Today in class we received a brief introduction to the course, regarding homework and notes on the portal, our books not being required in class everyday, and an outline of the history of calculus. Calculus was discovered by Leibniz and Newton, as they were looking to model things such as motion, especially planetary motion. We also drew two graphs modeling the motion of a ball and began a partner exercise also regarding motion that we need to finish for homework.
Today we spent class using the "delta process" to find mtan.In the delta process, we learned that h= change in X. Some people went up to the board to complete our first problem. At the beginning of class, we found that finding the slope of one secant line is the same as averaging two other ones that go through the same points. Towards the end of class, we reviewed pascal's triangle. For homework, we should do "week 2 assignment 1." Fun Class Everyone!
Today in class we continued studying limits. We reviewed the slope of a tangent line and the "delta method" to find mtan, and we started a worksheet on this. We redefined a tangent line as a line that is not limited to intersecting a curve once. In the case of sine/cosine graphs and other functions, they intersect in at least two points, if not infinity.
For homework, we have to do p. 145-146 (1,2,5aii,5b,15), p. 45 (1), and explain why calculus cannot define a tangent line as a line that only intersects a curve in exactly one point.
Class #6 (8-27-08): Students were given a position - time graph, and they then had to walk in the motion detector in a way that would create the prescribed position time graph. Students had to think carefully about acceleration and its impact on the position time graph. We then discussed the history of calculus; specifically, we discussed the fact that Newton was unable to clearly define the concept of "limit". The students started working on an investigation of how different functions behave near points where they are undefined.
(08-28-08) Today in class we took a quiz on calculating average and instantaneous velocity. After the quiz we went over the limit investigation worksheet. We learned that the limit of f(x) as x approaches a certain number (c) equals L. Using this limit we are sure that output or y-value is close to the value us close to the value of L. This logic is used when trying to find what the corresponding y-value is when x is close to the limit.
addition to 8-28-08 comment We also discussed three different reasons that the limit of f(x), as x approaches c, might fail to exist: (i) a jump (b) f(x) growing without bound (a vertical asymptote) (c) infinitely rapid oscillation of the f(x) values
On Friday, we practiced evaluating Lim f(x)as x approaches c and spent the last five minutes discussing the quizzes from Thursday. The primary focus of the class was that when considering the limit of f(x), it is O.K. to consider x close to a, but never consider x=a. There were several examples of Lim f(x) in which x=a was not present or in a position away from the rest of the graph so as to emphasize that x=a has no bearing on Lim f(x) as x approaches c. We then split up into groups to practice finding limits. Class ended with a discussion of the quiz and a notification that the homework had been changed. For the weekend, no problems were to be done, only reading.
Today we worked on evaluating limits using algebra if when substituting the value x is approaching into the equation you get (0/0). Using the theorem that if f(x)=g(x) for all values not equaling a, then the limit of f(x) as x approaches a will equal the limit of g(x) as x approaches a, we replaced an equation that was in indeterminate form with one that yields a viable answer. We factored and canceled out radicals in the numerator to do this. We also discussed the laws of limits, such as the limit of the sum of two functions equals the sum of the individual limits.
This Friday (Sept. 6) we took a quiz over limits. After we finished the quiz, we completed the continuity packet and went over some of the problems that were tricky. Mr. Hansen put up a problem on the board for anyone already finished with the packe that said find two functions f(x) and g(x) whose limits are not defined at x=4, and when the functions are added together the limit is defined at x=4.
9/8/2008 6:39:00 PM We got out quizzes back. :P We started of class by talking about mathematical models, coming to the conclusion that good models have to be continuous. This is the case because often when we try to predict a phenomena our values tend to be slightly off, and if there are discontinuities in our model, our prediction, could thus be terribly wrong. We also learned about a function that is not continuous at any point, a later inquired into the Intermediate Property Value. The objective was to determine if all graphs which display the IPV are continuous. Caroline Rouse, proved that this was not true, by drawing two parallel diagonal lines. Thus, all continuous functions display the IPV, but not all function that have this property are continuous.
We spent the day on the topic: evaluating the notion of behavior of functions as x -> infinity. We learned that as x approaches infinity for any function that is a constant / x raised to a positive power. Then, to evaluate the limit of functions as x approaches infinity, we used a variation of a polish 1 to get variables to be virtually 0 at infinity, giving us the answer.
