Use the fact that for a power function y=k x^n, for small...

Use the fact that for a power function $y=k x^{n},$ for small changes, the percent change in output $y$ is approximately $n$ times the percent change in input $x$. (See Problem 88.) The stopping distance $s,$ in feet, of a car traveling $v$ mph is $s=v^{2} / 20 .$ An increase of $10 \%$ in the speed of the car leads to what percent increase in the stopping distance?

Use the fact that for a power function $y=k x^{n},$ for small changes, the percent change in output $y$ is approximately $n$ times the percent change in input $x$. (See Problem 88.)
The stopping distance $s,$ in feet, of a car traveling $v$ mph is $s=v^{2} / 20 .$ An increase of $10 \%$ in the speed of the car leads to what percent increase in the stopping distance?

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Key Concepts

Power Functions

A power function is a mathematical function of the form y = k × x^n, where k is a constant and n is an exponent. This type of function illustrates how the output value changes in relation to the input variable raised to a power, providing insight into the scaling behavior of various phenomena.

Differential Approximation for Small Changes

For small changes in the input of a function, the differential or linear approximation estimates how the output changes. In a power function, this means that a small percent change in x leads to an approximately n times larger percent change in y, which simplifies calculations and predictions in applied problems.

Sensitivity Analysis via Exponents

The exponent n in a power function not only determines the curvature of the function but also quantifies the sensitivity of the output to input changes. This sensitivity analysis is crucial in many practical applications, as it indicates that even a modest change in the input can result in a significantly amplified change in the output when n is greater than one.

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