1. Read3. 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 – 48x2 + 4x3V = l w h

V = 144 – 96x + 12x2 = 12(12 –8x + x2)

V = -92 + 24x is negative at x = 2. There is a relative max. Box is 8 by 8 by 2 =128 in3.

12(12 –8x + x2) = 0

(6-x)(2-x) = 0

x = 6 or x = 2

Figure 3.46: The graph of A = 2 r 2 + 2000/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 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 area22 2S r rh 2

2 10001000 r h h

r where

2 12

22 21

2000000

2r

S r r r r

24 2000S r r

2

20004 0r

r

34 2000r

3500

5.42r

10.84h

Use the 2nd derivative test to show values give local minimums.

4.8 Business Terms

x = number of itemsp = unit priceC = Total cost for x itemsR = xp = revenue for x items= average cost for x unitsC

P = R – C or xp - C

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

2

500025C

x 5000

25C xx

2

500025 0

x

25000 25 0x

200 14.14x

14 items willminimize the dailyaverage cost.

4.10 Old problemGiven a function, find its derivative

Given the derivative, find the function..

function derivative

Inverse problem

Find a function that has a derivative y = 3x2

The answer is called the antiderivative

You can check your answer by differentiation

Curves with a derivative of 3x2

Each of these curves is an antiderivative of y = 3x2

Antiderivatives

ny x

( )y kf x

( ) ( )y f x g x

1

, 11

nxy C n

n

( )y kf x C

( ) ( )y f x g x

Derivative Antiderivative

Find an antiderivative

4 23 5 2x x x

1x

x

223 2x

Find antiderivatives4 23 5 2y x x x

5 3 2

3 5 25 3 2

x x xy x C

5 3 23 5 12

5 3 2y x x x x C

Check by differentiating

Find an antiderivative1 12 2

12

1 1x xy x x

xx

22 4 23 2 9 12 4y x x x

3 1

2 222

3y x x C

5 394 4

5y x x x C

5 3

9 12 45 3

x xy x C

sin cosd

x xdx

cos sind

x xdx

2tan secd

x xdx

2cot cscd

x xdx

sec sec tand

x x xdx

csc csc cotd

x x xdx

Trigonometric derivatives

siny x C

cosy x C

tany x C

coty x C

secy x C

cscy x C

Derivative Antiderivative

cosy x

siny x

2secy x

2cscy x

sec tany x x

csc coty x x