Question 2: What do the partial derivatives of a function tell us? x y. In the last question, we calculated the partial derivatives of

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1 Question : What do the partial derivatives of a function tell us? In the last question, we calculated the partial derivatives of portion of this surface is graphed in Figure 1. f(, ) 4. A Figure 1 The surface that surface. f(, ) 4 and the point (, 1, 4) on The partial derivatives with respect to at and 1 is f, f,1 6 1 The sign of the partial derivative is positive. Following what we learned in earlier chapters, this must indicate that the function is increasing in one direction. To see this, let s look at a slice of this function. The blue line represents a slice of the function in the direction located at 1. As values increase (moving right to left), the function gets larger. This can be seen more clearl b setting 1 in the function, 10

2 The slope of the tangent line at is 6. z f,1 4 f,1 Figure - The slope of the tangent line at = is equal to,1 f. This is the value of the partial derivative with respect to at the point and 1. It also indicates the rate at which z changes along the slice where is fied. Rate of Change If is held constant on the function z f,, then f, gives the rate of change of z with respect to. If is held constant on the function z f,, then f, gives the rate of change of z with respect to. On a graph, these rates ma be visualized as the slope of a tangent line on a slice where either or is fied. 11

3 Eample 5 Interpret a Partial Derivative Suppose f(, ) 4. Earlier in this section, we found that f, 4 f, Interpret the value of the partial derivative along the slice illustrated below. Solution Since the value of the partial derivative with respect to is negative, z must be decreasing as increases along the slice. To see this graphicall, we need to note that the red line is a slice in the direction. The equation of of the slice is obtained b setting, f(, )

4 z f, Visuall. we can see that the function is decreasing. The slope of the tangent line at 1 is -6. The derivative of a function of one variable ma also be interpreted in terms of economics and finance. At the beginning of this section, we introduced a profit function with two variables. Depending on the partial derivative taken, we can determine the rate at which profit is changing when production of one product or the other is increased. Eample 6 Profit A business produces thousand units of product A and thousand units of product B. Its profit function is P (, ) thousand dollars Use the profit function to answer the parts below. a. Find and interpret P 5,9. Solution Set 5 and 9 in the function, 13

5 P (5,9) thousand dollars This tells ou that when 5000 units of product A and 9000 units of product B are produced and sold, the profit is $469,000. b. Find and interpret 5,9 P. Solution Hold constant and calculate the partial derivative with respect to, P, The derivative of each of the other terms is zero since the are constants 0 Now set 5 and 9 in the partial derivative to ield P 5, Since this function indicates how profit P changes as the number of units of product A increased, the units on this rate are thousands of dollars dollars thousands of units of product A units of product A This means that if production of product A is increased b 1 unit, the profit will increase b $110. c. Find and interpret 5,9 P. Solution Hold constant and calculate the partial derivative with respect to, 14

6 P, Now substitute the numbers into the partial derivative, P 5, If the production of product B increases b 1 unit, the profit will decrease b $18. This eample demonstrates the process for calculating how the profit changes when the production of one product is increased (holding the other constant). In earlier chapters, we learned that this is called marginal profit. In the contet of multivariable functions, partial derivatives also represent margins. However we also need to specif the input which will change in the marginal profit. In part a of Eample 6, we might state that the marginal profit with respect to product A is $110 per unit of product A. Similar margins ma be calculated for other functions like revenue, cost, and productivit. 15