   Chapter 3.10, Problem 8E

Chapter
Section
Textbook Problem

Verify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1.8. (1 + x)–3 ≈ 1 – 3x

To determine

To verify: The linear approximation at a=0 and to determine the value of x for which the linear approximation is accurate to within 0.1.

Explanation

Given:

The function is (1+x)313x and the point a=0.

Result used:

The linear approximation of the function at x=a is g(x)g(a)+g(a)(xa).

Derivative rules: Chain rule

If y=f(u) and u=g(x)  are both differentiable function, then dydx=dydududx.

Calculation:

Consider the function f(x)=(1+x)3.

Differentiate with respect to x,

f(x)=ddx((1+x)3)

Let u=1+x, f(x)=u3.

f(x)=ddx((u)3)

Apply the chain rule and simplify the terms,

f(x)=ddu(u3)dudx=3u31dudx=3u4dudx

Substitute u=1+x in f(x),

f(x)=3(1+x)4ddx(1+x)=3(1+x)4[ddx(1)+ddx(x)]=3(1+x)4[0+1]=3(1+x)4

Thus, the derivative of the function is f(x)=3(1+x)4

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