Question

7-10: 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. \({\left( {1 + x} \right)^{ - 3}} \approx 1 - 3x\)

Short answer

Linear approximation is \(L\left( x \right) = 1 - 3x\). The values of \(x\) lies between \( - 0.116 < x < 0.144\).

Step-by-step solution

Linear Approximation of a Curve at a Given point

We can approximate a function \(f\left( x \right)\) of a curve at a point \(a\) by the tangent line at the given point \(\left( {a,f\left( a \right)} \right)\). To do that first we have to find the equation of the tangent line at that given point, which is

\(y - f\left( a \right) = f'\left( a \right)\left( {x - a} \right)\)

Hence the linear approximation of the curve is

\(L\left( x \right) = f\left( a \right) + f'\left( a \right)\left( {x - a} \right)\)

Finding the Linear approximation of the Curve

Given that \(f\left( x \right) = {\left( {1 + x} \right)^{ - 3}}\) and the point is \(a = 0\).

Now \(f'\left( x \right) = - 3{\left( {1 + x} \right)^{ - 4}}\). Hence slope of the curve at \(a = 0\) is \(f'\left( 0 \right) = - 3\).

Now the equation of the tangent line at \(\left( {0,f\left( 0 \right)} \right)\) is,

\(\begin{aligned}y &= f\left( 0 \right) + f'\left( 0 \right)\left( {x - 0} \right)\\ &= 1 - 3x\end{aligned}\)

Hence the linear approximation is \(L\left( x \right) = 1 - 3x\). So \({\left( {1 + x} \right)^{ - 3}} \approx 1 - 3x\).

For the accuracy of the linear approximation within 0.1 we have,

\({\left( {1 + x} \right)^{ - 3}} - 0.1 < x < {\left( {1 + x} \right)^{ - 3}} + 0.1\)

This is true when \( - 0.116 < x < 0.144\).

Plot the graph

The procedure to draw the graph of the above equation by using the graphing calculator is as follows:

To check the answer visually draw the graph of the function\({f_1}\left( x \right) = {\left( {1 + x} \right)^{ - 3}} - 0.1\),\({f_2}\left( x \right) = {\left( {1 + x} \right)^{ - 3}} + 0.1\)and \({f_3}\left( x \right) = x\) by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\({\left( {1 + x} \right)^{ - 3}} - 0.1\)in the\({Y_1}\)tab.
2. Select the “STAT PLOT” and enter the equation\({\left( {1 + x} \right)^{ - 3}} + 0.1\)in the\({Y_2}\)tab.
3. Select the “STAT PLOT” and enter the equation\(x\)in the\({Y_3}\)tab.
4. Enter the “GRAPH” button in the graphing calculator.

Visualization of graph of the function\({f_1}\left( x \right) = {\left( {1 + x} \right)^{ - 3}} - 0.1\),\({f_2}\left( x \right) = {\left( {1 + x} \right)^{ - 3}} + 0.1\)and \({f_3}\left( x \right) = x\) is shown below: