The wind-chill index W is the perceived temperature when the...

The wind-chill index $W$ is the perceived temperature when the actual temperature is $T$ and the wind speed is $v,$ so we can write $W=f(T, v) .$ The following table of values is an excerpt from Table 9.1 $.1 .$ Use the table to find a linear approximation to the wind-chill index function when $T$ is near $-15^{\circ} \mathrm{C}$ and $v$ is near 50 $\mathrm{km} / \mathrm{h}$ . Then estimate the wind-chill index when the temperature is $-17^{\circ} \mathrm{C}$ and the wind speed is 55 $\mathrm{km} / \mathrm{hr}$ .

Key Concepts

Total Differential

The total differential of a function is an expression that approximates the change in the function's value based on the changes in all of its independent variables. It is computed by multiplying each partial derivative by the corresponding change in the variable and summing the results, forming the basis for the tangent plane approximation.

Tangent Plane

The tangent plane is the geometric representation of the linear approximation for a function of two variables. It provides a flat surface that closely approximates the function near a specific point, making it easier to estimate the function’s value for small deviations in the input variables.

Linear Approximation

Linear approximation is a method used to estimate the value of a function at a point close to a known value by using the function’s value and its partial derivatives at the known point. In the context of multivariable functions, it relies on the idea that for small changes in each variable, the function behaves like its tangent plane.

Partial Derivatives

Partial derivatives measure the rate at which a function changes with respect to one variable while keeping the other variables constant. They are crucial when forming a linear approximation because they provide the slopes in the directions of each independent variable, allowing for an effective local linear model.

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