WARNING: these blogs puts the last blog on top of the page. I strongly suggest you start at the bottom with the first post on Calculus. This one is the fourth post on the subject and least friendly....lol. (Maybe my son or Uncle Mel can show me how to change the order of the posting ? That would be helpful)
Newton's Method of Approximation
This really does not have much value today because of the graphing calculator, but you will quickly learn that Calculus as we know it (the last 200 years) is pretty much obsolete. The graphing calculator does it ALL (except the reading of the problem and the setting up of the equations). Why we still do the Calculus by hand, i do not know. That is why there are FOUR different types of Calculus offered in universities today.
There is the Calculus for Math Majors (I would avoid this one unless you plan on being a mathematicians). This course starts off getting all bogged down with limits and proofs.
There is Calculus for the engineering majors. Again they do not waste too much time with proofs and go right into USING the derivative for solving physics problems.
There is Business Calculus which is my favorite. Again they do not waste too much time with proofs and go right into USING the derivative for solving business problems.
And there is Calculus for the Liberal Arts major. Usually called "Descriptive Calculus". It gives students an overview of Calculus and its uses for max and mins in graphs and word problems.
I strongly suggest that you take at least one of these Calculus. In today's world, many students are going on to graduate school. If you plan to apply to Law school or Medical school or a top notch MBA program, the admissions committee wants to see Calculus on your college transcript. It separated you from the "rest of the applicants". Even Wall Street financial institutes like to see Calculus on the transcript (by the way, they love hiring engineers).
Now the questions is why do we need to study Calculus ?
Well it started with Sputnik, the Russian satellite launched in the 1950's. The government asked the leading scientist why American was falling behind. The answer was we are not producing enough engineers. And why not ? Because most freshman fail Calculus when they come to college. Well what can we do about that ? Lets let the high school teachers teach them Calculus. And Calculus at the high school level was introduced. We have the students 5 days a week. And we can actually teach to a class of 30 or 40 students. At the university, they have 200 or 300 students in a lecture hall. Plus the college professors did not have the patients to show the algebra steps plus they got bogged down with limits and proofs.
Always remember learn math from a teacher who knows math, NOT from a mathematician they stuck in a classroom (sorry mathematicians....you guys are just too smart...both Einstein and Newton taught at the university, but had no students).
So you have the Russians to blame for having to take Calculus in high school or college. You might as well enjoy it while you are there and learn some neat stuff.
Who knows what the Chinese will force us to learn. Maybe Chinese !!
Ok getting back to Newton's Method of Approximation.
Lets say we want to find the square root of 2. (yea, yea i know we can just use our graping calculator...but assume we are stranded on some island without our calculator and our life depends on us finding an approximate value for the square root of 2). By the way the square root of 2 was very important to Pythagoras back in 800 BC. He was trying to find the length of the diameter of a square 1 by 1. Using his famous formula a^2 + b^2 = c^2, he was able to get the square root of 2. Of course back then they did not have algebra (which would not be invented until 1,000 AD by the Arabs and Hindu's). So since Pythagoras could not find the exact value of the square root of 2, he called it an unmeasurable number.
So we want to approximate the square root of 2.
We first must look at the function y = square root of 2...or y = x^(1/2).
We do know that the square root of 1 is equal to one...so we have at least one point (1,1) on the graph y = x^(1/2)
now what we want to do is find the ETL (equation of the tangent line....i told you to read the other blogs first....bottom of the page up !!)
the slope of the tangent line which is the derivative is y' = (1/2) x^(-1/2)
(remember the rule for derivative y = x^n, then y' = n x^(n-1) )
so if we plug in x = 1, we get slope = m = y' = (1/2) 1^(-1/2) = (1/2)
and remember the equation of the line y = mx + b
now we find b....by plugging in (1,1).... 1 = (1/2)(1) + b...therefore b = (1/2)
so the ETL at (1,1) on the graph is y = (1/2)x + (1/2)
now we are ready to use the ETL to approximate the square root of 2. we plug in 2 !!
