Chapter 4 | Applications of Derivatives
357
We see that
f ( x ) = sin x ⇒ f ⎛ ⎝ π 3 ⎞ f ′( x ) =cos x ⇒ f ′ ⎛ ⎝ π 3 ⎞
⎛ ⎝ π 3
⎞ ⎠ = 3 2
⎠ = sin
⎠ =cos ⎞ ⎠ = 1 2 Therefore, the linear approximation of f at x = π /3 is given by Figure 4.9 . L ( x ) = 3 2 + ⎛ ⎝ π 3 1 2 ⎞ ⎠ To estimate sin(62°) using L , we must first convert 62° to radians. We have 62° = 62 π 180 ⎛ ⎝ x − π 3
radians, so the
estimate for sin(62°) is given by sin(62°) = f ⎛ ⎝ 62 π 180 ⎞ ⎠ ≈ L ⎛
⎞ ⎠ = 3 2 +
⎛ ⎝ 62 π
⎞ ⎠ = 3 2 +
⎛ ⎝ 2 π
⎞ ⎠ = 3 2 +
⎝ 62 π 180
π 3
π 180 ≈ 0.88348.
1 2
1 2
180 −
180
Figure 4.9 The linear approximation to f ( x ) = sin x at x = π /3 provides an approximation to sin x for x near π /3.
Find the linear approximation for f ( x ) =cos x at x = π 2 .
4.6
Linear approximations may be used in estimating roots and powers. In the next example, we find the linear approximation for f ( x ) = (1+ x ) n at x =0, which can be used to estimate roots and powers for real numbers near 1. The same idea can be extended to a function of the form f ( x ) = ( m + x ) n to estimate roots and powers near a different number m .
Example 4.7 Approximating Roots and Powers
Find the linear approximation of f ( x ) = (1+ x ) n at x =0. Use this approximation to estimate (1.01) 3 .
Solution The linear approximation at x =0 is given by
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