 2 · 1 + 1
h
1
1
−
3 + 2h 3
h
3 − (3 + 2h)
3(3 + 2h)
h
1
−2h
·
h 3(3 + 2h)
−2
3(3 + 2h) 2
=−.
9
Thus the slope of the tangent line to the curve y = f (x) at the point (1, f (1)) is f (1) = −2/9.
Using the pointslope formula of the line through (1, f (1)) = (1, 1/3) with slope f (1) = −2/9,
y − f (1) = f (1)(x − 1) or, y− 1
2
= − (x − 1),
3
9 so the equation of the tangent line to the curve through the point (1, f (1)) is
y= 12
− (x − 1).
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.
 Fall '10
 KIHYUNHYUN
 Calculus, Derivative

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