**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 interactive animation. Calculus In Motion^{TM} animations are packaged on a CD-ROM (the basic license is for 1 computer, but other licenses also are available). They perform equally well on either the Windows or Macintosh platform. Although a detailed instruction manual is included on the CD-ROM (PDF format), most of the animations can be successfully run simply using the on-screen information.
NOTE: The animations are data files written in The Geometer’s Sketchpad so Sketchpad must reside on your computer for the files to run – see full explanation in “Requirements” on the home page.**

**Galleries below are titled:**

Arc Length

Area Between 2 Curves

Def. of a Derivative / Def. of Integrations / Inverse Functions

Graphers

Limits & Continuity

Maclaurin & Taylor Series

Optimization

Related Rates

Riemann Sums

Slope Fields & Euler’s Method

Theorems

Volumes on a Base

Volumes by Revolution

Animations to AP Calculus AB & BC Exams’ Released Free Response Questions 2018-1997 (Jump to FRs)

**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.

**DEFINITION OF A DERIVATIVE ~ DEFINITION OF INTEGRATION ~ INVERSE FUNCTIONS**

**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)

** LIMITS AND 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 AND 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.

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## 169 ANIMATIONS OF AP CALCULUS AB & BC EXAMS’

RELEASED FREE RESPONSE QUESTIONS 2018 – 1997

**Click below to link to AP Central for statements of the Free Response Questions from the exams for …**

AP CALCULUS AB 2017 & 2018 AP CALCULUS AB 2016 & older

AP CALCULUS BC 2017 & 2018 AP CALCULUS BC 2016 & older

*2018*

*2017 * animations to AP Calculus AB/BC Exams’ FRs

*2016 * animations to AP Calculus AB/BC Exams’ FRs

*2015 *animations to AP Calculus AB/BC Exams’ FRs

*2014 *animations to AP Calculus AB/BC Exams’ FRs

*2013 *animations to AP Calculus AB/BC Exams’ FRs

*2012 *animations to AP Calculus AB/BC Exams’ FRs

*2011 *animations to AP Calculus AB/BC Exams’ FRs

*2010 *animations to AP Calculus AB/BC Exams’ FRs

*2009 *animations to AP Calculus AB/BC Exams’ FRs

*2008 *animations to AP Calculus AB/BC Exams’ FRs

*2007 *animations to AP Calculus AB/BC Exams’ FRs

*2006 *animations to AP Calculus AB/BC Exams’ FRs

*2005 *animations to AP Calculus AB/BC Exams’ FRs

*2004 *animations to AP Calculus AB/BC Exams’ FRs

*2003 *animations to AP Calculus AB/BC Exams’ FRs

*2002 *animations to AP Calculus AB/BC Exams’ FRs

*2001 *animations to AP Calculus AB/BC Exams’ FRs

*2000 *animations to AP Calculus AB/BC Exams’ FRs

*1999 *animations to AP Calculus AB/BC Exams’ FRs

*1998 *animations to AP Calculus AB/BC Exams’ FRs

*1997 *animations to AP Calculus AB/BC Exams’ FRs