cheat page cheat page Classes     The Lecture Version of fX.f'X.f''X.intX.gsp
- Limits, Derivatives, Antiderivatives, FTC I, FTC II Presented Dynamically & Analytically



LOGISTICS

Links Claims and Purposes the fX.f'X.f''X.intX.gsp Sketchpad
  • It focuses on the topics of differentials, Reimann Boxes, and the FTC I & II, but,
    has review material on limits and dertivatives.
     
  • It suggests the early introduction and use of a term like a "cumulative function,"
    as in cumulative probability.
    It suggests the use of the "Mother Tongue" rather than just "mathematics."
    See The Languages of the Math Classroom
     
  • It facilitates an analytic and a numeric approach to topics.
     
Disclaimers about the fX.f'X.f''X.intX.gsp Sketchpad
  • It DOES NOT present an algebraic approach or deal with computation problems and solutions.
     
  • It DOES NOT REALLY TAKE A DERIVATIVE.
    It uses [ f(x +.005) - f(x - .005)]/.01 as an approximation of the derivative.
     
  • It DOES NOT INTEGRATE.
    It uses sums of Reimann boxes plotted to produce a plot of a cummulative curve.
     

        This talk is just a brief summary of what this Sketchpad can do.  Here we BRIEFLY examine "Limits, Derivatives, Antiderivatives, FTC I, FTC II Presented Dynamically & Analytically."



LIMIT by approach
  • Take a limit, as x approaches c, where f(x) is continuous/not continuous.
  • Take a limit, as x approaches infinity.


DERIVATIVES by definition, secant
  • Slide to make h go to infinity as the slope of the secant approaches the slope of the tangent, the derivative at (x, f(x)).
  • Use emojis to mark status of the function.
  • Find & mark other values of C, f(C), f' (C), f ' ' (C), intervals over which the function is increasing/decreasing/zero.
  • Plot the 1st, 2nd, 3rs, 4th derivative of a function using the derivatives of sin(x) as example.


PARTITION & SUMS 4 boxes
  • Adjust the partitioning of an interval.
  • State/show Reimann Boxes notes, left-hand boxes, right-hand boxes, midpoint boxes areas, sums
  • Show int, from a to b, [(f(x)) dx] using the "hump areas" of sine/cosine.
  • Show int, from a to b, [(f(x)-g(x)) dx]
  • Show cumulative probability distribution.


HISTORY
  • Note how a finite sum becomes an integral.
  • State history of symbols & vocabulary.
  • State & explain FTC I & FTC II.


INTEGRATION by dots
  • Examine how the plotted points & how curve is off by a constant.
  • Examine integral of f(y)


The talk ends here.
Below are listed the Links and Activities on each sheet of the Sketchpad as found in the Teachers' Manual web page
Review If Needed
0 - toc
Links
Functions, graphing, fancy, history (see 0 - toc)
Derivatives Web Page
Reimann Sums gsp
ReimannSumNotes.pdf
Intro to Antiderivatives
Derivative Calculator at https://www.derivative-calculator.net/
Integral Calculator at https://www.integral-calculator.com/
Activities
* Use the above pages & calculators to assist in writing a test.
1 - LIMIT by approach
Links
endbahavior.htm
limit.htm
limit.gsp
 
Activities
1st. Slide the x value to approach.
* Take a limit, as x approaches c, where f(c) is continuous
* Take a limit, as x approaches c, where f(c) is not continuous,
      as in x=c is a vertical asymptote
* Take a limit, as x approaches infinity
* Change the function & repeat the above
* Examine endbehavior
2 - DERIVATIVE by definition, secant
Links
limit.htm which includes limit.gsp
Show work
 
Activities
1st. Drag the red point to make h smaller, closer to 0, to make h approach 0, to obtain the derivative.
2nd. As h gets smaller the secant line EF becomes more like a tangent line.
3rd. Try to slide the red point so close to (x, f(x)) that the slope of the secant
equals the slope of the tangent, the derivative at (x, f(x)).
3 - DERIVATIVE by m of tangent line, x
Links
TABLE f, f(x), f '(x), f " " (x) DerAnyFx.gsp then page 5
also See No. 20 below reguarding "Derivative TABLE found in DerAnyFx.gsp"
Show computation & functions
 
Activities
* Use Ctrl + C to make more emojis as needed.
* Use emojis to mark status of the function.
* Find & mark the zeros of the function.
* Find & mark the zeros of the first derivative.
* Discuss the status of the function at these points/values.
* Find & mark the zeros of the second derivative.
* Discuss the status of the function at these points/values.
* Find & mark other values of C, f(C), f' (C), f ' ' (C).
* Discuss intervals over which the function is increasing/decreasing/zero.
* Discuss intervals over which the function is concave up/down.
* Summarize as desired.
* Change the function & repeat the questions/activities.
4 - DERIVATIVE by trace
Links
Hide/show trig memory trick
Hide/show derivative functions
Hide/show Teaching Activities.
 
