#### 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.

#### Answer

The values of *x* for which the linear approximation is accurate to within 0.1 is −0.1162<x<0.1441.

#### Explanation

**Given:**

The function is (1+x)−3≈1−3x and the point a=0.

**Result used:**

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

**Derivative rules:** Chain rule

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

**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)=u−3.

f′(x)=ddx((u)−3)

Apply the chain rule and simplify the terms,

f′(x)=ddu(u−3)dudx=−3u−3−1dudx=−3u−4dudx

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.

Substitute x=0 in f′(x)=−3(1+x)−4,

f′(0)=−3(1)−4=−3

Substitute x=0 in f(x)=(1+x)−3,

f(x)=(1+0)−3=1

Substitute the value a=0, f(x)≈f(a)+f′(a)(x−a),

f(x)≈f(0)+f′(0)(x−0)≈f(0)+f′(0)(x)

Substitute f′(0)=−3 and f(0)=1,

f(x)≈1−3(x−0)=1−3x

Here, the function f(x)=(1+x)−3,

(1+x)−3≈1−3x

Hence, the required result is verified.

Now to determine the value of *x* for which |f(x)−L(x)|<0.1.

Since the linear approximation at a=0 is (1+x)−3≈1−3x, the function f(x)=(1+x)−3 and the linearization L(x)=1−3x.

|(1+x)−3−(1−3x)|<0.1−0.1<(1+x)−3−1+3x<0.11−0.1<(1+x)−3+3x<1+0.10.9<(1+x)−3+3x<1.1

Here, y=(1+x)−3+3x.

Use the online graphing calculator to draw the y=(1+x)−3+3x as shown below in Figure 1.

Form Figure 1, it is observed that the function for which the linear approximation is accurate to within 0.1.

Use the online graphing calculator to zoom towards the value y=0.9 and y=1.1 as shown below in Figure 2.

From Figure 2, it is observed that the graph of the function y=(1+x)−3+3x lies between 0.9 and 1.1 whenever −0.1162<x<0.1441.

Therefore, the value of −0.1162<x<0.1441 for which the linear approximation of function (1+x)−3 is accurate to within 0.1.