Our Riemann Sum Geogebra exploration from class
Another Geogebra definite integral exploration
Friday, January 28, 2011
Wednesday, January 26, 2011
Scribe Post 1/26/2011
Scribe Post for 1/26/2011
We started off class today by getting into groups and correcting our midterm multiple choice.
It's important to remember the product rule and the chain rule. Also, looking at the answers sometimes can be helpful in figuring out whether or not a calculator is needed to answer the question, and in figuring out what work is really necessary.
Next, we went over the free response questions. Again, it's important to remember wording and proving things with calculus. Also, number lines are not counted as a justification, instead you have to use words in your conclusion. To get the marks you have to write out an answer, being careful about using ambiguous pronouns. There are 9 marks per free response question. Remember to verify and explain to get all the points. Also, be careful to follow the instructions and show all you work when the directions ask for it, and answer exactly what the question is asking.
Finally, we took a quiz which was a free response question.
Here's a link about finding derivatives of inverse functions, which is something we've had on homework and a test, and the midterm, but is still something we're finding a little tricky. http://www.analyzemath.com/calculus/Differentiation/derivative_inverse.html
If you're really interested in how the AP test is scored, here's a link: http://www.collegeboard.com/student/testing/ap/calculus_ab/samp.html
We started off class today by getting into groups and correcting our midterm multiple choice.
It's important to remember the product rule and the chain rule. Also, looking at the answers sometimes can be helpful in figuring out whether or not a calculator is needed to answer the question, and in figuring out what work is really necessary.
Next, we went over the free response questions. Again, it's important to remember wording and proving things with calculus. Also, number lines are not counted as a justification, instead you have to use words in your conclusion. To get the marks you have to write out an answer, being careful about using ambiguous pronouns. There are 9 marks per free response question. Remember to verify and explain to get all the points. Also, be careful to follow the instructions and show all you work when the directions ask for it, and answer exactly what the question is asking.
Finally, we took a quiz which was a free response question.
Here's a link about finding derivatives of inverse functions, which is something we've had on homework and a test, and the midterm, but is still something we're finding a little tricky. http://www.analyzemath.com/calculus/Differentiation/derivative_inverse.html
If you're really interested in how the AP test is scored, here's a link: http://www.collegeboard.com/student/testing/ap/calculus_ab/samp.html
Monday, January 24, 2011
Motion & Speed, 1/24/11
First order of business today was to talk about the midterm. O’B’s scale on this midterm corresponded to the AP scale:
93-100 on midterm --> 5 on AP Test
85-92 on midterm --> 4 on AP Test
76-84 on midterm --> 3 on AP Test
Then O’B gave us a quick pep talk about Second Semester Senior Slide...
Next we went over the most recent quiz. Some tips:
#2 required you to do some process of elimination (eliminating choices A, B, and E).
#3 required you to set the denominator of the derivative equal to 0 and solve for y.
- #5 was marked as an actual AP FRQ (out of 9 points). The diffy Q here was relatively simple and worked as long as you remembered not to use rounded values for calculations. Remember that part b) asks at what rate the amount of oil in the well is DECREASING - so if the amount of oil is decreasing, the rate of change of the amount of oil is POSITIVE.
Next we segued into talking about motion by looking at Worksheet 2. O’B gave us a lot of comments on this one and took it back at the end of class.
O’B stressed the follow-through when we are mathematically justifying answers. For example, when looking at #2, we said that the particle must be at rest at some point since the sign of velocity goes from negative to positive. In other words, because of the Intermediate Value Theorem - but the IVT assumes that the function is continuous. So why is THIS function continuous? Sophie pointed out that the function is differentiable - which means you can always take the derivative, which means that the function is continuous after all.
We graphed the table of values given to see what the velocity function really looked like:
Then we talked about the potential maxes and mins - AKA the points where the derivative (acceleration) of the function (velocity) is 0. But it was hard for us to say exactly where a = 0 because we don’t have a function - only a few points. Mr. O’B sketched several versions of a possible velocity curve to demonstrate this to us.
O'B showed us how to use the magical fitpoly to sketch the velocity curve. We found that increasing the degree of the function increased the reliability of the curve - until we got the “perfect” fit with a degree of 5. We then limited the domain to 0
Domingo earned huuuuuge bonus points for quoting the Mean Value Theorem - since there's a horizontal secant line connecting D and F, there's also a corresponding horizontal tangent line to the velocity curve at some point c between D and F (maybe a little before E?). In other words, MVT basically guarantees that yes, there is a time t=c where a(c)=0.
Then Molly countered by using the IVT to say that since acceleration went from positive to negative, there must be a place where a(c)=0. But do we know that a differentiable function (AKA velocity) always has a continuous derivative? The verdict is: NO. Extra credit: find a differentiable function whose derivative is not continuous.
What we learned from these graphs is to never assume anything based only on a table of values - unless the function is monotonic, meaning that there are no "surprises" between given points.
