Aim: How can we apply the first derivative Aim: How can we apply the first derivative to solve easy, medium and hard level to solve easy, medium and hard level

problems related to Optimization?problems related to Optimization?

Donow: Try these Donow: Try these compositions of functionscompositions of functions

If f(X)=2, f’(3)=-1, g(3)=3, If f(X)=2, f’(3)=-1, g(3)=3, g’(3)=0, Find F’(3)g’(3)=0, Find F’(3)

(question borrowed (question borrowed from Mrs. Dela from Mrs. Dela Cruz)Cruz)

a.a. F(x)= 2f(x)- g(x)F(x)= 2f(x)- g(x)

Just think here to Just think here to your self, is there your self, is there any product rule any product rule or division rule or division rule involved?involved?

(con).(con).

No right? Good. Let’s simply make the No right? Good. Let’s simply make the f(x) and g(x) into f’(x) and g’(x) f(x) and g(x) into f’(x) and g’(x) respectively. (dont’ worry about the respectively. (dont’ worry about the two!!!)two!!!)

2f’(x)-g’(x) = F’(x)2f’(x)-g’(x) = F’(x)

Now just plug in:Now just plug in:

F’(3)= 2f’(3)- g’(3)F’(3)= 2f’(3)- g’(3)

Based of our charts we see that f’(3)= -1Based of our charts we see that f’(3)= -1

g’(3)= 0g’(3)= 0

Continued…Continued…

So, its F’(3)= 2(-1)-0= -2So, its F’(3)= 2(-1)-0= -2

That’s it, see how easy it is…That’s it, see how easy it is…

Let’s try another one:Let’s try another one:

F(x)= 1/2f(x)g(x)F(x)= 1/2f(x)g(x)

Ask yourself: Does this question Ask yourself: Does this question involve the product rule or the involve the product rule or the division rule Yes!!!!division rule Yes!!!!

The Product RuleThe Product Rule

Well, don’t panic! There is a simple Well, don’t panic! There is a simple product rule we can apply:product rule we can apply:

F’(x)= f’g+g’f.F’(x)= f’g+g’f.You may say: Here we go again…You may say: Here we go again…Well, this derivative is simply the Well, this derivative is simply the

derivative of f times g added to the derivative of f times g added to the derivative of g mulitplied with the f derivative of g mulitplied with the f value.value.

Don’t get it? Well, lets approach the Don’t get it? Well, lets approach the example:example:

Solving: ½ f(x)g(x)…Solving: ½ f(x)g(x)…

ProblemProblem

F’(x)=F’(x)=

1/2[f(x)g’(x)+g(x)f’(x)1/2[f(x)g’(x)+g(x)f’(x)

F’(3) = F’(3) = 1/2[f’(3)+g(3)+g’(3)1/2[f’(3)+g(3)+g’(3)f(3)f(3)

F’(3)= ½[2(0)+3(-1)]F’(3)= ½[2(0)+3(-1)]

The algorithmThe algorithm

Application of the Application of the product ruleproduct rule

Placing what needs Placing what needs to be derived, in to be derived, in the equationthe equation

Plugging in with the Plugging in with the values of the values of the original equationoriginal equation

Continuation of the donow Continuation of the donow prob.prob.

F’(3)= -3/2F’(3)= -3/2 Evaluating the Evaluating the expression.expression.

Why the donow problem?Why the donow problem?

You see what I was trying to show you?You see what I was trying to show you?

Tell me, what was common about both Tell me, what was common about both things: The first derivative.things: The first derivative.

For all of them, I was trying to show you For all of them, I was trying to show you that the first derivative plays an integral that the first derivative plays an integral role in our finding of the derivatives.role in our finding of the derivatives.

In our lesson, it will play a bigger roleIn our lesson, it will play a bigger roleNow on to our lesson…Now on to our lesson…

What is Optimization?What is Optimization?

Optimization is basically trying to get Optimization is basically trying to get the most of something while getting the most of something while getting rid of the least. It can also be trying rid of the least. It can also be trying to get least amount of something.to get least amount of something.

You see the big oil companies: Exxon You see the big oil companies: Exxon Mobil, Chevron Corporation and BP Mobil, Chevron Corporation and BP trying to make the most amount of trying to make the most amount of money, but trying to cut their costs money, but trying to cut their costs in a highly competitive environment.in a highly competitive environment.