y = (1/2) (2) + (1/2)...or the approx value of 2^(1/2) = 1.5
the calculator tells us the square root of 2 is 1.4....we are very close...(neat huh? yea, if you love math...lol)
Lets try another one...try square root of 3...and this time we have to go to the point (4,2) because we know that the square root of 4 is equal to 2.
remember the slope = m = y' = (1/2) x^(-1/2)...plugging in 4 we get m = (1/4)
and then we get the ETL y = (1/4) x + 1
therefore when we plug in 3, we get y = 1.75 which is our aprox. for 3^(1/2)
the calculator says square root of 3 is 1.73...again very close
WAY TO GO NEWTON !!! and now YOU can do it !!
COMMENTS
(Thanks for sharing your blogs with me but you have gotten over my head but I still appreciate you sending them....30 year algebra teacher and friend.)
(I really enjoy playing with these numbers. I have not done Calculus since college. I notice that if you go further out your approximations get better. Using pencil and paper really helped me follow your blog much easier. Keep them coming...20 year algebra teacher and friend.)
Wednesday, January 26, 2011
Calculus - what is it good for ?
So now you know some of the basics, what can we do with the derivative ?
One big use is in physics or the real world when dealing with cars and rockets and baseballs and something we call speed or velocity.
It turns out that the change in distance over the change in time is SPEED or Velocity and that is the SLOPE we have been talking about called the derivative (neat huh ?).
Now when an object like a rocket or cannon ball or baseball hits its PEAK or maximum height, it has ZERO velocity.
(That is why those towers in the medieval times were so effective. By the time the spear reach the top of the tower it had no velocity...just pluck it out of the sky and use it as a weapon against the enemy below. The same was true with most arrows of that time. In today's world it would be a water balloon fight at the beach motel. If you are on the 3rd or 4th floor, by the time the balloon makes it up to our balcony, you can just pluck it our of the air and throw it back at your sons below you...lol)
Maximum Height
So now if we look at the typical parabola of a flight object the equation might be:
y = -5 x^2 + 10 X + 2
To find the velocity, all we need to do is take the derivative. ( if y = x^n, then y' = n * x^(n-1)....if y = x^2 then y' = 2x...if y = x^3, then y' = 3x^2 ).
So the derivative of the distant formula about would be
Vel. = -10 x + 10.
And if we want to find the maximum height of the ball or object all we need to do is set the velocity = 0.
so 0 = -10x + 10....now some simple algebra gives us that x = 1.
the ball reaches its maximum height at x = 1 and we plug that into the distance equation given and we get y = -5 (1)^2 + 10 (1) + 2 = 7. The object reaches the max height of 7 !!
We can check this with our graphing calculator by graphing y = -5x^2 +10x + 2 and then hitting the buttons 2nd function/Calc max...we also get x=1, y=7.
Minimum Cost
so we can also use the derivative to fine the minimum cost which is very useful in the business world.
Lets assume we have to make a box with a fixed volume of 8 cubic feet. The box has a square base and an open top. What dimensions will minimize or cost of material ?
Take my word for it the formula is SA (Surface Area) = x^2 + 32/x
the derivative SA' = 2x - 32/ x^2
so now we set the derivative equal to zero and we get x = 2.52 and SA = 19.
lets check: if the box was 2 X 2 X 2 (volume of 8) the SA would be SA = 4 + 16 = 20.
if we made it 1 X 1 X 8 (volume of 8) the SA = 1 + 32 = 33
if we made it 2.5 X 2.5 X 1.28 (vol. of 8) the SA = 6.25 + 12.8 = 19.05 (very close)
But the best is what the Calculus tells us using the derivative, x = 2.52
One big use is in physics or the real world when dealing with cars and rockets and baseballs and something we call speed or velocity.
It turns out that the change in distance over the change in time is SPEED or Velocity and that is the SLOPE we have been talking about called the derivative (neat huh ?).
Now when an object like a rocket or cannon ball or baseball hits its PEAK or maximum height, it has ZERO velocity.
