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Find the points, if any, guaranteed by the Mean Value Theorem for the closed interval [ a,b ]

Determine whether Rolle’s Theorem can be applied to f on the closed interval [ a,b ]. If Rolle’s theorem can be applied, find all values of c in the open interval ( a,b ) such that f′(c)=0 F(x) = ∣x – 2∣ - 2 [0,4].

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Find the points, if any, guaranteed by the Mean Value Theorem for the closed interval [ a,b ]

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  1. Determine whether Rolle’s Theorem can be applied to f on the closed interval [a,b]. If Rolle’s theorem can be applied, find all values of c in the open interval (a,b) such that f′(c)=0 F(x) = ∣x – 2∣ - 2 [0,4]

  2. Find the points, if any, guaranteed by the Mean Value Theorem for the closed interval [a,b] F(x) = √x – 2x [0,4]

  3. Use the derivative tests to determine any relative extrema G(x) = 2x²(1-x²)

  4. Sketch the graph of a function f, having the given characteristics: F(0)=f(6)=0 F′(3) = f′(5)=0 F′(x)>0 if x<3 F′(x) >0 if 3<x<5 F′(x) <0 if x>5 F′′(x) <0 if x<3 or x>4 F′′(x) >0 if 3<x<4

  5. The cost of inventory depends on the ordering and storage costs according to the inventory model: C = (Q/x)s + (x/2)r Determine the order size that will minimize the cost, assuming that sales occur at a constant rate, Q is the number of units sold per year, r is the cost of storing one unit for 1 year, s is the cost of placing an order, and x is the number of units per order.

  6. Analyze and sketch the graph of the function: F(x) = (x-1)³(x-3)²

  7. At noon, ship A is 100 km due east of ship B. Ship A is sailing west at 12 km per hour, and ship B is sailing south at 10 km per hour. At what time will the ships be nearest to each other, and what will this distance be?

  8. Use newton’s method to approximate any real zeros of the function accurate to 3 decimal places. Use the zero finder on calculator to verify. F(x) = x³ - 3x -1

  9. Use the information to evaluate and compare Δy and dy: F(x) = x4+1 x=-1 Δx = dx = .01

  10. The diameter of a sphere is measured to be 18 centimeters, with a maximum possible error of .05 cm. Use differentials to approximate the possible propagated error and percent error in calculating the surface area and the volume of the sphere.

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