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2008
(41)
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October
(10)
- Quotient Rule
- 10.28.08.Homework help and a note
- curve sketch from derivative information
- concave up VS. concave down (many perspectives)
- concavity and inflection points.Day 1
- statement, converse, contrapositve, f not differen...
- Derivative fails to exist
- Student notes for second quarter
- derive sine and cosine
- anti-derive polynomials
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October
(10)
5 comments:
Monday, October 20, 2008
Today, we were given a worksheet that began with review of topics from last week, such as the derivative of common functions such as cosine, sin, and exponents. Then we set our focus towards absolute value functions. We wanted to know what the derivative was at a specific point such as (0,0) on the graph f(x)=|x|. In order to try to solve that, we found the limit as h approaches 0 for both the left and the right side, and we got -1 and 1, respectively. Since they are not equal, we learned that there is no derivative at this point. We moved on with more examples that require us to use the same concept of finding the left and right limits to see if there is a derivative.
The second part of the worksheet had us graph a parabola and zooming in enough that it appears to be a line on the calculator. This showed us the importance of a function being "locally linear" in order to have a derivative at a point as opposed to the absolute value functions, which never appear to be locally linear.
Wednesday, October 22, 2008
Today in class we worked on a new worksheet which taught us more about differentiability and Continuity. Included on this worksheet were problems involving statments and their converses and contra-positives, and we had to determine which were true about differentiability. We also did a problem where we solved for unknown constants (a) and (b) in order to make f(x) both continuous and differentiable. There is a quiz tomorrow and here are some important things to remember:
1. the derivitives of :
sinX = cosX
cosX = -sinX
e^x = e^x
2. how to find higher order derivatives of cosX and sinX:
cosX: you know that cosX will eventually repeat, so you must evaluate the derivative until you find a pattern (cosX, -sinX, -cosX, sinX, cosX, etc)Then, you find the remainder of the higher order divided by 4 and the derivitive is that number in the sequence of derivitives of cosX
sinX: you do the same thing only with the patterns of sinX (sinX, cosX, -sinX, -cosX, sinX, etc.)
Also, be able to solve problems like those given on Monday and Tuesday's Worksheets.
Today we talked about graphs that have both tangent lines that appear below and above the graph. The tangent lines that appear below the graph could be underestimates of linearization while the lines that appear above could be overestimates in linearization.
More importantly, this brought us to the realization that if f prime is increasing (and the tangent line appears below the curve), then the graph CONCAVES UP. If the tangent lines from f prime are above the curve and decreasing then the graph CONCAVES DOWN.
Also, when concavity changes, there is an INFLECTION POINT on the graph. Graphically, this line is above the curve on oneside and below it on the other. Algebraically, this is the same x-value as where f prime reaches its maximum or minimum (because f prime goes from increasing to decreasing)
So the maximum of f IS f prime =0
The maximum of f prime IS f double prime= 0
Today we reviewed more about concavity. When f is concave up, f prime is increasing, the tangent lines lie beneath the curve, the shape of the graph is like a "U" or part of the "U", and the sign of f double prime is positive. When f is concave down, f prime is decreasing, the tangent lines lie above the curve, the shape of the graph is an upside down "U" or part of the upside down "U", and the sign of f double prime is negative. We then worked on a worksheet, and if you finished the worksheet, there is no homework. The date of the next test is Tuesday November 4.
Thursday, November 6, 2008
On Thursday we continued examining the topic of the Chain Rule. We directly applied our new knowledge of this rule when we worked on a handout in class with practice Chain Rule problems. This handout included problems where we apply the Chain Rule directly to a function in order to find the derivative, such as 3cos^5(x) and tan^3(6x). On the back, we used the Chain Rule to solve for derivatives when given a table of values for two functions or when shown a graph of the functions.
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