MAT 1221Survey of Calculus

Section 3.4Optimization Problems

http://myhome.spu.edu/lauw

Expectations Formally answer the problem in full

sentence with the expected information Do not include information unrelated to

the problem

Reading Read 4.3 for bonus points

Homework WebAssign HW 3.4 Review the examples in class carefully Do your HW ASAP

Preview If we need to find absolute max/min, the

closed interval method is the only method we know so far.

It does not work if the interval is not closed.

Introduce a new method if we have one and only one critical number.

Example 1A 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 the material used.

Step 1: Draw a diagram. 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 the material used.

Step 2: Define the variables and the function we need to optimize

Step 3: Simplify the function and state the range of the variableWhat is the range for ? (The domain of )

Example 1

V= 32,000 cm3

Minimize the surface area

xh

12 128000)( xxxf

The domain of is or To find the absolute min. of , the Closed

Interval Method does not applied.

First Derivative Test for Absolute Extreme ValuesSuppose that is the only critical no. of a continuous function defined on an interval(a) (similar for absolute max.)

(b) If for all and for all , then is the absolute min value of

x

y

c

Step 4: Use the appropriate tests to find the optimal value

80 140 , 40 4800f f

Step 5: Make a conclusion The dimensions required are …

Expectations 2 conclusions

• The absolute minimum value of is • The dimensions required are cmxcmxcm

Example 2Find the point(s) on the hyperbolathat are closest to the point .

422 xy

Step 1: Draw a diagram. Find the point(s) on the hyperbolathat are closest to the point .

422 xy

Step 2: Define the variables and the function we need to optimizeFind the point(s) on the hyperbolathat are closest to the point .

422 xy

Step 3: Simplify the function and state the range of the variableWhat is the range for ? (The domain of )

422 xy

Step 4: Use the appropriate tests to find the optimal value

Step 5: Make a conclusion The required points are …

Expectations 2 conclusions

• The absolute minimum value of is…• The required points are…