1.Read 3. Identify the known quantities and the unknowns. Use a variable. 2.Identify the quantity to be optimized. Write a model for this quantity. Use appropriate formulas. This is the primary function. 3.If too many variables are in the primary function write a secondary function and use it to eliminate extra variables. 4.Find the derivative of the primary function. 5.Set it equal to zero and solve. 6.Reread the problem and make sure you have answered the question. 4.7 Solving Max-Min Problems

Figure 3.43: An open box made by cutting the corners from a square sheet of tin. (Example 1) 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) 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? Maximize the volume V = (12 – 2x) (12 – 2x) x =144x – 48x 2 + 4x 3 V = l w h V = 144 – 96x + 12x 2 = 12(12 –8x + x 2 ) V = x is negative at x = 2. There is a relative max. Box is 8 by 8 by 2 =128 in 3. 12(12 –8x + x 2 ) = 0 (6-x)(2-x) = 0 x = 6 or x = 2

Figure 3.46: The graph of A = 2  r /r is concave up. Minimizing surface area 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?

Figure 3.46: The graph of A = 2  r /r is concave up. You have been asked to design a 1 liter oil (1 liter = 1000cm 3 ) can in the shape of a right cylinder. What dimensions will use the least material? Minimize surface area where Use the 2 nd derivative test to show values give local minimums.

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

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

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

Find a function that has a derivative y = 3x 2 The answer is called the antiderivative You can check your answer by differentiation

Curves with a derivative of 3x 2 Each of these curves is an antiderivative of y = 3x 2

Antiderivatives Derivative Antiderivative

Find an antiderivative

Find antiderivatives Check by differentiating

Find an antiderivative

Trigonometric derivatives

Derivative Antiderivative