We then worked on an "IVP and lim x-> infinity + review" packet, but did not finish it. For homework, we are to complete week 4, assignment 2.
We started out reviewing the homework, and then we focused on the squeeze theorem problem. We learned that for functions that contained a trig component (especially sine and cosine) we can use the other part of the function to create a boundary. When we evaluate the limits of the boundaries, we can determine the limit of the whole function. We also talked about the derivative function. We defined it as the equation of tangent line slopes at given x-values. We can derive it by making the fixed point in the delta process x, f(x).
Today we started off talking about the homework problem that asked to find what was the limit as X approaches zero for: ((2x^2+1)^sqrt)/(3x-5). After figuring that out, we discovered that changing x with -x, it makes the input change from infinity to negative infinity. For the second part of class, we worked on part 2 of the worksheet. Using the delta process,we wanted to find out what f'(x) (the derived function, or the derivative) is for f(x) using the basic quadratic equation. After completing that section, we were asked to find what the to use the delta process to find the derivatives of other powers. From there, we noticed a pattern: If f(x)=X^n, f'(x)=nx^(n-1). Using this basic equation, the delta process will no longer be necessary to find the derivative.
Friday, Sept. 12 Yesterday we worked in groups and found the derived function formulas for y=x^3, y=x^4 and y=x^5. Students then conjectured that if N is a positive integer, the derivative formula for f(x) = x^ N will always be f'(x) = N x^(N-1). Today in class we proved this conjecture, working together as a class. The work is also shown on the embedded slideshow. Fixed point (x, x^N) Moving point (x+h, (x+h)^N) msec = [(x+h)^N-x^N]/ h =[x^N + Nx^(N-1)h + terms with h^2, h^3,...,h^N - x^N] / h
the x^N terms subtract out and we then factor out h to get (h/h)[Nx^(N-1)+ terms with h, h^2, ...,h^(N-1)
we evaluate the limit as h-->0 (h/h) = 1 all terms that have h, h^2,...h^(N-1) will approach ZERO We are left with NX^(N-1)= f'(x)
Monday, sept. 15th We worked on test review for half of the class (since this weekend was homecoming). We then discussed some other derivative formulas: (1) dc / dx = 0 if c is a constant. We discussed this geometrically [the function y = c is a horizontal line whose slope is 0 at each point] and algebraically: Fixed point (x,c) moving point (x+h, c) y' = lim (as h->0) of [ c - c] / h = lim (as h->0) of 0 / h =0
(2) We also discussed the derivative of a constant multiple of a power function, f(x) = Kx^n We discussed how the factor of K stretches the graph vertically so that increases all of the slopes by a factor of K, and we also used the delta process and noticed how the K factors out: lim(h->0) [K(x+h)^n - K x^n] / h = lim(h->0) K [{(x+h)^n-x^n}/h] = K lim(h->0) [(x+h)^n - x^n]/h =K f'(x)
FOR THOSE STUDENTS WHO WERE NOT HERE: #3c is wrong, the answer should be 10 cm/sec not 12 cm/sec
Today in class, we started with what Velocity and Acceleration values mean when they carry similar and different signs. When velocity and acceleration are both positive, speed is increasing. When velocity and acceleration have alternating signs, speed is decreasing.
We were also reminded that output values for f '(x) give us the slope of tangent for f(x). Similarly, if f(x) is a position function, f '(x) gives us the V inst.
We also discussed that tangency means that two things have to be shared: 1) a point 2) same slope
We continued into discussing how to solve 2 different equations with different variables if they share a point.
From this, we discovered that f '(x) can help us in discovering tangency and related equations.
PS: Horizontal tangents have a slope of 0. That might help with the homework or handout questions and further simplifications.
Today in class we talked about how to determine when a ball rolling up a ramp changes direction, how to find the total distance traveled, and how to determine displacement/net change in position, which tells us how far away from the origin of where the ball started the ball is. We also figured out that the power rule works for the roots. In groups on our own we worked on the worksheet from 9/19/08 side one #3 and side two #26.
Today in class we talked about how to determine when a ball rolling up a ramp changes direction, how to find the total distance traveled, and how to determine displacement/net change in position, which tells us how far away from the origin of where the ball started the ball is. We also figured out that the power rule works for the roots. In groups on our own we worked on the worksheet from 9/19/08 side one #3 and side two #26.