(That is why those towers in the medieval times were so effective. By the time the spear reach the top of the tower it had no velocity...just pluck it out of the sky and use it as a weapon against the enemy below. The same was true with most arrows of that time. In today's world it would be a water balloon fight at the beach motel. If you are on the 3rd or 4th floor, by the time the balloon makes it up to our balcony, you can just pluck it our of the air and throw it back at your sons below you...lol)
Maximum Height
So now if we look at the typical parabola of a flight object the equation might be:
y = -5 x^2 + 10 X + 2
To find the velocity, all we need to do is take the derivative. ( if y = x^n, then y' = n * x^(n-1)....if y = x^2 then y' = 2x...if y = x^3, then y' = 3x^2 ).
So the derivative of the distant formula about would be
Vel. = -10 x + 10.
And if we want to find the maximum height of the ball or object all we need to do is set the velocity = 0.
so 0 = -10x + 10....now some simple algebra gives us that x = 1.
the ball reaches its maximum height at x = 1 and we plug that into the distance equation given and we get y = -5 (1)^2 + 10 (1) + 2 = 7. The object reaches the max height of 7 !!
We can check this with our graphing calculator by graphing y = -5x^2 +10x + 2 and then hitting the buttons 2nd function/Calc max...we also get x=1, y=7.
Minimum Cost
so we can also use the derivative to fine the minimum cost which is very useful in the business world.
Lets assume we have to make a box with a fixed volume of 8 cubic feet. The box has a square base and an open top. What dimensions will minimize or cost of material ?
Take my word for it the formula is SA (Surface Area) = x^2 + 32/x
the derivative SA' = 2x - 32/ x^2
so now we set the derivative equal to zero and we get x = 2.52 and SA = 19.
lets check: if the box was 2 X 2 X 2 (volume of 8) the SA would be SA = 4 + 16 = 20.
if we made it 1 X 1 X 8 (volume of 8) the SA = 1 + 32 = 33
if we made it 2.5 X 2.5 X 1.28 (vol. of 8) the SA = 6.25 + 12.8 = 19.05 (very close)
But the best is what the Calculus tells us using the derivative, x = 2.52
ETL - Equation of the Tangent Line
So what is the first thing we are going to do with this new concept, the derivative. Well since the derivative is the slope of the tangent line, why don't we find the equation of this tangent line (ETL).
A historical note first. Calculus was discovered or refined by two mean. One was British (the apple fell on his head), Newton and the other was German, Leibniz.
(A funny note, if Germany had won WWII, we would probably be studying Leibniz's Calculus, but since the British and the Americans won WWII, we studying Newton's calculus.) Children in Germany study Leibniz's calculus. I am sure other civilizations like the Chinese and American Indians (South and North) were developing their own mathematical ideas, but once Christopher Columbus sailed the ocean blues, the world got a lot smaller...or like they like to say these days, Flatter.
Another interest historical note about Newton. At the time of many of his discoveries, Europe was experiencing a plague and in those days people left the city and went out to their estates in the country. The university was closed for 6 months. It just happened that Newton's father had an extensive library and Newton had time to kill. In that time he developed some of his greatest ideas, including the new math called Calculus.
This phenomenon occurs many time throughout history. Today our greatest mind is Steven Hawkins who is paralyzed from the neck down. All he can move is his finger. But thanks to computers, he has been able to write several books. The most famous being the Short History of Time (which everyone has on their book shelves, but no one understands including yours truly...i have tried three or four times). But the idea is that Hawkins has his mind free from every day duties to just think. One of his big ideas is the Big Bang theory of the Universe. And his final conclusion is that there is a God out there.
Another person in history who was confined was Galileo. He was put under house arrest (not for his ideas of the sun being the center which was a concept accepted long ago by most learned men of Europe including the Pope) because he published a book after being ORDERED by the Church NOT to publish.
So Galileo had a lot of time on his hands to observe the moons of Jupiter. And he was visited by other great thinkers of the time and they developed many of the ideas of the heavens that we follow today.
