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)
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