Hide/show 1st derivitive in green
Hide/show 2nd derivitive in purple
Hide/show 3rd derivitive in orange
Hide/show 4th derivitive in red
 
Activities
Trace the derivatives.
1st: Turn OFF then ON in Display Menu "Trace Point" OR use Ctrl + T.
2nd: Drag the DOT ON THE AXIS to trace that color derivative.
3rd: Erace the trace with Display Menu OR Shift + Ctrl + E.
4th: Trace the derivatives.
Teaching activities
1. Enable the tracing, trace, name the dot-drawn derivative, record it on the screen w/pen
2. Repeat step 1 with second derivative.
3. Repeat step 1 with the next derivative.
4. Repeat step 1 with the next derivative.
5. Reflect on/Discuss the result of all graphs.
6. Unhide the memory trick.


Partition, Differential, Boxes
5 - PARTITION & SUMS 4 boxes
Links
Hide/Show Boxes
Hide/Show g(x)
Hide/Show left-hand sums
Hide/Show g(x)
Go to ReimannSums.gsp
 
Notes
* Reimann.htm exists and is formated like this page and includes
ReimannSumNotes.pdf and ReimannSums.gsp
* The next page shows example of area and sum computation.
 
Activities
* Show boxes.
* Discuss and position left-hand boxes, right-hand boxes, midpoint boxes.
* Discuss the areas and sum of area of
left-hand boxes, right-hand boxes, midpoint boxes.
* Move boxes from picture.
* Change a and b and perhaps f(x).
* Show boxes and discuss areas.
* Move boxes from picture.
6 - REIMANN & SUMS
Links
* ReimannSumNotes.pdf
* Reimann.htm including ReimannSums.gsp
 
Activities
* Discuss areas, summation meaning & format
* Perhaps go to links
7 - SUMS f(x) - g(x), slide end points to change [a, b]
Links
Show height and area computation
Show midpoint coputation
Show g(x)
 
Activities
1. Start w/g(x) = 0
2. Ask can an area be negative?
* Show negative & positive areas.
* Show the left vs midpoint vs right turning negative as x is dragged.
3. Ask how would increasing the number of boxes effect the area?
4. Ask which is the best approach for computation & why?
5. Reveal g(x) and discuss height.
6. Change f(x), g(x), or both.
7. N is small (so the differences in the 3 different sums is large).
Use the midpoint sum to answer this question.
As is, the sum is an approximation of the area under f(x) from a to b.
What is the result if one wishes the sum from b to a?
8 SUMS f(x)- g(x), input [a,b] in the boxes for a and b
Note
This page only provides the left-sum, but provides a
more accurate way to name a and b.
 
Links
Show g(x)
 
Activities
* Change f(x) to an odd function and ask questions.
* Change f(x) to an even function and ask questions.
9 - CUMULATIVE AREA probability distribution
Notes
My calc students don't usually take statistics because they are taking calc.
This sheet serves as a background and shows a real life example of using an integral.
N is 32. This is small, compaired to infinite, but large enough to provide a good estimate.
 
Two images are provided. The top is what the sheet usually looks like.
The bottom shows notes.
 
This also illustrates the use of the term "cumulative" to indicate the integral
of a function and provide/state cumulative results.
 
Links
* Show function f(x, mu, sigma)
* Show constant 1.
* Show constant 2.
* Download statistics spreadsheet - use sheet cum
This has an integration function feature.
* Link to "Probability for Calc I"
This provides a summary of statistics and the normal probability distribution.
 
 
Activities
* Play.
10 - HISTORY sum 2 integral
Links
  1st. Symbols & vocabulary
  2nd: Exactly how do they relate?
  3rd: How Does the finite go to the infinite?
  4th: FTC I and II
Activities
* In order reveal and discuss the notes.
11 - INTEGRATION by dots
Links
FTC I and II
Plots on the next page
 
Activities
* In order reveal and discuss the notes.
12 - INTEGRATION FTC I & FTC II
Notes:  The plotted points are off by a constant. It is an indefinate integral.
Change the parameter "plus c" to adjust the plotted antiderivative points.
 
Links
* Show g(x) and height = h(x)
* Show ordered pairs of plotted points.
* Show "Why plus c?"
* Show blue Suggested Functions box.
 
Activities
* Change the function as desired.
* Functions that fit on the screen are suggested.


Extras
13 - INTEGRATION f(y)
Links
 
Activities
* Play.
20 - Derivative TABLE found in DerAnyFx.gsp
Links
DerAnyFx.gsp -- derivatives of any function -- includes building a TABLE
 
Activities
1st: Drag the blue point to change x and show a new tangent line.
2nd: Below, select, left to right each of the measurements.
3rd: In Number Menu, select Tabulate and a table is created.
4th: With Tabulate selected, click once and go to Properties.
5th: Drag the blue point to a new x value (abscissa).
6th: Double click on table to add another row with the new abscissa.
7th: Repeat as desired.
8th: To delete a row, Shift + Click + click.


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