The Athenian were the first (we know of) who understood that men need leisure time in order to think. They needed shelter and food provide. Today we have great univeristies like Harvard where half of their faculty do not teach anymore in the classrooms, but instead do research with grad students. MIT is that was too. As is Stanford and other great universities. It helps to have great endowments from generous people to help pay for these men and women, shelter and food.
Now getting back to my ETL (Equation of the Tangent Line).
We know the equation of the line is y = mx + b.
And now we know that m = slope = the derivative !!
Going back to our original equation y = x^2, remember that the derivative is y' = 2x.
Therefore the slope = m = y' = 2x.
(By the way, Newton used y' for the derivative. Leibniz used dy/dx. The dy/dx is has more practical uses in the long run, but here we will use y'. In your calculator TI-84 the notation under 2nd function/Calc/ is dy/dx.)
Now we want to know at WHAT POINT are we looking for this tangent line. Lets used the point (1,1).
Since the slope = y' = 2x, when we plug in one for x, we get m = 2 !!
We know the equation of any line is y = mx + b. So now we find b, by plugging in the point (1,1).
So 1 = (2)(1) + b...or 1 = 2 + b...subtract 2 from both sides and we get b = -1.
And therefore the ETL is y = 2x - 1.
Time for pencil and paper. Draw the graph y = x^2 and draw the line y = 2x - 1.
(or use your child's graphing calculator.)
Now you see the parabola y = x^2 and the ETL y = 2x - 1.
And that is CALCULUS !!
A historical note first. Calculus was discovered or refined by two mean. One was British (the apple fell on his head), Newton and the other was German, Leibniz.
(A funny note, if Germany had won WWII, we would probably be studying Leibniz's Calculus, but since the British and the Americans won WWII, we studying Newton's calculus.) Children in Germany study Leibniz's calculus. I am sure other civilizations like the Chinese and American Indians (South and North) were developing their own mathematical ideas, but once Christopher Columbus sailed the ocean blues, the world got a lot smaller...or like they like to say these days, Flatter.
Another interest historical note about Newton. At the time of many of his discoveries, Europe was experiencing a plague and in those days people left the city and went out to their estates in the country. The university was closed for 6 months. It just happened that Newton's father had an extensive library and Newton had time to kill. In that time he developed some of his greatest ideas, including the new math called Calculus.
This phenomenon occurs many time throughout history. Today our greatest mind is Steven Hawkins who is paralyzed from the neck down. All he can move is his finger. But thanks to computers, he has been able to write several books. The most famous being the Short History of Time (which everyone has on their book shelves, but no one understands including yours truly...i have tried three or four times). But the idea is that Hawkins has his mind free from every day duties to just think. One of his big ideas is the Big Bang theory of the Universe. And his final conclusion is that there is a God out there.
Another person in history who was confined was Galileo. He was put under house arrest (not for his ideas of the sun being the center which was a concept accepted long ago by most learned men of Europe including the Pope) because he published a book after being ORDERED by the Church NOT to publish.
So Galileo had a lot of time on his hands to observe the moons of Jupiter. And he was visited by other great thinkers of the time and they developed many of the ideas of the heavens that we follow today.
The Athenian were the first (we know of) who understood that men need leisure time in order to think. They needed shelter and food provide. Today we have great univeristies like Harvard where half of their faculty do not teach anymore in the classrooms, but instead do research with grad students. MIT is that was too. As is Stanford and other great universities. It helps to have great endowments from generous people to help pay for these men and women, shelter and food.
Now getting back to my ETL (Equation of the Tangent Line).
We know the equation of the line is y = mx + b.
And now we know that m = slope = the derivative !!
Going back to our original equation y = x^2, remember that the derivative is y' = 2x.
Therefore the slope = m = y' = 2x.
(By the way, Newton used y' for the derivative. Leibniz used dy/dx. The dy/dx is has more practical uses in the long run, but here we will use y'. In your calculator TI-84 the notation under 2nd function/Calc/ is dy/dx.)
Now we want to know at WHAT POINT are we looking for this tangent line. Lets used the point (1,1).
