math16B_suppl_notes_1 - Math 16B S06 Supplementary Notes 1...

Math 16B – S06 – Supplementary Notes 1
The Derivative and Local Linear Approximation
For a function g ( x ) of one variable, the value g ( a ) of the derivative of g at the point x = a is
the slope of the tangent line to the graph of g at the point ( a, g ( a )). Near that point, the tangent
line is a “good” approximation to the graph, in the following sense. The equation of the tangent
line is
y = g ( a ) + g ( a )( x - a ) .
The difference
(1) R ( x ) = g ( x ) - g ( a ) - g ( a )( x - a )
is the error you get at x when you approximate g by the function whose graph is the tangent line
(see Figure 1 . 1). The approximation is good near a in the sense that R ( x ) is small for x near a
compared to x - a , vanishingly small in the limit:
(2) lim
x →∞
R ( x )
x - a
= 0 .
To see this, simply divide (1), the equality that defines R ( x ), by x - a , to get
R ( x )
x - a
=
g ( x ) - g ( x )
x - a
- g ( a ) .
As x approaches a , the difference quotient on the right side of the equality approaches g ( a ), so the
ratio on the left side tends to 0.
Among all straight lines through the point ( a, g ( a )), the tangent line gives the best linear approx-
imation near a to the graph of g . For the line with slope c = g ( a ), the error in the approximation
at x , divided by x - a , has the limit g ( a ) - c , not 0, as x approaches a .
The relation (2), while it says you can expect the error in a certain approximation to be small,
does not tell you how small. It does not tell you, for example, how close x should be to a to
guarantee that the approximation is accurate to, say, three decimal places.