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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 [link] of the function y = sin x + cos x . Describe its overall shape. Is it periodic? How do you know?
    An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of the function “y = sin(x) + cos(x)”, a curved wave function. The graph of the function decreases until it hits the approximate point (-(3pi/4), -1.4), where it increases until the approximate point ((pi/4), 1.4), where it begins to decrease again. The x intercepts shown on this graph of the function are at (-(5pi/4), 0), (-(pi/4), 0), and ((3pi/4), 0). The y intercept is at (0, 1).
    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 = 2 and 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 :
    A B x y A B x y
    0 1 3 1
    1 0 1 3
    1 1 12 5
    1 2 5 12
    2 1
    2 2
    3 4
    4 3
  5. Try to figure out the formula for the y -values.
  6. The formula for the x -values is a little harder. The most helpful points from the table are ( 1 , 1 ) , ( 1 , 3 ) , ( 3 , 1 ) . ( Hint : Consider inverse trigonometric functions.)
  7. If you found formulas for parts (5) and (6), show that they work together. That is, substitute the x -value formula you found into y = A sin x + B cos x and simplify it to arrive at the y -value formula you found.

Key concepts

  • For a function to have an inverse, the function must be one-to-one. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test.
  • If a function is not one-to-one, we can restrict the domain to a smaller domain where the function is one-to-one and then define the inverse of the function on the smaller domain.
  • For a function f and its inverse f −1 , f ( f −1 ( x ) ) = x for all x in the domain of f −1 and f −1 ( f ( x ) ) = x for all x in the domain of f .
  • Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions.
  • The graph of a function f and its inverse f −1 are symmetric about the line y = x .

Key equations

  • Inverse functions
    f −1 ( f ( x ) ) = x for all x in D , and f ( f −1 ( y ) ) = y for all y in R .

For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.

For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.

f ( x ) = x 2 4 , x 0

a. f −1 ( x ) = x + 4 b. Domain : x −4 , range : y 0

Got questions? Get instant answers now!

Questions & Answers

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You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject. Well, this is what I guess so.
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Practice Key Terms 5

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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