Friday, November 30, 2007
Thursday, November 29, 2007
Optimization Problems
Well i am finally back to scribe, it has been a pretty long time since i had last scribed so here it is. Today in class, Mr. K introduced a new topic Optimization problems. Optimization problems are problems that deal with either finding the maximum or minimum of the function in the problem. although very rarely is the function given so you as a calculus student must use all your knowledge you've learnt up to now and create a formula/function that can be used to answer the question that is being asked. With that said Mr. K gave the class the 6 rules to live by when doing optimization problems, although do not really live by them, just follow them loosely.
6 Rules to Optimization Problems
1. Find what you are trying to either maximize or minimize.
2. Write an equation for it, make sure to use descriptive variables so you won't get confused when doing the problem.
3. Try to get the equation into a two variable form, where you are finding one of the variables, because you want to find the derivative.
4. Write the derivative of the equation in the two variable form.
5. Set the derivative to zero, and solve for the remaining values. (First Derivative Test or Second derivative Test)
6. If needed depending on what the question asks, Plug the value of that variable into the one or (two) equations to find all dimensions.
Now here is the explanation of the question done in class.
Well first we identified it was a maximum problems, because it asks for the box with dimensions with the largest volume.
Next the equation is made by using V=h*w*L, so h=x, w=16-2x, and L=21-2x. So the equation looks something like this, x(16-2x)(21-2x).
Then we multiply it out. 336x - 74x^2 + 4x^3
Then we take the derivative. Which is 336 - 148x + 12x^2
then by using the quadratic equation we find that x = 3 and 9.3, so we do a line graph and find it is positive before 3 and negative after 3 meaning that by the use of first derivative test, 3 is the maximum x can be.
Well, that was it for today's class. Tomorrow's Scribe will be M@rk.
6 Rules to Optimization Problems
1. Find what you are trying to either maximize or minimize.
2. Write an equation for it, make sure to use descriptive variables so you won't get confused when doing the problem.
3. Try to get the equation into a two variable form, where you are finding one of the variables, because you want to find the derivative.
4. Write the derivative of the equation in the two variable form.
5. Set the derivative to zero, and solve for the remaining values. (First Derivative Test or Second derivative Test)
6. If needed depending on what the question asks, Plug the value of that variable into the one or (two) equations to find all dimensions.
Now here is the explanation of the question done in class.
Well first we identified it was a maximum problems, because it asks for the box with dimensions with the largest volume.
Next the equation is made by using V=h*w*L, so h=x, w=16-2x, and L=21-2x. So the equation looks something like this, x(16-2x)(21-2x).
Then we multiply it out. 336x - 74x^2 + 4x^3
Then we take the derivative. Which is 336 - 148x + 12x^2
then by using the quadratic equation we find that x = 3 and 9.3, so we do a line graph and find it is positive before 3 and negative after 3 meaning that by the use of first derivative test, 3 is the maximum x can be.
Well, that was it for today's class. Tomorrow's Scribe will be M@rk.
Wednesday, November 28, 2007
Limits Involving Infinity
Hello all! It’s Robert here and I’m your scribe for today.
Today’s class was all about Limits Involving Infinity. Finding limits of infinity is all about finding the asymptotes of a graph.
Some things you need to remember when finding limits (finding asymptotes):
Today’s class was all about Limits Involving Infinity. Finding limits of infinity is all about finding the asymptotes of a graph.
Some things you need to remember when finding limits (finding asymptotes):
- When the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at y=0.
- When the degree of the denominator is less than the degree of the numerator, there is no horizontal asymptote.
- When the degree of the numerator is equal to the degree of the denominator, you look to the leading coefficients of the term that has the highest exponent on x.
Well here are the slides for today's class. Take a look and see the different steps used in solving the problems.
That's it for today folks. The next scribe will be........Dino!!!
Tuesday, November 27, 2007
Scribe #5 out of A LOT!
Today's amazing class started off with a bit of side-tracking. Mr. K. introduced us to a couple anecdotes that will be helpful to everyone in their overall learning skills. Now of course Mr. K. got them from a source that I am quite lazy enough not to discover... but they were:
LEVELS OF LEARNING (Bloom's Taxonomy)
Basically describes the different ways we can learn. They are Remember, Understand, Apply, Analyze, Evaluate, and Create. When we begin to learn in elementary, we start at the beginning and eventually climb up the levels of learning. Currently we are at the levels of Analyze and Evaluate, which still require all the others, and use them in class everyday.
