Animations of Calculus for GSP 4 or GSP5
(80 concepts & 160 FRs from past AP exams)

Whether teaching calculus at the introductory or AP level, at a high school or college, there is no better way to explore this rich study of movement and change than through dynamic animation.  Calculus In Motion  animations are packaged on a CD and perform equally well on either the Windows or Macintosh platform.  An instruction booklet is included.  The animations described below must be opened by The Geometer's Sketchpad v4 or v5 (no prior versions), owned and sold by McGraw-Hill Education, on either Windows or Macintosh platforms.  Although a detailed instruction manual is included on the CD-ROM (PDF format), most of the animations can be run successfully using only the on-screen information.

ARC LENGTH (set of 3)
AREA BETWEEN 2 CURVES (set of 11)
DEFINITION OF A DERIVATIVE
DEFINITION OF INTEGRATION
GRAPHERS (set of 9)
INVERSE FUNCTIONS
LIMITS + CONTINUITY (set of 4)
MACLAURIN & TAYLOR SERIES
OPTIMIZATION (Set of 7)
RELATED RATES (set of 17)
RIEMANN SUMS (set of 3)
SLOPE FIELDS + EULER'S (set of 3)
THEOREMS (set of 6)
VOLUMES ON A BASE (set of 5)
VOLUMES BY REVOLUTION (set of 7)

 

AP EXAM FREE RESPONSE
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....1997
   
  



















 
 


ARC LENGTH

Develop the idea of arc length using any f(x), parametric, or polar curve & any number of partitions.

       


AREA BETWEEN 2 CURVES

Sweeping horizontally or vertically, the first animation explains the main idea, then 8 specific examples
follow with changeable intervals, and finally, 2 animations (one for vertical sweeps and one for horizontal)
you can enter any desired curves as well as the boundaries of integration.

gsp69.gif (80594 bytes) gsp70.gif (61540 bytes) gsp71.gif (64359 bytes) gsp72.gif (61576 bytes)
 gsp73.gif (60817 bytes) gsp74.gif (89836 bytes) gsp75.gif (72526 bytes)
gsp76.gif (62222 bytes) gsp77.gif (63827 bytes) gsp78.gif (61782 bytes) gsp79.gif (102810 bytes)
 

 
DEF. OF A DERIVATIVE

DEF. OF INTEGRATION

INVERSE FUNCTIONS

Drag h to 0 to see PQ become the tangent line. See the limit process in action. Also, create the numerical derivative.

Sweep left or right to accumulate the integral using standard changeable geometric shapes. Also vary the start and stop points.

Using animated tangent lines, compare the derivatives of inverse functions. “Morph” the curves using sliders.
(*also for precalculus)

 
GRAPHERS
Explore slope using animated tangent lines. See any desired combination of f ‘, f ‘’, area, and F.
“Morph” each graph using sliders. A 7th animation (not shown below) allows the user to enter any
desired function and applies all of the same animated features to it. (*also for precalculus)
  polynomial trigonometric logarithmic
exponential parametric polar
 
Newton's method piecewise w/tangents chain rule
 



LIMITS + CONTINUITY
Explore the ε, ∂ definition of limits and the definition of continuity.
Evaluate the limits (full, left-hand or right-hand) of any function (including piece-wise defined) as x →a or as x→±∞


 


MACLAURIN & TAYLOR SERIES
Enter any f(x). Overlay a Maclaurin or Taylor Series polynomial of degree n & use it to approximate the value of f(x) at any point t. Vertical gray bands show where the power series is within a chosen tolerance to f(x). As n increases, the band widens.

 

    
 


OPTIMIZATION

Interact with various classic applications to find the most, lerasdy, cheapest, fastest, etc.
Graphed data includes tangent line & derivative analysis

 



RELATED RATES

A click of a button advances time to commence the action to these classic problems.
Other buttons reveal the values and graphs of the rates.
       

     
       
     
 



RIEMANN SUMS
Choose rectangles using left endpoints, right endpoints, or midpoints; or trapezoids to approximate an integral for any number
partitions from 1 to 80!  Functions can be morphed by dragging sliders, or use the first page to type in any desired function for f(x).



SLOPE FIELDS + EULER'S METHOD

To introduce what a slope field is, use the graph of f ’ to see its values controlling a gliding dynamic “slope column”.
Snapshots of this column are the slope field. A tangent segment “pilots” the field to draw f.
Once understood, a different animation allows any differential equation to be entered and generates the slope field.
Manually follow the field to draw f or use Euler’s Method (includes explanation of E.M. and numerical table of data). Easily adjustable.



THEOREMS
Mean-Value Theorem for Derivatives, Mean-Value Thm. for Derivatives vs. Rolle’s Thm., Intermediate-Value Theorem
Mean-Value Theorem for Integrals (a little game),1st Fundamental Theorem of Calculus (a visual proof)
Connect the 1st FTC to both Mean-Value Theorems

   
 



VOLUMES ON A BASE
Visualize these shapes one step at a time. Start by rotating the xy-plane to horizontal.
View a few stationary slices, then a sweeping slice, and finally, an accumulating slice.
Rotate the solid any time for other viewing angles. Choose from an assortment of bases and cross-sections.



VOLUMES BY REVOLUTION
These animations cover both the disk/washer technique and the cylindrical shell technique.
Develop the process by first revolving one lone rectangle. Next, revolve several rectangles in a region and stack or nest the results.
Finally, revolve any desired region (bounded by 1 or 2 functions of choice) on an interval of choice, about any horizontal or vertical axis.
 


 

 


160 Animations of AP Calculus AB & BC Exams'
Free Response Questions    2017-1997


 
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AB1/BC1
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AB1/BC1
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AB1/BC1
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AB5/BC5
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AB1/BC1
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