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

 Intro & Purpose of This Page     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 .

 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 * 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.