The Calculus in a freshman core terminal class!!
The idea of slope extended beyond the straight line
Contrast a line with a parabola graphically and with a table on the TI-84 and equation wise (algebraically) so that students can see that the rate of change for a nonlinear function changes in a nonconstant manner)
Show the slope of line is the "m"
Write the slope formula using dy/dx or as delta y / delta x (using the Greek letter delta triangle thingie)
Showing posts with label calculus. Show all posts
Showing posts with label calculus. Show all posts
Wednesday, April 04, 2007
Friday, March 09, 2007
Related Rates
These calculus problems can be a lot of fun -- ok, I am the Math Lady.
Found an awesome web site:
http://www2.scc-fl.com/lvosbury/CalculusI_Folder/RelatedRateProblems.htm
The best demonstration of the concept is the balloon problem which shows that the rate of change in volume changes depending on how full the balloon is -- the two rates are related ;)
Strong algebra skills are a true asset here as some of the solving involves negative and fractional exponents and dividing by fractions.
These calculus problems can be a lot of fun -- ok, I am the Math Lady.
Found an awesome web site:
http://www2.scc-fl.com/lvosbury/CalculusI_Folder/RelatedRateProblems.htm
The best demonstration of the concept is the balloon problem which shows that the rate of change in volume changes depending on how full the balloon is -- the two rates are related ;)
Strong algebra skills are a true asset here as some of the solving involves negative and fractional exponents and dividing by fractions.
Thursday, February 22, 2007
What Do Students Need Help With?
Algebra
Generalizing from y = mx + b so that the result is an equation like y = 2x + 5.
Students benefit from being reminded that they are finding the equation that describes the entire line not just a point.
I like to ask them "How many points are on the line?" -- the answer I like the most is "Infinite" but "Too many to count" or "A real lot" are very acceptable answers.
It takes a while to see that (x-5)/(5-x) = -1!!
Calculus and Number Lines
The idea of limits and that inscribing an n-sided polygon in a circle, the more sides you have, the closer the area is to the area of the circle. So as n approaches infinity using a circle with a radius of 1, the area approaches pi (3.14159265....).
Secant and tangent lines have nothing to do with trig functions secant and tangent!!
A really cool problem was f(x) = [x] + [-x] which involves the greatest integer function.
The TI-84 function for greatest integer function is Int. The greatest integer function is best demonstrated with a number line and asking the student what was the last number that you passed. If you are exactly at an integer, then the answer is that number.
For example, [2.5] = 2 we passed 2 on the way to 2.5.
and [4] = 4
But [-2.5] = -3 we passed -3 on the way to -.25 (but we have not yet passed -2!!)
The cool thing about this problem is that for most x's, such as x = 2.5,
f(x) = [2.5] + [-2.5] which comes out to
f(x) = 2 + -3 = -1
The exception is at exactly an integer -- f(4) = [4] + [-4] = 4 + -4 = 0.
This graph looks like a straight line like y = -1, however at each integer value, there is a discontinuity, as y jumps to 0 at -3, -2, -1, 0, 1, 2, 3, etc.
This is best seen by setting xmin = 0.99 and xmax = 1.01 on the window menu -- at x = 1 there is a dot on the x axis!!
Trigonometry
The Law of Cosines is really the Pythagorean Theorem with a little extra.
We have always been using it except that if we can use the Pythagorean Theorem then we must have a right triangle!
If we rewrite a^2 + b^2 = c^2 as
c^2 = a^2 + b^2 - 2ab cos C (cos 90 = 0)
so we are used to seeing it as
c^2 = a^2 + b^2
General Math (Percents)
I love to work with students on 'trick questions' .
For example, you buy a stock at $100 -- it goes up 10% and then down 10%, what is its final price?
You buy a stock for $40 -- it goes up 50% and then down 50%, what is its final price?
And, you buy a stock for $60 -- it goes up 100% and down 100%, what is its final price?
Algebra
Generalizing from y = mx + b so that the result is an equation like y = 2x + 5.
Students benefit from being reminded that they are finding the equation that describes the entire line not just a point.
I like to ask them "How many points are on the line?" -- the answer I like the most is "Infinite" but "Too many to count" or "A real lot" are very acceptable answers.
It takes a while to see that (x-5)/(5-x) = -1!!
Calculus and Number Lines
The idea of limits and that inscribing an n-sided polygon in a circle, the more sides you have, the closer the area is to the area of the circle. So as n approaches infinity using a circle with a radius of 1, the area approaches pi (3.14159265....).
Secant and tangent lines have nothing to do with trig functions secant and tangent!!
A really cool problem was f(x) = [x] + [-x] which involves the greatest integer function.
The TI-84 function for greatest integer function is Int. The greatest integer function is best demonstrated with a number line and asking the student what was the last number that you passed. If you are exactly at an integer, then the answer is that number.
For example, [2.5] = 2 we passed 2 on the way to 2.5.
and [4] = 4
But [-2.5] = -3 we passed -3 on the way to -.25 (but we have not yet passed -2!!)
The cool thing about this problem is that for most x's, such as x = 2.5,
f(x) = [2.5] + [-2.5] which comes out to
f(x) = 2 + -3 = -1
The exception is at exactly an integer -- f(4) = [4] + [-4] = 4 + -4 = 0.
This graph looks like a straight line like y = -1, however at each integer value, there is a discontinuity, as y jumps to 0 at -3, -2, -1, 0, 1, 2, 3, etc.
This is best seen by setting xmin = 0.99 and xmax = 1.01 on the window menu -- at x = 1 there is a dot on the x axis!!
Trigonometry
The Law of Cosines is really the Pythagorean Theorem with a little extra.
We have always been using it except that if we can use the Pythagorean Theorem then we must have a right triangle!
If we rewrite a^2 + b^2 = c^2 as
c^2 = a^2 + b^2 - 2ab cos C (cos 90 = 0)
so we are used to seeing it as
c^2 = a^2 + b^2
General Math (Percents)
I love to work with students on 'trick questions' .
For example, you buy a stock at $100 -- it goes up 10% and then down 10%, what is its final price?
You buy a stock for $40 -- it goes up 50% and then down 50%, what is its final price?
And, you buy a stock for $60 -- it goes up 100% and down 100%, what is its final price?
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