Download 9-3 optimizing with constraints

IB Math SL Year 2
SWITCH #1
and #3
Name: ________________________
Date: _________
9-3: Optimization Day 2
Today’s Goals: What is optimization? How do you maximize/minimize quantities using calculus?
What is optimization?
It involves finding the ______________ or ______________ value of a function subjected to a ________________.
Constraint - _______________________________________________________________________________________
General Approach to solving optimization question using calculus:
1.
2.
3.
4.
Read the problem at least ________(Carefully) and draw a _______, if appropriate
Identify your givens and what you are looking for
Assign a variable for each quantity involved in the problem.
Write an equation involving the quantity you are trying to make as ____ or as ______ as possible
(Equation to be ____________________________)
5. Write an equation that involves the restrictions to the problems (____________________ equation)
6. Use the ___________________equation to rewrite the equation to be optimized so that it is only in terms
of _________variable
7. Find the ____________ points (derivative = ______)
Determine the max or min value
8.
Example 1) Find the dimensions of a rectangle with a perimeter of 100 ft whose area is as large as possible.
1. Identify what we will optimize:
2. Identify constraint:
3. Sketch:
4. Write an equation of the quantity to optimize :
5. Write an equation for the constraint, only in terms of one variable.
6. Use constraint equation, to rewrite:
7. Find critical values.
8. Answer the question.
9.
IB Math SL Year 2
Example 2) An open box is to be made from a 16in by 30in piece of cardboard by cutting squares of equal size from the
four corners and bending up the sides. What size should the length of the squares be to obtain a box with largest
possible volume?
Example 3) Find the two numbers whose sum is 40 and whose product is as large as possible?
1. Identify what we will optimize:
2. Identify constraint:
3. Write an equation of the quantity to optimize :
4. Write an equation for the constraint, only in terms of one variable.
5. Use constraint equation, to rewrite:
6. Find critical values.
7. Answer the question.
IB Math SL Year 2
You Try!
4. You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum
area that you can enclose?
5. We need to enclose a rectangular field with a fence. We have 500 feet of fencing material and a building is
on one side of the field and so won’t need any fencing. Determine the dimensions of the field that will
enclose the largest area.
IB Math SL Year 2
6. The sum of one positive number and twice a second positive number is 200. Find the two numbers so that
their product is a maximum.
7. Let x be the number of thousands of units of an item produced. The revenue for selling x units is r(x) = 4√𝑥
and the cost of producing x units is c(x) = 2x2.
a. The profit p(x) = r(x) – c(x). Write an expression for p(x).
𝑑𝑝
𝑑2 𝑝
b. Find 𝑑𝑥 and 𝒅𝑥 2 .
c. Hence, find the number of units that should be produced in order to maximize profit.