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Unformatted text preview: Section 15.3 Partial Derivatives Differentiating Functions of more than one Variable 1. Basic Definitions In single variable calculus, the derivative is defined to be the instanta- neous rate of change of a function f ( x ). In multivariable, this definition no longer makes sense because there are many different directions in which one could move, so the rate of change will depend not only upon the point we are at, but also the direction we choose to move. We illustrate with an example. Example 1.1. A sheet of unevenly heated metal lies in the xy-plane with the lower left corner at the origin. The temperature at any point of the sheet is a function of x and y , T ( x, y ). After taking some mea- surements, you gather the following information: 3 85 90 110 135 155 180 2 100 110 120 145 190 170 1 125 128 135 160 175 160 120 135 155 160 160 150 y/x 1 2 3 4 5 If we are stood at the point (2 , 1) in the xy-plane, the temperature changes depending upon the direction we choose to mover. If we fix the y-value at 1 and move in the positive x-direction, then the function becomes a function of a single variable, so we can consider its rate of change in the x-direction. Specifically, we have T ( x, 1) T (3 , 1)- T (2 , 1) 1 = 160- 135 1 = 25 , so the rate of change of temperature in the x-direction is approximately 25 degrees per unit moved. This means that the temperature is increas- ing in the x-direction quite quickly. Likewise, in the y-direction we can consider the rate of change like with a single variable function. Specifically, fixing x = 2, we have T (2 , y ) T (2 , 2)- T (2 , 1) 1 = 120- 135 1 =- 15 , so the rate of change is negative in the y-direction (meaning the tem- perature is dropping in the y-direction)....
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