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Chapter 1 | Functions and Graphs
The Maximum Value of a Function In many areas of science, engineering, and mathematics, it is useful to know the maximum value a function can obtain, even if we don’t know its exact value at a given instant. For instance, if we have a function describing the strength of a roof beam, we would want to know the maximum weight the beam can support without breaking. If we have a function that describes the speed of a train, we would want to know its maximum speed before it jumps off the rails. Safe design often depends on knowing maximum values. This project describes a simple example of a function with a maximum value that depends on two equation coefficients. We will see that maximum values can depend on several factors other than the independent variable x . 1. Consider the graph in Figure 1.42 of the function y = sin x +cos x . Describe its overall shape. Is it periodic? How do you know?
Figure 1.42 The graph of y = sin x +cos x .
Using a graphing calculator or other graphing device, estimate the x - and y -values of the maximum point for the graph (the first such point where x > 0). It may be helpful to express the x -value as a multiple of π. 2. Now consider other graphs of the form y = A sin x + B cos x for various values of A and B . Sketch the graph when A =2and B = 1, and find the x - and y -values for the maximum point. (Remember to express the x -value as a multiple of π, if possible.) Has it moved?
3. Repeat for A = 1, B = 2. Is there any relationship to what you found in part (2)? 4. Complete the following table, adding a few choices of your own for A and B :
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