4.7 Applied Optimization Fri Oct 30 Do Now Differentiate 1) A(x) = x(20 - x) 2) f(x) = x^3 - 3x^2 +...
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Transcript of 4.7 Applied Optimization Fri Oct 30 Do Now Differentiate 1) A(x) = x(20 - x) 2) f(x) = x^3 - 3x^2 +...
4.7 Applied OptimizationFri Oct 30
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
Optimization
• 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 6in. 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
• P.262 #9,15,23,29,43,52,57,63
4.7 OptimizationMon Dec 22
• 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 OptimizationMon 12/22
• 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