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 Subject: My favorite calculus tools Author: gene Date: Aug 5 2004
Here are my favorite (as of the moment) calculus tools. At least I liked them
enough after searching around to use in my calculus course last year. Some are
not yet in Math Tools--but they will be soon! (In most areas Math Tools has
now added others covering similar material--I'll have to check which I like
best.)

I've left in the instructions I gave students for using them.

LIMITS
Experiment with the definition of limit as follows:
Go to  http://mathforum.org/mathtools/tool/2129/ , the Epsilon-Delta Applet.
Scroll down and push the "Launch EpsilonDelta" button. For the given value of
epsilon (which corresponds to half the height of the yellow box) move the delta
slider below the graph to obtain a delta (corresponding to half the width of the
box) for which 0<|x-a|

At the top of the applet window is a pull-down window. Pull down to Examples
3-5 and see what you can learn from them. Your only hand-in part of this
applet exercise is to articulate carefully what your experiments with Examples
3-5 actually show.
---

If you're having a hard time with the limit notion and think you might benefit
from a step-by-step approach, go to
http://mathforum.org/mathtools/tool.html?co=c&tp=15.1&id=1146&new_id=113
and click on the image. This takes you step-by-step through a discussion of
what a limit is, replete with examples—and you can go back step-by-step as
well.
---

TANGENT LINE AND DERIVATIVE

SECANTS BECOMING TANGENTS
Here's an applet that can give you a visceral experience at moving secant lines
into tangent lines: http://mathforum.org/mathtools/support/2148/ (I like the
version at
http://www.calvin.edu/~rpruim/courses/m161/F01/java/SecantTangent.shtml )
Experiment with all the items in the pull-down menu, and hand in answers the
questions posed by the titles.
---

VISCERAL FEEL FOR TANGENTS
In class we used the Surfing Applet:
http://mathforum.org/mathtools/tool/1002/
---

DERIVATIVE AND TANGENT LINE
http://mathforum.org/mathtools/tool/2319/
It's like the surfing applet with numbers. You can use the pull-down functions
or you can enter functions by typing. Experiment as you wish on the pull-down
functions, but be sure to try on x^3, x^4, and x^5. Of all functions with their
general shapes, what's special about the derivatives of these? Write up what are
the key visual properties of the derivatives of each and draw another candidate
for the derivative of each.
---
DERIVATIVE PUZZLE
Play with the Derivative Puzzle applets
http://mathforum.org/mathtools/tool/2197/
until you're confident about getting the shape of the derivative from the shape
of the function.
---

FUNCTION COMPOSITION AND CHAIN RULE
The function composition applet I showed in class is
http://mathforum.org/mathtools/tool/2149/
Notice that this uses g(f(x)) while the book more frequently talks about
f(g(x)). In any event, if you move the red dot on the first graph you see there
the point (X,f(X)) say. The middle graph obligingly shows the point
(f(X),g(f(X))), (got that?) while the final panel is (X,g(f(x))). If you wish
you can chose functions g and f for which you feel comfy with the composition
and put these in.

A little below the above applet you come to Launch Function Composition. Do so
and you get the above figure with tangent lines. Miraculously (for those
skeptical of the Chain Rule) the product of the slopes of the functions in the
first two graphs is the slope of the function in the third graph.

There’s a pull-down menu at the top with some examples worth mulling about
(the first couple have absolute values which are quite illustrative in this
case). Note that you have to click Launch Applet after making the pull-down
menu choice. And you can always put in functions you're curious about.
---

TRIG FUNCTIONS
The applet I used to graph the trig functions is to be found on
http://mathforum.org/mathtools/tool/12217/ and called "graphs of sin, etc.". The
surrounding applets are very good practice in recognizing the graphs of
functions, with trig fun following the graphs applet.
---
INVERSE TRIG FUNCTIONS AND DERIVATIVES
For help with inverse trig functions and their derivatives, here is some nice
text with embedded applets:
(Note: if you follow the link at the top of the page to "The Zebra Danio and its
escape response" you come to a nice application of inverse trig functions to
biology.)
(Not yet in Math Tools.)
----

DIFFERENTIATION PRACTICE
Here are a couple sites that give you practice in basic differentiation along
with instant feedback:
http://www.csvrgs.k12.va.us/math/ Choose any of the Surfin' topics.
http://www.math.sjsu.edu/~valdes/calcreview/derivativequiz1/derivativequiz1.html
/ This has the basic formulas and examples available on the upper right.
(Not yet in Math Tools.)
---

FAMILIES OF CURVES
For this section a very good applet is "Slider Graph,"
http://mathforum.org/mathtools/tool.html?co=pc&new_id=600
Use this applet to write a paragraph about the family of curves a*sin(b*x+c),
which is how you'd enter A sin (Bx+C). What are the effects of the size of A, B,
and C? Of the sign of A, B, and C? (Warning: the applet starts out with
A=B=C=0).

Note that to graph the function on the top of p. 177 you'd enter
"exp((-(x-a)^2)/b)." Write a paragraph about the effect of the size and the
sign of a and b.
---

OPTIMIZATION PROBLEMS
Use the Ladder http://mathforum.org/mathtools/tool.html?new_id=1402 (note that
you must use the sliders to change lengths) .
---
Here are applets relating to some of these optimization problems which might be
insightful:

The fastest route through the desert and along the road to the town
http://mathforum.org/mathtools/tool.html?&new_id=1403

The Shortest Ladder over a Fence
http://mathforum.org/mathtools/tool.html?&new_id=1404

Area of Largest Rectangle Inscribed in a Semicircle
http://mathforum.org/mathtools/tool.html?&new_id=1401
---
For these use the Integrator applet I showed in class,
http://mathforum.org/mathtools/tool.html?id=1065&context=tool&new_id=1061.
---

FUNDAMENTAL THEOREM OF CALCULUS
http://archives.math.utk.edu/visual.calculus/4/ftc.2/
http://www.dean.usma.edu/math/research/mathtech/java/FTCProject/ftcprojectpage.html
(Not yet in Math Tools.)
---

CHANGE OF VARIABLE
Look at http://math.furman.edu/~dcs/java/change.html and figure out how the
integrals of the two functions can be equal.
(Not yet in Math Tools)
---

RIEMANN SUMS
1. For the function f(x) = 4/(1+x^2) approximate the integral of f between 0 and
1 by hand using
LEFT(4), RIGHT(4), MID(4), and TRAP(4).

2. Using any means you choose (including the new applet at
http://www.math.ucla.edu/~ronmiech/Java_Applets/Riemann/index.html ) calculate
these approximations for n=6 and n=10.
(Not yet in Math Tools)

3. In like manner, make these approximations for the integral of 1/x from 1 to 2
for n=6 and n=10 and compare with ln 2.

4. In 3 for n = 10 how do the various approximations compare in terms of
accuracy?
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