Since the slope = y' = 2x, when we plug in one for x, we get m = 2 !!
We know the equation of any line is y = mx + b. So now we find b, by plugging in the point (1,1).
So 1 = (2)(1) + b...or 1 = 2 + b...subtract 2 from both sides and we get b = -1.
And therefore the ETL is y = 2x - 1.
Time for pencil and paper. Draw the graph y = x^2 and draw the line y = 2x - 1.
(or use your child's graphing calculator.)
Now you see the parabola y = x^2 and the ETL y = 2x - 1.
And that is CALCULUS !!
Tuesday, January 25, 2011
Extra Help - Calculus
Ah, my favorite subject, Calculus or like Lou Diamond-Phillips said in the movie Stand and Deliver (a must see for all math teachers and teachers in general)...
"Cal-coo-coo-lus ? what is this cal-coo-coo-lus ?"
The movie is about a computer science teacher hired to teach computers in a high school in LA. The population of the school is about 95% hispanic. Mexicans from very poor families. Many migrant workers.
But when the teacher gets to the school, they do NOT have any computers, so they assigned him to teach "Boom-boom Math" to 16 and 17 yrs old. Of course the classroom is full of graffiti and many of the students in the class "no speaka no english". There is also a little problem with some gang members.
After a year, this teacher (ex-computer analysis man) decides he can teach the Burros algebra. Then he gets excited when some of the students get fired up, and he teaches them Geometry during the SUMMER (the same thing i have done at my school in Tampa for 30 years). Alg II and again Precalculus during the summer. Then he proclaims he will teach them Calculus !!
Now they are down to maybe 8 or 10 students out of a school of thousands. And in order to do well in Calculus (an AP course...AP stands for Advance Placement and if you do well enough on the AP test some colleges will give you college credit...side note: my son got 15 college credit hours for his AP's he took in high school).
To make a long story short (oops too late), the students do well on the AP exam and are accused of cheating by the College Board who administers the test. Apparently since they had the same teacher, they all missed pretty much the same questions, plus there is NO WAY that Burros from LA could learn Calculus the FIRST year it is offered at their high school.
In Florida, over half of the AP students in public schools do NOT pass the AP exams.
Well the students in this movie have to re-take the test under the strict supervision of Andy Garcia. And sure enough they all pass the second time !!
Now back to my student today. We start with the concept of SLOPE that we covered in the first extra help lesson on equations of lines. y = mx + b
What we want now is the slope of the tangent line to a curve. And in order to do this we need some basic concepts of limits (the hardest concept in Calculus...why Calculus start killing the kids with the limit concept is beyond me...why not teach them the practical application of the "derivative" (the slope of the tangent line) and save the limit concept for later ? That is what i am going to do here).
So we learn how to take a "derivative"...it is easy as multiplying and then subtracting.
y = x^2. The rule says that you bring the exponent (in this case the 2) in front of the x and then subtract one from the exponent.
So the derivative of y = x^2, is just y' = 2x^1 = 2x.
The derivative of y = x^3, is just y' = 3x^2.
The derivative of y = x^4, is just y' = 4x^3
OR using some symbols. y = x^n, is just y' = n x^(n-1). That is IT !! you got it !!
Here are some challenges:
y = x is the same as y = x^1, so y' = 1x^0, (but x^0 = 1) so y' = 1.
y = 2x, y' = 2
y = mx, y' = m. (don't get scared...we are just having some fun)
y = 1/x, in algebra 1/x = x^-1, so y = x^-1, then y' = -1 x^-2 or -1/x^2
(remember that 2^(-1) = 1/2...or 2^(-2) = 1/(2^2) = 1/4...the negative exponent puts the number on the bottom)
Ok, one more
y = square root of x, in algebra we say y = x^(1/2)
y = x^(1/2) then y' = (1/2) x^(-1/2)....yes, yes, (1/2) - (1) = -(1/2)
That is enough for our purposes, but if you followed it this far you have enter the TOP 1% of the population !!