Today we did mostly partner work concerning graphing y' when given the graph of y. We also went over one of the homework problems: the ten-part question where we had to find the total distance an object that was moving in both the positive and negative direction traveled over a certain interval.
Yesterday in class we took a brief quiz at the beginning of the period covering linearization of a function, differentials, and Newton's Method. After the quiz, we stated that "the values of f prime equal the slopes on f." We then applied this statement as we did a work sheet for the remainder of the period that asked us to draw a possible graph of f(x) given a graph of its derivative.
Today, we reviewed homework, then continued in proving the derivatives of trigonometric functions.
Since we proved lim (x-> 0) sin(x)/x=1 yesterday, we started by proving today that lim (x-> 0) (1-cos(x))/x=0. To do this, we multiplied the left side of the equation by its conjugate radical and split the result into lim (x-> 0) sin(x)/x * lim (x-> 0) sin(x)/(1+cos(x)). We then found that the right side of this limit equals 0, so therefore, the limit as x approaches 0 for the function (1-cos(x))/x is 0.
Next, we proved d/dx(sinx) = cosx, using the delta process and trigonometric identities. Finally, we used the same method to prove d/dx(cosx) = -sinx.
Date: 10.8.2008 We walked into class to find a WS on our desks. The front side contained answers to the Monday quiz, and the back side had trig practice problems. We were handed back our quizzes and discussed the problems with on which some of us struggled. A second worksheet was handed out which dealt with the derivatives of sinx and cosx. Another section of the worksheet focused on finding the limit of indeterminate functions by using our calculators. The WS also asked us to prove that lim as x –0 of sinx/x = 1 for x –0+. We solved for the area of a particular triangle inscribed within and circle and the area of a sector, after which the squeeze theorem was applied. Using the squeeze theorem we figured out that lim x-0 cosx< lim x-0 sinx/x < lim x-0 1.
Today we spent half of the class doing a quiz on the chain rule. After that, we did a worksheet involving the relationship between two focus points and a point on an ellipse. If a beam of light is shot from the first focal point to the point on the ellipse, it will hit the other focal point.
24 comments:
Today in class we received a brief introduction to the course, regarding homework and notes on the portal, our books not being required in class everyday, and an outline of the history of calculus. Calculus was discovered by Leibniz and Newton, as they were looking to model things such as motion, especially planetary motion. We also drew two graphs modeling the motion of a ball and began a partner exercise also regarding motion that we need to finish for homework.
Today we spent class using the "delta process" to find mtan.In the delta process, we learned that h= change in X. Some people went up to the board to complete our first problem. At the beginning of class, we found that finding the slope of one secant line is the same as averaging two other ones that go through the same points. Towards the end of class, we reviewed pascal's triangle. For homework, we should do "week 2 assignment 1." Fun Class Everyone!
Today in class we continued studying limits. We reviewed the slope of a tangent line and the "delta method" to find mtan, and we started a worksheet on this. We redefined a tangent line as a line that is not limited to intersecting a curve once. In the case of sine/cosine graphs and other functions, they intersect in at least two points, if not infinity.
For homework, we have to do p. 145-146 (1,2,5aii,5b,15), p. 45 (1), and explain why calculus cannot define a tangent line as a line that only intersects a curve in exactly one point.
Class #6 (8-27-08):
Students were given a position - time graph, and they then had to walk
in the motion detector in a way that would create the prescribed position time graph. Students had to think carefully about acceleration and its impact on the position time graph.
We then discussed the history of calculus; specifically, we discussed the fact that Newton was unable to clearly define the concept of "limit". The students started working on an investigation of how different functions behave near points where they are undefined.
(08-28-08)
Today in class we took a quiz on calculating average and instantaneous velocity. After the quiz we went over the limit investigation worksheet. We learned that the limit of f(x) as x approaches a certain number (c) equals L. Using this limit we are sure that output or y-value is close to the value us close to the value of L. This logic is used when trying to find what the corresponding y-value is when x is close to the limit.
addition to 8-28-08 comment
We also discussed three different reasons that the limit of f(x), as x approaches c, might fail to exist:
(i) a jump
(b) f(x) growing without bound (a vertical asymptote)
(c) infinitely rapid oscillation of the f(x) values
On Friday, we practiced evaluating Lim f(x)as x approaches c and spent the last five minutes discussing the quizzes from Thursday. The primary focus of the class was that when considering the limit of f(x), it is O.K. to consider x close to a, but never consider x=a. There were several examples of Lim f(x) in which x=a was not present or in a position away from the rest of the graph so as to emphasize that x=a has no bearing on Lim f(x) as x approaches c. We then split up into groups to practice finding limits. Class ended with a discussion of the quiz and a notification that the homework had been changed. For the weekend, no problems were to be done, only reading.
