Optimization Problems in Calculus
This worksheet focuses on applying calculus techniques to solve optimization problems, including finding maximum and minimum values of functions.
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Optimization Problems in Calculus
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Read each problem carefully and use calculus techniques to find the optimal solution. Show all your work.
1. A farmer has 1200 meters of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area?
2. A cylindrical can is to be made to hold 1000 cm³ of oil. Find the dimensions (radius r and height h) that will minimize the cost of the metal to make the can. Assume the top and bottom are included. Which of the following is the correct relationship between r and h for minimum surface area?
h = r
h = 2r
r = 2h
h = πr
3. Find the point on the graph of y = √x that is closest to the point (3, 0). The x-coordinate of this point is and the y-coordinate is .
4. A box with a square base and open top must have a volume of 32,000 cm³. Find the dimensions of the box that minimize the amount of material used.
5. A critical point of a function f(x) always corresponds to a local maximum or local minimum.
True
False