4.7 Applied OptimizationWed Dec 17

Do Now

Differentiate

1) A(x) = x(20 - x)

2) f(x) = x^3 - 3x^2 + 6x - 12

Optimization

• One of the most useful applications of derivatives is to find optimal designs– Most cost efficient, maximized profit, etc

• Finding maximum and minimums solve these optimzation problems

Optimzation

• 1) Draw a picture (if possible)• 2) Determine what quantity needs to be

maximized or minimized• 3) Determine what variables are related to your

max/min• 4) Write a function that describes the max/min• 5) Use derivatives to find the max/min• 6) Solve

Ex 1

• A piece of wire of length L is bent into the shape of a rectangle. Which dimensions produce the rectangle of maximum area?

Ex 2

• Your task is to build a road joining a ranch to a highway that enables drivers to reach the city in the shortest time. How should this be done if the speed limit is 60km/h on the road and 110km/h on the highway? The perpendicular distance from the range to the highway is 30km, and the city is 50km down the highway

Ex 3

• All units in a 30-unit apartment building are rented out when the monthly rent is set at r = $1000/month. A survey reveals that one unit becomes vacant with each $40 increase in rent. Suppose each occupied unit costs $120/month in maintenance. Which rent r maximizes monthly profit?

More examples if time

Closure

• A three-sided fence is to be built next to a straight section of river, which forms the 4th side of the rectangular region. Given 90 ft of fencing, find the maximum area and the dimensions of the corresponding enclosure

4.7 OptimizationThurs Dec 18

• Do Now• An open box is to be made from a 3 ft by 8 ft piece of

sheet metal by cutting out squares of equal size from the four corners and bending up the sides. Find the maximum volume the box can have

HW Review: p.262 #1-59

• n/a

Ex 2

• A three-sided fence is to be built next to a straight section of river, which forms the 4th side of the rectangular region. Given 60 ft of fencing, find the maximum area and the dimensions of the corresponding enclosure

Ex 3

• A square sheet of cardboard 18 in. on a side is made into an open box (no top) by cutting squares of equal size out of each corner and folding up the sides. Find the dimensions of the box with the maximum volume.

You try

• You have 40 (linear) feet of fencing with which to enclose a rectangular space for a garden. Find the largest area that can be enclosed with this much fencing and the dimensions of the corresponding garden

Example 7.3

• Find the point on the parabola

closest to the point (3, 9)

Closure

• Journal Entry: How can we use derivatives to find an optimal design to a situation?

• HW: p.262 #1-59 EOO Try to get 6-7 correct (skip any CAS, GU problems)

4.7 OptimizationFri Dec 19

• Do Now• A child’s rectangular play yard is to be built

next to the house. To make the three sides of the playpen, 24 ft of fencing are available. What should be the dimensions of the sides to make a maximum area?

HW Review: p.262 #1-59 EOO

Worksheet

• HF

Closure

• Hand in: A box with no top is to be built by taking a 12 inch by 16 inch sheet of cardboard and cutting x-in. squares out of each corner and folding up the sides. Find the dimensions of the box that maximizes the volume of the box

• HW: p.262 #1-59 AOO Try to get 6-7 more correct

4.7 OptimizationMon Dec 22

• Do Now – Optimization problem we haven’t done yet

HW Review