Today we worked on evaluating limits using algebra if when substituting the value x is approaching into the equation you get (0/0). Using the theorem that if f(x)=g(x) for all values not equaling a, then the limit of f(x) as x approaches a will equal the limit of g(x) as x approaches a, we replaced an equation that was in indeterminate form with one that yields a viable answer. We factored and canceled out radicals in the numerator to do this. We also discussed the laws of limits, such as the limit of the sum of two functions equals the sum of the individual limits.
This Friday (Sept. 6) we took a quiz over limits. After we finished the quiz, we completed the continuity packet and went over some of the problems that were tricky. Mr. Hansen put up a problem on the board for anyone already finished with the packe that said find two functions f(x) and g(x) whose limits are not defined at x=4, and when the functions are added together the limit is defined at x=4.
9/8/2008 6:39:00 PM
We got out quizzes back. :P
We started of class by talking about mathematical models, coming to the conclusion that good models have to be continuous. This is the case because often when we try to predict a phenomena our values tend to be slightly off, and if there are discontinuities in our model, our prediction, could thus be terribly wrong. We also learned about a function that is not continuous at any point, a later inquired into the Intermediate Property Value. The objective was to determine if all graphs which display the IPV are continuous. Caroline Rouse, proved that this was not true, by drawing two parallel diagonal lines.
Thus, all continuous functions display the IPV, but not all function that have this property are continuous.
September 9th Class
We reviewed homework first... :D
We spent the day on the topic: evaluating the notion of behavior of functions as x -> infinity. We learned that as x approaches infinity for any function that is a constant / x raised to a positive power. Then, to evaluate the limit of functions as x approaches infinity, we used a variation of a polish 1 to get variables to be virtually 0 at infinity, giving us the answer.
We then worked on an "IVP and lim x-> infinity + review" packet, but did not finish it. For homework, we are to complete week 4, assignment 2.
We started out reviewing the homework, and then we focused on the squeeze theorem problem. We learned that for functions that contained a trig component (especially sine and cosine) we can use the other part of the function to create a boundary. When we evaluate the limits of the boundaries, we can determine the limit of the whole function. We also talked about the derivative function. We defined it as the equation of tangent line slopes at given x-values. We can derive it by making the fixed point in the delta process x, f(x).
Today we started off talking about the homework problem that asked to find what was the limit as X approaches zero for: ((2x^2+1)^sqrt)/(3x-5). After figuring that out, we discovered that changing x with -x, it makes the input change from infinity to negative infinity.
For the second part of class, we worked on part 2 of the worksheet. Using the delta process,we wanted to find out what f'(x) (the derived function, or the derivative) is for f(x) using the basic quadratic equation. After completing that section, we were asked to find what the to use the delta process to find the derivatives of other powers. From there, we noticed a pattern: If f(x)=X^n, f'(x)=nx^(n-1). Using this basic equation, the delta process will no longer be necessary to find the derivative.
Friday, Sept. 12
Yesterday we worked in groups and found the derived function formulas for y=x^3, y=x^4 and y=x^5. Students then conjectured that if N is a positive integer, the derivative formula for f(x) = x^ N will always be f'(x) = N x^(N-1).
Today in class we proved this conjecture, working together as a class. The work is also shown on the embedded slideshow.
Fixed point (x, x^N)
Moving point (x+h, (x+h)^N)
msec = [(x+h)^N-x^N]/ h
=[x^N + Nx^(N-1)h + terms with h^2, h^3,...,h^N - x^N] / h
the x^N terms subtract out and we then factor out h to get
(h/h)[Nx^(N-1)+ terms with h, h^2, ...,h^(N-1)
we evaluate the limit as h-->0
(h/h) = 1
all terms that have h, h^2,...h^(N-1) will approach ZERO
We are left with
NX^(N-1)= f'(x)
Monday, sept. 15th
We worked on test review for half of the class (since this weekend was homecoming). We then discussed some other derivative formulas:
(1) dc / dx = 0 if c is a constant.
