#    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.

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