We then went over Examples 2 and 3. Remember:
- be careful with words... for example, "decreasing" vs. "negative". Decreasing describes a function while negative describes a slope of that function
- always justify your answers with a number or example to prove your point
Assignment was given out in class.
If you're still a little shaky on velocity, check out this helpful link: http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/applications/velocity.html
This one's got a few quizzes at the end and an interactive applet: http://www.edinformatics.com/math_science/acceleration.htm
UPDATE: Talking About the AP Test
Sophie had some questions on specificity of answers. O'B answered by saying that if you want to justify something on the AP test, you have to justify it with calculus. It's not enough to say that a graph is decreasing or increasing - you need to use analytical reasoning to say that a derivative is positive or negative. Also, you should never use general pronouns like "it" to refer to something, especially when the difference between a function and its derivative could mean the loss or gain of a few points on an FRQ.
Another thing to know for the AP test is what you can do on your calculator.
1. Make a graph (but don't use it to calculate max/mins or intersect, etc.)
2. Solving an equation (solver function)
3. Take a derivative at a point (nDeriv)
4. Find a definite integral (stay tuned - fnInt???)
As long as you clearly show how you came to your answer (labeling parts, showing the process, maybe even saying that you used your calculator), you will get full credit.
93-100 on midterm --> 5 on AP Test
85-92 on midterm --> 4 on AP Test
76-84 on midterm --> 3 on AP Test
Then O’B gave us a quick pep talk about Second Semester Senior Slide...
Next we went over the most recent quiz. Some tips:
#2 required you to do some process of elimination (eliminating choices A, B, and E).
#3 required you to set the denominator of the derivative equal to 0 and solve for y.
- #5 was marked as an actual AP FRQ (out of 9 points). The diffy Q here was relatively simple and worked as long as you remembered not to use rounded values for calculations. Remember that part b) asks at what rate the amount of oil in the well is DECREASING - so if the amount of oil is decreasing, the rate of change of the amount of oil is POSITIVE.
Next we segued into talking about motion by looking at Worksheet 2. O’B gave us a lot of comments on this one and took it back at the end of class.
O’B stressed the follow-through when we are mathematically justifying answers. For example, when looking at #2, we said that the particle must be at rest at some point since the sign of velocity goes from negative to positive. In other words, because of the Intermediate Value Theorem - but the IVT assumes that the function is continuous. So why is THIS function continuous? Sophie pointed out that the function is differentiable - which means you can always take the derivative, which means that the function is continuous after all.
We graphed the table of values given to see what the velocity function really looked like:
Then we talked about the potential maxes and mins - AKA the points where the derivative (acceleration) of the function (velocity) is 0. But it was hard for us to say exactly where a = 0 because we don’t have a function - only a few points. Mr. O’B sketched several versions of a possible velocity curve to demonstrate this to us.
O'B showed us how to use the magical fitpoly to sketch the velocity curve. We found that increasing the degree of the function increased the reliability of the curve - until we got the “perfect” fit with a degree of 5. We then limited the domain to 0
Domingo earned huuuuuge bonus points for quoting the Mean Value Theorem - since there's a horizontal secant line connecting D and F, there's also a corresponding horizontal tangent line to the velocity curve at some point c between D and F (maybe a little before E?). In other words, MVT basically guarantees that yes, there is a time t=c where a(c)=0.
Then Molly countered by using the IVT to say that since acceleration went from positive to negative, there must be a place where a(c)=0. But do we know that a differentiable function (AKA velocity) always has a continuous derivative? The verdict is: NO. Extra credit: find a differentiable function whose derivative is not continuous.
What we learned from these graphs is to never assume anything based only on a table of values - unless the function is monotonic, meaning that there are no "surprises" between given points.
We then went over Examples 2 and 3. Remember:
- be careful with words... for example, "decreasing" vs. "negative". Decreasing describes a function while negative describes a slope of that function
- always justify your answers with a number or example to prove your point
Assignment was given out in class.
If you're still a little shaky on velocity, check out this helpful link: http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/applications/velocity.html
This one's got a few quizzes at the end and an interactive applet: http://www.edinformatics.com/math_science/acceleration.htm
UPDATE: Talking About the AP Test
Sophie had some questions on specificity of answers. O'B answered by saying that if you want to justify something on the AP test, you have to justify it with calculus. It's not enough to say that a graph is decreasing or increasing - you need to use analytical reasoning to say that a derivative is positive or negative. Also, you should never use general pronouns like "it" to refer to something, especially when the difference between a function and its derivative could mean the loss or gain of a few points on an FRQ.
Another thing to know for the AP test is what you can do on your calculator.
1. Make a graph (but don't use it to calculate max/mins or intersect, etc.)
2. Solving an equation (solver function)
3. Take a derivative at a point (nDeriv)
4. Find a definite integral (stay tuned - fnInt???)
As long as you clearly show how you came to your answer (labeling parts, showing the process, maybe even saying that you used your calculator), you will get full credit.
Wednesday, January 19, 2011
Tuesday, January 11, 2011
Thursday, January 6, 2011
Wednesday, January 5, 2011
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