We discussed this geometrically [the function y = c is a horizontal line whose slope is 0 at each point] and algebraically: Fixed point (x,c)
moving point (x+h, c)
y' = lim (as h->0) of [ c - c] / h
= lim (as h->0) of 0 / h
=0
(2) We also discussed the derivative of a constant multiple of a power function, f(x) = Kx^n
We discussed how the factor of K
stretches the graph vertically so that increases all of the slopes by a factor of K, and we also used the delta process and noticed how the K factors out:
lim(h->0) [K(x+h)^n - K x^n] / h
= lim(h->0) K [{(x+h)^n-x^n}/h]
= K lim(h->0) [(x+h)^n - x^n]/h
=K f'(x)
Notes for September 19, 2008:
FOR THOSE STUDENTS WHO WERE NOT HERE: #3c is wrong, the answer should be 10 cm/sec not 12 cm/sec
Today in class, we started with what Velocity and Acceleration values mean when they carry similar and different signs. When velocity and acceleration are both positive, speed is increasing. When velocity and acceleration have alternating signs, speed is decreasing.
We were also reminded that output values for f '(x) give us the slope of tangent for f(x). Similarly, if f(x) is a position function, f '(x) gives us the V inst.
We also discussed that tangency means that two things have to be shared: 1) a point 2) same slope
We continued into discussing how to solve 2 different equations with different variables if they share a point.
From this, we discovered that f '(x) can help us in discovering tangency and related equations.
PS: Horizontal tangents have a slope of 0. That might help with the homework or handout questions and further simplifications.
Today in class we talked about how to determine when a ball rolling up a ramp changes direction, how to find the total distance traveled, and how to determine displacement/net change in position, which tells us how far away from the origin of where the ball started the ball is. We also figured out that the power rule works for the roots. In groups on our own we worked on the worksheet from 9/19/08 side one #3 and side two #26.
Today in class we talked about how to determine when a ball rolling up a ramp changes direction, how to find the total distance traveled, and how to determine displacement/net change in position, which tells us how far away from the origin of where the ball started the ball is. We also figured out that the power rule works for the roots. In groups on our own we worked on the worksheet from 9/19/08 side one #3 and side two #26.
Today we did mostly partner work concerning graphing y' when given the graph of y. We also went over one of the homework problems: the ten-part question where we had to find the total distance an object that was moving in both the positive and negative direction traveled over a certain interval.
Yesterday in class we took a brief quiz at the beginning of the period covering linearization of a function, differentials, and Newton's Method. After the quiz, we stated that "the values of f prime equal the slopes on f." We then applied this statement as we did a work sheet for the remainder of the period that asked us to draw a possible graph of f(x) given a graph of its derivative.
Today, we reviewed homework, then continued in proving the derivatives of trigonometric functions.
Since we proved lim (x-> 0) sin(x)/x=1 yesterday, we started by proving today that lim (x-> 0) (1-cos(x))/x=0. To do this, we multiplied the left side of the equation by its conjugate radical and split the result into lim (x-> 0) sin(x)/x * lim (x-> 0) sin(x)/(1+cos(x)). We then found that the right side of this limit equals 0, so therefore, the limit as x approaches 0 for the function (1-cos(x))/x is 0.
Next, we proved d/dx(sinx) = cosx, using the delta process and trigonometric identities. Finally, we used the same method to prove d/dx(cosx) = -sinx.
Remember: cos(x+h) = cosxcosh - sinxsinh and sin(x+h) = sinxcosh + sinhcos
x.
Homework is Assignment 3, Week 8
Date: 10.8.2008
We walked into class to find a WS on our desks. The front side contained answers to the Monday quiz, and the back side had trig practice problems. We were handed back our quizzes and discussed the problems with on which some of us struggled.
A second worksheet was handed out which dealt with the derivatives of sinx and cosx. Another section of the worksheet focused on finding the limit of indeterminate functions by using our calculators. The WS also asked us to prove that lim as x –0 of sinx/x = 1 for x –0+. We solved for the area of a particular triangle inscribed within and circle and the area of a sector, after which the squeeze theorem was applied. Using the squeeze theorem we figured out that lim x-0 cosx< lim x-0 sinx/x < lim x-0 1.
Today we spent half of the class doing a quiz on the chain rule. After that, we did a worksheet involving the relationship between two focus points and a point on an ellipse. If a beam of light is shot from the first focal point to the point on the ellipse, it will hit the other focal point.
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