We started out class by talking about how to get a 5 on the AP test this may. Really all you need to get is 30 points on the FRQ. For the free response question, you have 90 minutes to answer 6 questions, which averages to 15 minutes per question. Nine points are allocated for very specific things.
We worked on questions D and C from the midterm exam, starting with question D, which was fairly straightforward in its processing and grading:
To solve for part a, you could receive points for the following:
For part b, you could receive points for the following:
And finally, for part c, you could receive points for the following:
We then went on to work on part C, which was more tricky in understanding what was expected in your answer. The question is posted below:
We started by analyzing what we know about the functions. Function f is a quadratic function, function g is a cubic, and function h is a power function. What determines that function h is a power function is the variables location in the base, not the power. Power functions are fairly easy to solve, like a cubic or a quartic, and you can use the power rules to find the derivative. To answer part a, there were a couple of ways to show that it is not possible to find a value for a so that f meets the second requirement. One way to solve this is shown below:
The points in this problem are allocated for doing some sort of calculation to find a value of a, and some calculation to show that it does not work for another qualification. To solve part b, you need to find a value for c to receive the point.
In part c, the best way to go about this problem is to look at the critical points, because that is where the derivative changes signs. Using some calculus you can find the derivative, shown below.
An important piece of information here is in the explanation. On these free response questions it is really important to justify your answers with calculus. Use of language is very important. Saying something like "the function" or "it" won't be accepted, because it is too ambiguous and could refer to anything. You have to be careful to check over your sentences and justifications to avoid ambiguity, and make sure to use calculus to prove your answer is correct!
Finally, in part d, four points were available. To solve, you need to take the function and evaluate it at four, but also take the derivative at four and set that equal to one. Using the power rule, you can get a system of equations: two equations with two variables. This can be solved with the graphing calculator or using substitution to find n=4. Then you can solve for k with your n value of 4 and get 256. Points are given not only for finding values and setting the equations equal to one, but also in verifying your answers (h(0)=0, h'(0)=0, and h'>0 for 0 < x < 4). This is given below:
If you're worried about these FRQs, don't be! Over the next few months we will be doing a lot of these problems to prepare for the test.
After going over the midterm free response questions, we went over the answers to the multiple choice questions. The multiple choice can't be released, so we won't go over that, but if you have questions about how to solve the problems I'm sure Mr. O'Brien would allow you to come in for help.
To finish off class, we started looking at a more physics oriented side of calculus. In unit four, we're taking time to go back and look at motion, including but not limited to position, velocity, displacement, distance, and acceleration. For this unit, it's important to know the physics students so that the class can make sure who to ask for help and also be aware of those in the class that have never seen this sort of problem before.
There are a few definitions that may be found helpful at this point. Position, x(t) or s(t), is the location of a particle at time t. Velocity, v(t)=s'(t), is how fast the position is changing. Acceleration, a(t)=v'(t)=s''(t), is how fast the velocity is changing.
We did a fairly confusing exercise where Eliot and Connor were particles, and Mr. O'Brien posed several questions that were more confusing then elucidating. What we did determine was at time 0, the position of Eliot was x(0)=7. For Connor at time 5, the position of Connor was x(5)=-4. Position is a particular number representing a particular place in a line, relative to the point, either to the right or left of the origin. We determined that distance divided by time is equal to average speed, or velocity. The average speed in this case is 11/5, which is also the velocity. However if Eliot moved around before moving towards Connor, it doesn't make sense to take the distance and divide it by the time, because we're not talking about a moment in time. We then started to talk about displacement which is the amount by which a thing is moved from its normal position. We tossed around the idea of whether the displacement of Eliot to Connor is any different from the displacement of Connor to Eliot, and determined that the displacement from Eliot to Connor is -11, while from Connor to Eliot it is 11. This is because Connor is moving in a positive direction while Eliot is moving in a negative direction. Finally, we answered the question of whether the velocity is different from the average speed of Eliot moving to Connor. The average velocity of Eliot moving to Connor is thus -11/5, while the average speed is in fact 11/5.
