Processing math: 100%
SOLUTION 18: Begin with the function
f(x)=log(100−x)
and choose
x−values:0→h4
so that
Δx=h4−0=h4
(Recall that D{log(g(x))}=1g(x)⋅g′(x)⋅1ln(10).)
The derivative of y=f(x) is
f′(x)=1100−x⋅(−1)⋅1ln(10)
=−1(−1)(x−100)⋅1ln(10)
=1x−100⋅1ln(10)
The exact change of y−values is
Δy=f(h4)−f(0)
=log(100−(h4))−log(100+(0))
=log(100−h4)−log(100)
( Recall that log(10n)=n.)
=log(100−h4)−log(102)
=log(100−h4)−2
The Differential is
dy=f′(0) Δx
=1((0)4)−100⋅1ln(10)⋅(h4)
=1−100⋅1ln(10)⋅(h4)
=−1100ln(10)h4
Since h is "small" we will assume that
Δy≈dy ⟶
log(100−h4)−2≈−1100ln(10)h4 ⟶
log(100−h4)≈2−1100ln(10)h4
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