LEARNING PYRAMID
Shows the different media (or methods) from which one can learn and the percentage of that knowledge that one would retain from each method. These methods are (in order):Lectures, Reading, Audio-Visual, Demonstrations, Discussions, Practicing by Doing, and Teaching. It turns out we only retain about 5% of knowledge from Lectures and a whopping 90% from teaching... that's a huge gap!
So in conclusion, Mr. K. told us that when we do our DEV (Developing Expert Voices projects)
, we are combining the Create and Teaching methods. That is one heck of a lot of learning that we do!!!
But now I will jump into the actual lesson of the day...
Well, we first did a quick review of the First Derivative Test, which is:
Next, Mr. K. introduced, the Second Derivative Test, which goes as follows:
We then jumped into some sample questions using this test. These can be seen on the slides for today's class. As well, you can click on the links underneath the definition of the tests on the green pages for more practice problems.
Next Mr. K. showed us a little loop-hole... It is thought that when the second derivative (ƒ''(x)) is equal to 0 (zero), there is a point of inflection on the parent function right??? The thing is, it might not be. It may in fact be only a candidate for the point of inflection. Then we must test the concavity on either side of this point (whether ƒ'' is positive or negative). If the concavity is different on each siide, it is a point of inflection, but if it is the same on both sides, the point is not a point of inflection, it is a local extreme that is relatively flat in a certain interval. AN EXAMPLE OF THIS IS THE FUNCTION ƒ(x)=x^4
We then continued with practice problems (which can be seen on the slides; link is above) until the end of class.
Quick thing to add to our understanding of these tests:
First Derivative Test: Interval test; find the candidates for global extrema and check the intervals on each side of them to prove they are in fact extrema.
Second Derivative Test: Point test; find candidates of global extrema and find the concavity of the parent function at each candidate.
Mr. K. has said, that one is not necessarily better than the other, they are just DIFFERENT... meaning in different situations, different tests will be better to use. (I like the Second one better)
That just about wraps up my Scribe for the evening...
REMEMBER GUYS: 5.2 ODDS FOR HOMEWORK!!!
Next Scribe is.......
LEVELS OF LEARNING (Bloom's Taxonomy)
Basically describes the different ways we can learn. They are Remember, Understand, Apply, Analyze, Evaluate, and Create. When we begin to learn in elementary, we start at the beginning and eventually climb up the levels of learning. Currently we are at the levels of Analyze and Evaluate, which still require all the others, and use them in class everyday.
LEARNING PYRAMID
Shows the different media (or methods) from which one can learn and the percentage of that knowledge that one would retain from each method. These methods are (in order):Lectures, Reading, Audio-Visual, Demonstrations, Discussions, Practicing by Doing, and Teaching. It turns out we only retain about 5% of knowledge from Lectures and a whopping 90% from teaching... that's a huge gap!
So in conclusion, Mr. K. told us that when we do our DEV (Developing Expert Voices projects)
, we are combining the Create and Teaching methods. That is one heck of a lot of learning that we do!!!
But now I will jump into the actual lesson of the day...
Well, we first did a quick review of the First Derivative Test, which is:
Next, Mr. K. introduced, the Second Derivative Test, which goes as follows:
We then jumped into some sample questions using this test. These can be seen on the slides for today's class. As well, you can click on the links underneath the definition of the tests on the green pages for more practice problems.Next Mr. K. showed us a little loop-hole... It is thought that when the second derivative (ƒ''(x)) is equal to 0 (zero), there is a point of inflection on the parent function right??? The thing is, it might not be. It may in fact be only a candidate for the point of inflection. Then we must test the concavity on either side of this point (whether ƒ'' is positive or negative). If the concavity is different on each siide, it is a point of inflection, but if it is the same on both sides, the point is not a point of inflection, it is a local extreme that is relatively flat in a certain interval. AN EXAMPLE OF THIS IS THE FUNCTION ƒ(x)=x^4
We then continued with practice problems (which can be seen on the slides; link is above) until the end of class.
Quick thing to add to our understanding of these tests:
First Derivative Test: Interval test; find the candidates for global extrema and check the intervals on each side of them to prove they are in fact extrema.
Second Derivative Test: Point test; find candidates of global extrema and find the concavity of the parent function at each candidate.
Mr. K. has said, that one is not necessarily better than the other, they are just DIFFERENT... meaning in different situations, different tests will be better to use. (I like the Second one better)
That just about wraps up my Scribe for the evening...
REMEMBER GUYS: 5.2 ODDS FOR HOMEWORK!!!
Next Scribe is.......
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