Optimization functions (con)Optimization functions (con)

Sometimes, we want to get the least Sometimes, we want to get the least volume with the most amount of volume with the most amount of material, so that we don’t waste any.material, so that we don’t waste any.

As you can see, Optimization has a As you can see, Optimization has a wide array of applications from big wide array of applications from big name gas tankers to simple cubic name gas tankers to simple cubic volumes.volumes.

I’ll help you dissect the problems…I’ll help you dissect the problems…

Question 1Question 1

Intimidated? Relax, we’ll start with Intimidated? Relax, we’ll start with something easy: something easy:

(borrowed from (borrowed from www.math.ucdavis.eduwww.math.ucdavis.edu))

Find two nonnegative numbers whose Find two nonnegative numbers whose sum is 9 and so that the product of sum is 9 and so that the product of one number and the square of the one number and the square of the other number is a maximum. other number is a maximum.

Approaching the problemApproaching the problem

You may have difficulty if you don’t break You may have difficulty if you don’t break the problem down into simple parts:the problem down into simple parts:

The problem states: “nonnegative The problem states: “nonnegative numbers”numbers”

So I tell myself “Only positive numbers”So I tell myself “Only positive numbers”

Next, it says “ sum of 9” We don’t know Next, it says “ sum of 9” We don’t know what the sum of the problem, is so we what the sum of the problem, is so we just have to make them up. just have to make them up.

Approaching the problem…Approaching the problem…con.con.

Let’s say that “x” is the first number, Let’s say that “x” is the first number, and “y is the second number. So, it and “y is the second number. So, it follows that x+y=9. Simple enough?follows that x+y=9. Simple enough?

Ok, take a deep breath…Ok, take a deep breath…

Now, read the second part of the Now, read the second part of the problem… problem…

(the product of one number and the (the product of one number and the square of the other is a maximum).square of the other is a maximum).

Seems a little tricky, right?Seems a little tricky, right?

Continued.Continued.

We know that one number is “x” and the We know that one number is “x” and the other number is the “square,” So, this other number is the “square,” So, this means that the “y” is the y squared.means that the “y” is the y squared.

Let P represent the product of the equation:Let P represent the product of the equation:

Now, we must attempt to connect both parts, Now, we must attempt to connect both parts, which is probably the toughest part, that I which is probably the toughest part, that I had trouble with…had trouble with…

2xyP

The connection:The connection:

Well, we want to find Well, we want to find the the product product of the of the equation,so our goal is equation,so our goal is to make both of the to make both of the variables in the variables in the product the same( You product the same( You can select either can select either variable you want to variable you want to put in terms of, but put in terms of, but lets use “y”, for lets use “y”, for simplicitysimplicity

It follows that:It follows that:

2)9(

9

xxP

xy

Derivative Time:)Derivative Time:)

Remember what I had told you before Remember what I had told you before about the first derivative, well now it’s about the first derivative, well now it’s time to apply it…time to apply it…

P’ = f’g+g’fP’ = f’g+g’f PP' = ' = xx (2) ( 9- (2) ( 9-xx)(-1) + (1) ( 9-)(-1) + (1) ( 9-xx)2 )2 = ( 9-= ( 9-xx) [ -2) [ -2xx + ( 9- + ( 9-xx) ] ) ] = ( 9-= ( 9-xx) [ 9-3) [ 9-3xx ] ] = ( 9-= ( 9-xx) (3)[ 3-) (3)[ 3-xx ] ] = 0 = 0 X=3 or 9.X=3 or 9.

Plugging back…Plugging back…

Remember the plugging back, now we Remember the plugging back, now we do the same…do the same…

If we do 9*0 or 0*9, well have 0. So the If we do 9*0 or 0*9, well have 0. So the only ones that work is 3 and 6, which only ones that work is 3 and 6, which well get 108 as the maximum well get 108 as the maximum product. product.

Example 2:Example 2:

Borrowed from Borrowed from http://www.math.ucdavis.eduhttp://www.math.ucdavis.edu Medium Problem: Medium Problem:

A sheet of cardboard 3 ft. by 4 ft. will A sheet of cardboard 3 ft. by 4 ft. will be made into a box by cutting equal-be made into a box by cutting equal-sized squares from each corner and sized squares from each corner and folding up the four edges. What will folding up the four edges. What will be the dimensions of the box with be the dimensions of the box with largest volume ? largest volume ?

Solving the problemSolving the problem

Drawing a picture is the safest way to Drawing a picture is the safest way to start, draw a picture, and label the start, draw a picture, and label the sides!sides!

