The depth, h (in mm ), of the water runoff down a slope...

The depth, $h$ (in $\mathrm{mm}$ ), of the water runoff down a slope during steady rain $^{3}$ is a function of the distance, $x$ (in meters), from the top of the slope, $h=f(x)=0.07 x^{2 / 3}$ (a) Find $f^{\prime}(x)$. (b) Find $f^{\prime}(30) .$ Include units. (c) Explain how you can use your answer to part (b) to estimate the difference in runoff depths between a point 30 meters down the slope and a point 6 meters farther down.

The depth, $h$ (in $\mathrm{mm}$ ), of the water runoff down a slope during steady rain $^{3}$ is a function of the distance, $x$ (in meters), from the top of the slope, $h=f(x)=0.07 x^{2 / 3}$
(a) Find $f^{\prime}(x)$.
(b) Find $f^{\prime}(30) .$ Include units.
(c) Explain how you can use your answer to part (b) to estimate the difference in runoff depths between a point 30 meters down the slope and a point 6 meters farther down.

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function's value changes with respect to changes in its input variable. In the context of the problem, differentiation is used to determine the instantaneous rate of change of the depth of water runoff with respect to the distance down the slope.

Power Rule

The power rule is a fundamental technique in calculus used for differentiating functions of the form x^n. It states that the derivative of x^n with respect to x is n*x^(n-1). In the given problem, the power rule is applied to the function f(x) = 0.07 * x^(2/3) to compute its derivative.

Evaluating Derivatives at Specific Points

After finding a general expression for the derivative, it is often necessary to evaluate it at a specific point to understand the behavior of the function at that location. In the problem, after computing f?'(x), evaluating it at x = 30 provides the rate of change of the water depth at that point, along with the corresponding units.

Units in Derivatives

When differentiating a function that describes physical quantities, it is important to carefully consider the units of both the original function and its derivative. The derivative's units are determined by dividing the units of the function by the units of the independent variable. In this problem, the function's units (millimeters) divided by the distance (meters) give the derivative units, which are essential for interpreting the rate of change in a physical context.

Linear Approximation

Linear approximation is the method of using the tangent line at a point to estimate the value of a function close to that point. The derivative at a specific point provides the slope of this tangent line. In the problem, the value of the derivative at x = 30 is used to estimate the change in runoff depth when moving a small distance down the slope (e.g., 6 meters) by multiplying the derivative by the change in x.

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