The following questions illustrate the procedure used to find the rectangle inscribed under the graph of f(x) = sin x in the first quadrant on the interval [0,
] that has maximum area. Here x stands for the x-coordinate of the lower left corner of the rectangle. The area of a rectangle is A = hw, where h and w are the height and width of the rectangle, respectively.
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Define the height as a function of x.
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Define the width as a function of x. (Hint: Use the symmetry of the rectangle's base in the interval [0,
].)
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Enter the height, width and area functions in Y1, Y2 and Y3 of the Y= editor of your TI-83. Unselect Y1 and Y2.
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Define the derivative of the area function in Y4.
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Graph the area function and its derivative. Describe the correspondence between the derivative and the maximum of the function.
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Find the zeros of the derivative on the interval
.
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Verify that the critical point found in Question 5 produces the absolute maximum of the area function and find the maximum area of the rectangle.
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