Other steps:Other steps:

Here is the Rectangular Box without Here is the Rectangular Box without the cover:the cover:

Steps:Steps:

Observe that the problems that we solve Observe that the problems that we solve are similar to that of the previous problem. are similar to that of the previous problem. These are: writing the equation, getting the These are: writing the equation, getting the “solved for” equation in terms of the same “solved for” equation in terms of the same variable. Finding the first derivative, setting variable. Finding the first derivative, setting it equal to 0, and then solving for the it equal to 0, and then solving for the variable. If we want to check whether the variable. If we want to check whether the answer is maximum, we plug our answer answer is maximum, we plug our answer into the second dx/dy. If the answer is less into the second dx/dy. If the answer is less than 0, were okay, otherwise, it does not than 0, were okay, otherwise, it does not exist.exist.

Finally solved…Finally solved… VV =LWH = (4-2 =LWH = (4-2xx) (3-2) (3-2xx) () (xx) . ) . VV' = (-2) (3-2' = (-2) (3-2xx) () (xx) + (4-2) + (4-2xx) (-2) () (-2) (xx) + (4-2) + (4-2xx) (3-2) (3-2xx) (1) ) (1) = -6= -6xx + 4 + 4xx2 - 82 - 8xx + 4 + 4xx2 + 42 + 4xx2 - 142 - 14xx + 12 + 12 = 12= 12xx2 - 282 - 28xx + 12 + 12 = 4 ( 3= 4 ( 3xx2 - 72 - 7xx + 3 ) + 3 ) = 0 (notice we had to use the product rule three = 0 (notice we had to use the product rule three

times, since there are 3 “partstimes, since there are 3 “parts

Now, you must use the quadratic quation to solve for Now, you must use the quadratic quation to solve for the x value. You will find that the answer comes out to the x value. You will find that the answer comes out to x= 0.57 or 1.77. Now, you must plug back in to the x= 0.57 or 1.77. Now, you must plug back in to the original equation. You will find that the largest volume original equation. You will find that the largest volume is 3.03 cubic feet at x=0.57.is 3.03 cubic feet at x=0.57.

          or           .

Hard problemHard problem

Is it getting easier? I certainly hope so. Is it getting easier? I certainly hope so. Now will come the hardest one of all…Now will come the hardest one of all…

Borrowed from Borrowed from http://http://www.math.ucdavis.eduwww.math.ucdavis.edu

Consider a rectangle of perimeter 12 Consider a rectangle of perimeter 12 inches. Form a cylinder by revolving inches. Form a cylinder by revolving this rectangle about one of its edges. this rectangle about one of its edges. What dimensions of the rectangle will What dimensions of the rectangle will result in a cylinder of maximum result in a cylinder of maximum volume ? volume ?

Approaching the toughie…Approaching the toughie…

Let’s draw a picture, we know that Let’s draw a picture, we know that the rectangle is inside the cylinder, the rectangle is inside the cylinder, and only half way. So here it is. and only half way. So here it is.

Dissecting the problemDissecting the problem

We know that the perimeter is the sum of We know that the perimeter is the sum of twice the width and twice the length.twice the width and twice the length.

So, its 12 = 2So, its 12 = 2rr + 2 + 2hh , now, we need to find the , now, we need to find the equation of the volume of the cylinder. Let p equation of the volume of the cylinder. Let p represent pi.represent pi.

V=prV=pr22h. Now, notice that we have “h in both h. Now, notice that we have “h in both equations. Since we have this, we can make equations. Since we have this, we can make the perimeter equation in terms of r. the perimeter equation in terms of r.

Now, if we do that, all we have to do, is follow Now, if we do that, all we have to do, is follow the aforementioned rules of finding the first the aforementioned rules of finding the first derivative, setting it equal to 0, and then derivative, setting it equal to 0, and then plugging in. We should then have the plugging in. We should then have the maximum volume.maximum volume.

Solved answer.Solved answer.

V’=p(12r-3rV’=p(12r-3r22 ) )p(3r)(4-r)p(3r)(4-r)3pr(4-r)=03pr(4-r)=0r= 0 or 4r= 0 or 4

Now, you have to guess different values. Now, you have to guess different values. You'll find that when r=4 and when You'll find that when r=4 and when h=2, there is the highest volume. This h=2, there is the highest volume. This volume will be 100 feet cubed. volume will be 100 feet cubed.