Math 151 Section 3.11
Differentials, Linear and Quadratic Approximations
Lines and quadratics are easy to compute. In this section we approximate function values and changes in
function values using linear and quadratic approximations.
 f  f ( x )  f ( a ) is the exact change in the function v alue
 x  x  a  dx
f
is close to f ' ( a ).
x
 f  f ' ( a )  x  f ' ( a ) dx We define f ' ( a ) dx  df .
If x  a is small enough, then
Note that df depends on both x and a.
The linearizat ion of f about a is L ( x )  f ( a )  f ' ( a )( x  a )
is the equation of the tangent line to f at ( a , f ( a )).
We can get a closer approximation with a quadratic. We find a quadratic function that agrees with our
function at a , has the same derivative at a , and has the same second derivative at a .
This function is the quadratic approximation to
Q ( x )  f ( a )  f ' ( a )( x  a ) 
f about a ,
1
f " ( a )( x  a ) 2
2
Examples:
1 . Use a differenti al or linearizat ion about a  25 to approximat e 24 .
2 . Use a quadratic approximat ion to approximat e tan(5  ).
3 . A cylindrica l can with no top or bootom has a radius of 6 with an error of  .02
and a height of 4. Approximat e the percent relative error in the volume of the can.
(
dV
)100 %
V