Answer
a) f ( x ) ≈ 1 - 2x
g ( x ) ≈ 1 - 2x
h ( x ) ≈ 1 - 2x
The 3 functions can be reduced to the same linearization. This happened because many complicated curves can reduced to the same simple linear approximation.
b) The function f ( x ) is closest to the tangent line therefore the tangent line is a better approximation for f ( x ) than g ( x ) or h ( x ).
Work Step by Step
a = 0
f ( x ) = ( x – 1 )²
f ( 0 ) = ( ( 0 ) – 1 )² = 1
f ‘ ( x ) = 2( x – 1 )
f ‘ ( 0 ) = 2( ( 0 ) – 1 )
f ‘ ( 0 ) = - 2
32a) L ( x ) = 1 – 2x
g ( x ) = $e^{-2x}$
g ( 0 ) = $e^{-2( 0 )}$ = 1
g ‘ ( x ) = - 2 $e^{-2x}$
g ‘ ( 0 ) = - 2 $e^{-2( 0 )}$ = - 2
32a) L ( x ) = 1 – 2x
h ( x ) = 1 + ln( 1 – 2x )
h ( 0 ) = 1 + ln( 1 – 2( 0 ) ) = 1 + ln 1 = 1 + 0 = 1
h ‘ ( x ) = $\frac{-2}{1 - 2x}$
h ‘ ( 0 ) = $\frac{-2}{1 - 2( 0 )}$= $\frac{-2}{1}$= - 2
32a) L ( x ) = 1 – 2x
32a) The linearization for f ( x ) , g ( x ) and h ( x ) are the same. The happened because many complicated curves can be reduced to a simple linear expression.
32b) f ( x ) is the curve that is closest to the tangent line so the tangent line approximates f ( x ) the closest of the 3 functions.
