Section 3.11: Linear and Quadratic Approximations
Definition: The linear approximation or linearization of f at x = a is the equation of
the tangent line to the curve y = f (x) at x = a. That is,
L(x) = f (a) + f 0 (a)(x − a).
The linearization is a good approximation of f (x) for values of x near x = a.
Example: Find the linearization of f (x) = x3 + 3x2 at a = 1.
Example: Find the linearization of f (x) = √
1
at a = 0.
x+2
1
Example: Use a linear approximation to estimate
√
36.1.
Example: Use a linear approximation to estimate (1.97)4 .
Example: Use a linear approximation to estimate cos(31◦ ).
2
Example: Find the linear approximation of f (x) =
√
√
the values of 0.9 and 1.01.
√
1 − x at a = 0 and use it to approximate
Example: Suppose that we don’t have a formula for f (x), but we know that f (1) = 2 and
√
f 0 (x) = x3 + 1. Use a linear approximation to estimate the value of f (1.1).
3
Example: Suppose the linear approximation of f (x) at a = 2 is given by y = 10 − 3x.
(a) Find f (2) and f 0 (2).
(b) If g(x) = [f (x)]2 , find the linear approximation for g(x) at x = 2.
Definition: The quadratic approximation of f (x) at x = a is
Q(x) = f (a) + f 0 (a)(x − a) +
f 00 (a)
(x − a)2 .
2
The quadratic approximation is a better estimate of f (x) than the linearization.
4
Example: Find the quadratic approximation of f (x) =
√
x at x = 4.
Example: Find the quadratic approximation of f (x) =
1
at x = 3.
x
5