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Interpretations of the Derivative

Interpretations of the Derivative - Interpretations of the...

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Interpretations of the Derivative Before moving on to the section where we learn how to compute derivatives by avoiding the limits we were evaluating in the previous section we need to take a quick look at some of the interpretations of the derivative. All of these interpretations arise from recalling how our definition of the derivative came about. The definition came about by noticing that all the problems that we worked in the first section in the chapter on limits required us to evaluate the same limit. Rate of Change The first interpretation of a derivative is rate of change. This was not the first problem that we looked at in the limit chapter, but it is the most important interpretation of the derivative. If represents a quantity at any x then the derivative represents the instantaneous rate of change of at . Example 1 Suppose that the amount of water in a holding tank at t minutes is given by . Determine each of the following. (a) Is the volume of water in the tank increasing or decreasing at minute? [ Solution ] (b) Is the volume of water in the tank increasing or decreasing at minutes? [ Solution ] (c) Is the volume of water in the tank changing faster at or minutes? [ Solution ] (d) Is the volume of water in the tank ever not changing? If so, when? [ Solution ] Solution In the solution to this example we will use both notations for the derivative just to get you familiar with the different notations. We are going to need the rate of change of the volume to answer these questions. This means that we will need the derivative of this function since that will give us a formula for the rate of change at any time t . Now, notice that the function giving the volume of water in the tank is the same function that we saw in Example 1 in the last section except the letters have changed. The change in letters between the function in this example versus the function in the example from the last section won’t affect the work and so we can just use the answer from that example with an appropriate change in letters.
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The derivative is. Recall from our work in the first limits section that we determined that if the rate of change was positive then the quantity was increasing and if the rate of change was negative then the quantity was decreasing. We can now work the problem. (a) Is the volume of water in the tank increasing or decreasing at minute? In this case all that we need is the rate of change of the volume at or, So, at the rate of change is negative and so the volume must be decreasing at this time. [ Return to Problems ] (b) Is the volume of water in the tank increasing or decreasing at minutes? Again, we will need the rate of change at . In this case the rate of change is positive and so the volume must be increasing at .
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