w09e (2) - Math 151 Workshop 9 1 A square and a circle are...

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Unformatted text preview: Math 151, Workshop 9 1. A square and a circle are placed so that the circle is outside the square and tangent to a side of the square. The sum of the length of one side of the square and the circle's diameter is 12 feet, as shown. Suppose the length of one side of the square is x feet. a) Write a formula for f (x), the sum of the total area of the square and the circle. What is the domain of this function when used to describe this problem? (The domain should be 12 related to the problem statement.) Sketch a graph of f (x) on its domain. 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 x b) Suppose that the object (square or circle) with larger area is painted red, and the object (square or circle) with smaller area is painted green. The cost of red paint to cover 1 square foot is $4, and the cost of green paint to cover 1 square foot is $10. Let g(x) be the function which gives the cost of painting the squares. Describe the function g(x). Sketch a graph of g(x) on its domain. c) Where is the function g(x) continuous? Where is it differentiable? Which value of x gives the least cost? 2. A number x0 is called a fixed point of the function f if and only if f (x0 ) = x0 . a) Find all the fixed points of the following functions to three-place accuracy. Illustrate your answers graphically f (x) = x2 g(x) = 3ex - 2e-x h(x) = 2 arctan x. 3 b) Suppose that f is a differentiable function and f (x) < 1 for all x. Using the mean value theorem, prove that f can have no more than one fixed point. c) To which of the functions in a) does the general statement in b) apply? 3. Suppose that the picture to the right is the graph of a polynomial. What is the lowest degree that this polynomial could have? Explain your assertion carefully. 4. Suppose 5x3 y - 3xy 2 + y 3 = 6 . (1, 2) is a point on this curve. Is the curve concave up or concave down at (1, 2)? Hint. Use implicit differentiation. x y ...
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This note was uploaded on 03/24/2011 for the course PHYSICS 123 taught by Professor Madey during the Spring '08 term at Rutgers.

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