Question

Suppose that the only information we have about a function \(f\) is that \(f\left( 1 \right) = 5\) and the graph of its derivative is as shown.

(a) Use a linear approximation to estimate \(f\left( {0.9} \right)\) and \(f\left( {1.1} \right)\).

(b) Are your estimates in part (a) too large or too small? Explain.

Short answer

1. The values are \(f\left( {0.9} \right) \approx 4.8\) and \(f\left( {1.1} \right) \approx 5.2\).
2. The slope of the \(f\left( x \right)\) are positive, but the tangents to the function \(f\left( x \right)\)are becoming narrower. So the tangent lines should lie above the curve of \(f\left( x \right)\). So, the estimates in part (a) are too large.

Step-by-step solution

Find the linearization of function \(f\left( x \right)\)

It is given that\(f\left( 1 \right) = 5\). On observing the graph carefully, it can be deduced that\(f'\left( 1 \right) = 2\). The linearization of the function \(f\left( x \right)\)at \(x = 1\)is written as \(L\left( x \right) = f\left( 1 \right) + f'\left( 1 \right)\left( {x - 1} \right)\). On plugging the values of \(f'\left( 1 \right)\), \(f\left( 1 \right)\) in \(L\left( x \right)\) and simplifying, we obtain :

^.\(\begin{aligned}{c}L\left( x \right) = f\left( 1 \right) + f'\left( 1 \right)\left( {x - 1} \right)\\ &= 5 + 2\left( {x - 1} \right)\\ &= 5 + 2x - 2\\ &= 3 + 2x\end{aligned}\)

Find \(f\left( {0.9} \right)\)and \(f\left( {1.1} \right)\)

The value of the function\(f\left( x \right)\)at \(x = 0.9\)and \(x = 1.1\) can be approximated as\(f\left( {0.9} \right) \approx L\left( {0.9} \right)\), \(f\left( {1.1} \right) \approx L\left( {1.1} \right)\) . On plugging \(x = 0.9,\,1.1\) in \(L\left( x \right)\)and simplifying, we obtain as follows:

\(\begin{aligned}{c}f\left( {0.9} \right) \approx L\left( {0.9} \right)\\ &= 2\left( {0.9} \right) + 3\\ &= 4.8\end{aligned}\)

\(\begin{aligned}{c}f\left( {1.1} \right) \approx L\left( {1.1} \right)\\ &= 2\left( {1.1} \right) + 3\\ &= 5.2\end{aligned}\)

Thus, \(f\left( {0.9} \right) \approx 4.8\) and \(f\left( {1.1} \right) \approx 5.2\).

Draw the conclusion

On reading the graph, it can be concluded that \(f'\left( x \right)\) is decreasing but remains positive for all \(x\). This implies that the slope of \(f\left( x \right)\)are positive, but the tangents to the function \(f\left( x \right)\)are becoming narrower. So the tangent lines should lie above the curve of \(f\left( x \right)\). So, the estimates in part (a) are too large.