MATH 131-505 Spring 2015
c
Wen
Liu
3.9
3.9 Linear Approximations and Differentials
Given a curve y = f (x), we use the tangent line at (a, f (a)) as an approximation to the curve f when
x is near a. An equation of this tangent line is
y = f (a) + f 0 (a)(x − a)
and the approximation
f (x) ≈ f (a) + f 0 (a)(x − a)
is called the linear approximation or tangent line approximation of f at a. The linear function
whose graph is this tangent line, that is,
L(x) = f (a) + f 0 (a)(x − a)
is called the linearization of f at a.
√
Example 1: (p. 241) Find the linearization of the function f (x) = x + 3 at a = 1 and use it to
√
√
approximate the numbers 3.98 and 4.05. Are these approximations overestimates or underestimates?
Applications to Physics:
1. For pendulum, the linearization of f (x) = sin x at a = 0 is L(x) = x.
2. For paraxial rays, the linear approximations
sin x ≈ x and
cos x ≈ 1
are used because x is close to 0.
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MATH 131-505 Spring 2015
3.9
c
Wen
Liu
Differentials
The ideas behind linear approximations are sometimes
formulated in the terminology and notation of differentials. If y = f (x), where f is a differentiable function,
then the differential dx is an independent variable; that
is, dx can be given the value of any real number. The
differential dy is then defined in terms of dx by the
equation
dy = f 0 (x)dx
The geometric meaning of differentials is shown on the
left.
If we let dx = x − a, then x = a + dx and we can rewrite the linear approximation
f (x) ≈ f (a) + f 0 (a)(x − a)
in the notation of differentials:
f (a + dx) ≈ f (a) + dy
Examples:
2. (p. 246) Find the differential of y =
√
1 + ln z.
3. (p. 244) Use differentials to estimate
√
4.05.
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MATH 131-505 Spring 2015
3.9
c
Wen
Liu
4. (p. 245) Use a linear approximation (or differentials) to estimate
(a) 8.062/3
(b) 1/1002
5. (p. 246) Suppose that the only information we have about a function f is that f (1) = 5 and the
graph of its derivative is as shown. Use a linear approximation to estimate f (0.9).
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