2.1. Tangent Lines and Rates of Change http://www.ck12.org
Example 2:
Iff(x) =x^2 − 3 ,findf(x)and use the result to find the slope of the tangent line atx=2 andx=− 1.
Solution:
Since
f′(x) =limh→ 0 f(x+hh)−f(x),then
f′(x) =limh→ 0 [(x+h)(^2) − 3 ]−[x (^2) − 3 ]
h
=limh→ 0 x
(^2) + 2 xh+h (^2) − 3 −x (^2) + 3
h
=limh→ 02 xh+h
2
h
=limh→ 0 ( 2 x+h)
= 2 x
To find the slope, we simply substitutex=2 into the resultf(x),
f′(x) = 2 x
f′( 2 ) = 2 ( 2 )
= 4
and
f′(x) = 2 x
f′(− 1 ) = 2 (− 1 )
=− 2
Thus slopes of the tangent lines atx=2 andx=−1 are 4 and− 2 ,respectively.
Example 3:
Find the slope of the tangent line to the curvey=^1 xthat passes through the point( 1 , 1 ).
Solution:
Using the slope of the tangent formula
f′(x) =hlim→ 0 f(x+hh)−f(x)
and substitutingy=^1 x,