4.7 Solving Max-Min Problems

1. 4.7 Solving Max-Min Problems • Read3. Identify the known quantities and the unknowns. Use a variable. • Identify the quantity to be optimized. Write a model for this quantity. Use appropriate formulas. This is the primary function. • If too many variables are in the primary function write a secondary function and use it to eliminate extra variables. • Find the derivative of the primary function. • Set it equal to zero and solve. • Reread the problem and make sure you have answered the question.

2. An open box is to be made by cutting squares from the corner of a 12 by 12 inch sheet and bending up the sides. How large should the squares be cut to make the box hold as much as possible? Figure 3.43: An open box made by cutting the corners from a square sheet of tin. (Example 1)

3. An open box is to be made by cutting squares from the corner of a 12 by 12 inch sheet and bending up the sides. How large should the squares be cut to make the box hold as much as possible? Figure 3.43: An open box made by cutting the corners from a square sheet of tin. (Example 1) Maximize the volume V =l w h V =(12 – 2x) (12 – 2x) x =144x – 48x2 + 4x3 V  = 144 – 96x+ 12x2 = 12(12 –8x+ x2) 12(12 –8x+ x2) = 0 (6-x)(2-x) = 0 x = 6 or x = 2 V  = -92+ 24x is negative at x = 2. There is a relative max. Box is 8 by 8 by 2 =128 in3.

4. Minimizing surface area Figure 3.46: The graph of A = 2 r 2 + 2000/r is concave up. You have been asked to design a 1 liter oil can in the shape of a right cylinder. What dimensions will use the least material?

5. Figure 3.46: The graph of A = 2 r 2 + 2000/r is concave up. You have been asked to design a 1 liter oil (1 liter = 1000cm3) can in the shape of a right cylinder. What dimensions will use the least material? Minimize surface area where Use the 2nd derivative test to show values give local minimums.

6. 4.8 Business Terms x = number of items p = unit price C = Total cost for x items R = xp = revenue for x items = average cost for x units P = R – C or xp - C

7. The daily cost to manufacture x items is C = 5000 + 25x 2. How many items should manufactured to minimize the average daily cost. 14 items will minimize the daily average cost.

8. 4.10 Old problem Given a function, find its derivative function derivative Inverse problem Given the derivative, find thefunction. .

9. Find a function that has a derivative y = 3x2 The answer is called the antiderivative You can check your answer by differentiation

10. Curves with a derivative of 3x2 Each of these curves is an antiderivative of y = 3x2

11. Antiderivatives Derivative Antiderivative

12. Find an antiderivative

13. Find antiderivatives Check by differentiating

14. Find an antiderivative

15. Trigonometric derivatives

16. Derivative Antiderivative