Congradulations, you did NOT faint. Blood pressure might be a little high, but that is always good for the heart and even better for the BRAIN. We must keep it active.
COMMENTS:
(Read your Calculus "perspective" with great interest. Good stuff and with your permission like to share with my fellow math teachers here at Jefferson. I also enjoyed reading your experiences from your recent trip back to the homeland.
Next time I see you, that'll be the topic of discussion....30 year Math Teacher)
"Cal-coo-coo-lus ? what is this cal-coo-coo-lus ?"
The movie is about a computer science teacher hired to teach computers in a high school in LA. The population of the school is about 95% hispanic. Mexicans from very poor families. Many migrant workers.
But when the teacher gets to the school, they do NOT have any computers, so they assigned him to teach "Boom-boom Math" to 16 and 17 yrs old. Of course the classroom is full of graffiti and many of the students in the class "no speaka no english". There is also a little problem with some gang members.
After a year, this teacher (ex-computer analysis man) decides he can teach the Burros algebra. Then he gets excited when some of the students get fired up, and he teaches them Geometry during the SUMMER (the same thing i have done at my school in Tampa for 30 years). Alg II and again Precalculus during the summer. Then he proclaims he will teach them Calculus !!
Now they are down to maybe 8 or 10 students out of a school of thousands. And in order to do well in Calculus (an AP course...AP stands for Advance Placement and if you do well enough on the AP test some colleges will give you college credit...side note: my son got 15 college credit hours for his AP's he took in high school).
To make a long story short (oops too late), the students do well on the AP exam and are accused of cheating by the College Board who administers the test. Apparently since they had the same teacher, they all missed pretty much the same questions, plus there is NO WAY that Burros from LA could learn Calculus the FIRST year it is offered at their high school.
In Florida, over half of the AP students in public schools do NOT pass the AP exams.
Well the students in this movie have to re-take the test under the strict supervision of Andy Garcia. And sure enough they all pass the second time !!
Now back to my student today. We start with the concept of SLOPE that we covered in the first extra help lesson on equations of lines. y = mx + b
What we want now is the slope of the tangent line to a curve. And in order to do this we need some basic concepts of limits (the hardest concept in Calculus...why Calculus start killing the kids with the limit concept is beyond me...why not teach them the practical application of the "derivative" (the slope of the tangent line) and save the limit concept for later ? That is what i am going to do here).
So we learn how to take a "derivative"...it is easy as multiplying and then subtracting.
y = x^2. The rule says that you bring the exponent (in this case the 2) in front of the x and then subtract one from the exponent.
So the derivative of y = x^2, is just y' = 2x^1 = 2x.
The derivative of y = x^3, is just y' = 3x^2.
The derivative of y = x^4, is just y' = 4x^3
OR using some symbols. y = x^n, is just y' = n x^(n-1). That is IT !! you got it !!
Here are some challenges:
y = x is the same as y = x^1, so y' = 1x^0, (but x^0 = 1) so y' = 1.
y = 2x, y' = 2
y = mx, y' = m. (don't get scared...we are just having some fun)
y = 1/x, in algebra 1/x = x^-1, so y = x^-1, then y' = -1 x^-2 or -1/x^2
(remember that 2^(-1) = 1/2...or 2^(-2) = 1/(2^2) = 1/4...the negative exponent puts the number on the bottom)
Ok, one more
y = square root of x, in algebra we say y = x^(1/2)
y = x^(1/2) then y' = (1/2) x^(-1/2)....yes, yes, (1/2) - (1) = -(1/2)
That is enough for our purposes, but if you followed it this far you have enter the TOP 1% of the population !!
Congradulations, you did NOT faint. Blood pressure might be a little high, but that is always good for the heart and even better for the BRAIN. We must keep it active.
COMMENTS:
(Read your Calculus "perspective" with great interest. Good stuff and with your permission like to share with my fellow math teachers here at Jefferson. I also enjoyed reading your experiences from your recent trip back to the homeland.
Next time I see you, that'll be the topic of discussion....30 year Math Teacher)
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