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GO01962-Smith-v1.cls
July 4, 2016
UAE_Math_Grade_12_Vol_1_SE_718383_ch3
DEFINITION 1.1 13:38
The linear (or tangent line) approximation of f(x) at x = x is the function
0
′
L(x) = f(x ) + f (x )(x − x ).
0
0
0
Observe that the y-coordinate y of the point on the tangent line corresponding to
1
x = x is simply found by substituting x = x in equation (1.1), so that
1
1
′
y = f(x ) + f (x )(x − x ). (1.2)
0
0
1
1
0
We define the increments Δx and Δy by
Δx = x − x 0
1
and Δy = f(x ) − f(x ).
1
0
Using this notation, equation (1.2) gives us the approximation
′
f(x ) ≈ y = f(x ) + f (x )Δx. (1.3)
1
0
0
1
We illustrate this in Figure 4.2. We sometimes rewrite (1.3) by subtracting f(x ) from
0
both sides, to yield
′
Δy = f(x ) − f(x ) ≈ f (x ) Δx = dy, (1.4)
0
0
1
′
where dy = f (x )Δx is called the differential of y. When using this notation, we also
0
define dx, the differential of x, by dx =Δx, so that by (1.4),
′
dy = f (x ) dx.
0
y
y = f(x)
f(x )
1
y = f(x 0 ) + f �(x )(x - x )
0
0
Δy
y 1
dy
f(x )
0
Δx
x
x 0 x 1
FIGURE 4.2
Increments and differentials

We can use linear approximations to produce approximate values of transcenden-
tal functions, as in example 1.1.
Copyright © McGraw-Hill Education EXAMPLE 1.1 Finding a Linear Approximation

Find the linear approximation to f(x) = cos x at x = ∕3 and use it to approximate
0
cos(1).

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