fX.f'X.f''X.intX.gsp - The Derivative & Antiderivative Sketchpad

    This page supports a Calc I prof's inclass presentation of content using fX.f'X.f''X.intX.gsp.

• It facilitates an analytic and a numeric approach to topics.
   
• 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.
   
• It DOES NOT present an algebraic approach or deal with computation problems and solutions.
   
• 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, to facilitate use of the "Mother Tongue" rather than just "mathematics." See The Languages of the Math Classroom

• It DOES NOT really present volumes or other Calc II topics. Use other internet-available software like the 3-Dimensional Graphing Calculator - Desmos at https://www.desmos.com/calculator/9jnamzxtjh for this.
   
• It encourages movement of items on the screen and the changing of functions and constants such as a, b, n, delta x.
   

    All material on this site is free. (Please cite the source.)  The sketchpads/resources from which this material is taken include Reimann.htm, limit.gsp, DerAnyFx.gsp, and ReimannSums.gsp.  Other Calc I material is found at: and .

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

* 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

Show work

limit.gsp

 

Activities

FOR THIS PAGE DO NOT CHANGE FUNCTION G(X).

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.

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.

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.