Sir Isaac Newton1643 – 1727

Sir Isaac Newton1643 – 1727

Isaac Newton was the greatest English mathematician of his generation. He laid the foundation for differential and integral calculus. His work on optics and gravitation make him one of the greatest scientists the world has known.

Isaac Newton was the greatest English mathematician of his generation. He laid the foundation for differential and integral calculus. His work on optics and gravitation make him one of the greatest scientists the world has known.

A Classic Problem

You have 40 feet of fence to enclose a rectangular corral along the side of a barn to house some chickens. What is the maximum area that you can enclose?

x x

40 2x

40 2A x x

240 2A x x

40 4A x 0 40 4x

4 40x

10x 40 2 Lengthx

Widthx

By The Extreme Value Theorem, the absolute maximum occurs here, since the endpoints give zero area.

0,20xI) Objective: Maximize Area of corral

A lwII) Objective Function

III) Domain

IV) “Calculus Time”

A Classic Problem

You have 40 feet of fence to enclose a rectangular corral along the side of a barn to house some chickens. What is the maximum area that you can enclose?

x x

40 2x

40 2A x x

240 2A x x

40 4A x 0 40 4x

4 40x 10x

40 2 Lengthx

Widthx

0,20xI) Objective: Maximize Area of corral

A lwII) Objective Function

III) Domain

IV) “Calculus Time”

10 40 2 10A

10 20A

V) Conclusion: By EVT, A = 200 ft2

is the absolute maximum area.

200A

Optimization ProblemsMini/Max Problems

Optimization concerns the minimization or maximization of functions. Itis used to solve complex design problems to improve cost, reliability, andperformance in a wide range of applications.

I Understand the Problem The first step is to read the problem carefully until it is clearly understood. Ask yourself: What is the objective? What are the constraints? Identify the given quantities, and conditions?

Draw a Diagram In most problems it is useful to draw and identify the given and required quantities on the diagram.

Introduce Notation Assign a symbol to the quantity that is to be maximized or minimized (let’s call it Q for now). Also select symbols (a, b, c, …, x, y, z) for other unknown quantities and label the diagram with these symbols. It may help to use initials as suggestive symbols – for example, A for area, h for height, t for time.

II Write an Objective Function Express Q in terms of some of the other symbols from Step 3.

Express Your Objective Function in One Variable If Q has been expressed as a function of more than one variable in Step 4, use the given information to find relationships (in the form of equations) among these variables. Then use these equations to eliminate all but one of the variables in the objective equation for Q. Thus , Q will be expressed as a function of one variable, say x, Q = f(x).

IV Find the Extrema of Q(x) Find the absolute maximum or minimum value for your objective function Q. In particular, if the domain of Q is a closed interval, then use the Extreme Value Theorem.

V Conclusion State your results, make sure you are a name dropper.

III Write the domain of this function!!

Extreme Value Theorem

Lets recall.

Extreme Value Theorem:

Absolute Maximum and Minimum points are interior points

Absolute Maximum and Minimum points are endpoints

Absolute Maximum point is an interior point and Absolute Minimum point is an endpoint

If is continuous on , , then attains an

absolute maximum value ( ) and an absolute

minimum value ( ) where and are in , .

Note: Check the endpoints for extreme values.

f a b f

f c

f d c d a b

Absolute Minimum point is an interior point and Absolute Maximum point is an endpoint

If the domain is not a closed interval, we use:

FDTAEV

Examples

An open box is to be made from 16 inch by 30 inch piece of cardboard by cutting out squares of equal size from the corners and bending up the sides. What size should the squares be to obtain a box with largest possible volume?

2 2Find all points on the curve 1 closest to (0, 2).x y

Given the figure below, find the radius and the height of the right cylinder of largest volume that can be inscribed in a right circular cone with radius 6 inches and height 10 inches.

The figure below shows an offshore oil well located at a point W that is 5 miles from the closest point A on a straight shoreline. Oil is to be piped from W to a shore point B that is 8 miles from point A by piping it on a straight line under water from W to some point P between A and B and then on to B via pipe along the shoreline. If the cost of laying pipe is $1,000,000 per mile under water and $500,000 over land, where should the point P be located to minimize the cost of laying the pipe?

A right cylinder is inscribed in a sphere of radius r. Find the largest possible volume of such a cylinder.