Unformatted text preview: Example
Let f (x) = √ 3x . Use the deﬁnition of the derivative to ﬁnd
f (3) = lim f (3 + h) − f (3)
h h→0 , (1) and then write the equation of the tangent line to the curve y = f (x) at the point
(3, f (3)).
Solution: Making the substitution f (3 + h) =
f (3) = lim 3(3 + h) into (1), it follows that
3(3 + h) − √ 3·3 h
√
9 + 3h − 3
.
= lim
h→0
h
h→0 Multiplying numerator and denominator in the above by the conjugate expression,
√
to 9 + 3h − 3, we have
√
√
9 + 3h − 3
9 + 3h + 3
f (3) = lim
·√
h→0
h
9 + 3h + 3
(9 + 3h) − 9
= lim
√
h→0 h
9 + 3h + 3 √
9 + 3h + 3, 3h
√
h→0 h
9 + 3h + 3
3
= lim √
h→0
9 + 3h + 3
3
=√
9+0+3
3
1
=
=.
6
2
= lim Thus the slope of the tangent line to the curve y = f (x) at the point (3, f (3)) is f (3) = 1/2.
Using the pointslope formula of the line through (3, f (3)) = (3, 3) with slope f (3) = 1/2,
y − f (3) = f (3)(x − 3) or, 1
y − 3 = (x − 3),
2 so the equation of the tangent line to the curve through the point (3, f (3)) is
1
y = 3 + (x − 3).
2 ...
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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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