We finished up class with a few minutes to work on 11 questions from the link on iCal. The answers are posted here:
http://ssh.springbranchisd.com/LinkClick.aspx?fileticket=cm531BaGtCw%3D&tabid=17567&mid=78702
Due for Thursday: 11 questions
Due for Friday: IW#1
There will be no extension for IW#1 past Friday.
We then went on to work on part C, which was more tricky in understanding what was expected in your answer. The question is posted below:
The points in this problem are allocated for doing some sort of calculation to find a value of a, and some calculation to show that it does not work for another qualification. To solve part b, you need to find a value for c to receive the point.
In part c, the best way to go about this problem is to look at the critical points, because that is where the derivative changes signs. Using some calculus you can find the derivative, shown below.
An important piece of information here is in the explanation. On these free response questions it is really important to justify your answers with calculus. Use of language is very important. Saying something like "the function" or "it" won't be accepted, because it is too ambiguous and could refer to anything. You have to be careful to check over your sentences and justifications to avoid ambiguity, and make sure to use calculus to prove your answer is correct!
Finally, in part d, four points were available. To solve, you need to take the function and evaluate it at four, but also take the derivative at four and set that equal to one. Using the power rule, you can get a system of equations: two equations with two variables. This can be solved with the graphing calculator or using substitution to find n=4. Then you can solve for k with your n value of 4 and get 256. Points are given not only for finding values and setting the equations equal to one, but also in verifying your answers (h(0)=0, h'(0)=0, and h'>0 for 0 < x < 4). This is given below:
If you're worried about these FRQs, don't be! Over the next few months we will be doing a lot of these problems to prepare for the test.
After going over the midterm free response questions, we went over the answers to the multiple choice questions. The multiple choice can't be released, so we won't go over that, but if you have questions about how to solve the problems I'm sure Mr. O'Brien would allow you to come in for help.
To finish off class, we started looking at a more physics oriented side of calculus. In unit four, we're taking time to go back and look at motion, including but not limited to position, velocity, displacement, distance, and acceleration. For this unit, it's important to know the physics students so that the class can make sure who to ask for help and also be aware of those in the class that have never seen this sort of problem before.
There are a few definitions that may be found helpful at this point. Position, x(t) or s(t), is the location of a particle at time t. Velocity, v(t)=s'(t), is how fast the position is changing. Acceleration, a(t)=v'(t)=s''(t), is how fast the velocity is changing.
We did a fairly confusing exercise where Eliot and Connor were particles, and Mr. O'Brien posed several questions that were more confusing then elucidating. What we did determine was at time 0, the position of Eliot was x(0)=7. For Connor at time 5, the position of Connor was x(5)=-4. Position is a particular number representing a particular place in a line, relative to the point, either to the right or left of the origin. We determined that distance divided by time is equal to average speed, or velocity. The average speed in this case is 11/5, which is also the velocity. However if Eliot moved around before moving towards Connor, it doesn't make sense to take the distance and divide it by the time, because we're not talking about a moment in time. We then started to talk about displacement which is the amount by which a thing is moved from its normal position. We tossed around the idea of whether the displacement of Eliot to Connor is any different from the displacement of Connor to Eliot, and determined that the displacement from Eliot to Connor is -11, while from Connor to Eliot it is 11. This is because Connor is moving in a positive direction while Eliot is moving in a negative direction. Finally, we answered the question of whether the velocity is different from the average speed of Eliot moving to Connor. The average velocity of Eliot moving to Connor is thus -11/5, while the average speed is in fact 11/5.
We finished up class with a few minutes to work on 11 questions from the link on iCal. The answers are posted here:
http://ssh.springbranchisd.com/LinkClick.aspx?fileticket=cm531BaGtCw%3D&tabid=17567&mid=78702
Due for Thursday: 11 questions
Due for Friday: IW#1
There will be no extension for IW#1 past Friday.
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