Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 13 - Partial Derivatives - 13.1 Functions Of Two Or More Variables - Exercises Set 13.1 - Page 917: 77

Answer

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Work Step by Step

Step 1 Most real-world phenomena are modeled by multivariate functions, usually in two or three variables. These multivariate functions take more than one input or variable and produce an output variable that is unique to the combination of inputs. Step 2 A physical application of a function in two variables includes mapping a location on Earth to its temperature. The temperature \(T\) can be expressed as a function of the latitude \(L_1\) and longitude \(L_2\): \[ T = G(L_1, L_2) \] Step 3 Different combinations of \(L_1\) and \(L_2\) result in a unique value of temperature, but multiple locations might share the same temperature. The domain of the function is related to the range of longitude and latitude values. For longitude, the range can be assigned \(0^\circ \leq L_2 < 360^\circ\), and for latitude, \(-90^\circ \leq L_1 \leq 90^\circ\). Step 4 The location-to-temperature mapping can be extended to a function in three variables. A function assigns a temperature \(T\) to a point in 3D space defined by the coordinates \(x\), \(y\), and \(z\): \[ T = G(x, y, z) \] The domain of this function is limited by the region under study. Step 5 In the geological sciences, another application of a function in two variables is modeling the elevation of geological features. Elevation \(E\) can be assigned to a point's coordinates \((x, y)\): \[ E = F(x, y) \] The domain of this function depends on the size of the studied region. Step 6 Multivariate functions have applications outside the physical sciences. In data analytics, they are used to analyze trends and create AI models for predicting consumer behavior. For instance, the probability of a person buying a product can be a function of their age and location: \[ P = F(a, l) \] Here, \(a\) is the age and \(l\) is the location. Adding more factors or independent variables can extend the function to three or more variables.
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