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MISCELLANEOUS CALCULUS ANIMATIONS
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On the left is a picture of Newton's
Method being applied to the function f(x) = cos(x) to approximate one of
its zeroes. The initial guess was xo = 0.5. Click
on the picture to see an animation. |
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Click here to
see an animation of the Mean Value Theorem being applied to a quadratic
function and click here to see it
applied to a cubic function. Click on the image on the left to see
an animation of that image. |
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Click on the picture on the right to see an
animation relating the sign
of the second derivative to the changing value of the first
derivative. Click here to
see a graph drawn in blue where it is concave up and red where it is
concave down along with animated "+" and "-" signs
indicating the sign of the second derivative. |
Click on the picture for an animation.
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Click on the picture for an animation.
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Area under a curve using rectangles
and a left approximation. Click on the picture on the left to see an
animation. |
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Lawn Sprinkler
Here is an example of the lawn sprinkler problem
found in the exercises for Section 3.1. In this example the speed of
the water is 16 ft/sec so the distance the water travels horizontally is
given by
and the path the water takes through the air is
given by
Click here
to see an animation for this problem and click
here for an animation with scales. Can you see the answer to the
questions posed in the text and can you support your answer analytically?
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| Imagine a teeter totter (red) that is
4 meters long and sqrt(2) meters above the ground at its center (where it
is supported--green support). Initially the teeter totter is at a
angle of 45o below the horizontal (in blue). The motion
of the teeter totter is such that on its first rotation upward it goes
through an angle of (17/18)(90) = 85o. On its subsequent
rotation downward it passes through an angle of (17/18)(17/18)(90)
degrees. On its next rotation upward it passes through an angle of
(17/18)(17/18)(17/18)(90) degrees. This pattern continues
indefinitely. What is the limit to the distance traveled by a point
on the extreme tip of the teeter totter? Click here to see an
animation. The animation goes through a few rotations of the teeter
totter. It is not designed to be completely accurate but should give
you a rough idea of what is happening. |
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The plane would be
descending at the most rapid rate at the point on the path where f '(x)
(which is negative) is a minimum, i.e., where the absolute value of f '(x)
is a maximum. This would occur at the point where the derivative of
f '(x) is zero, i.e., at the point where f "(x) = 0 which is the
inflection point (-2,1/2). The article "How Not to Land at Lake
Tahoe!" can be read by following the Matharticles link below and
looking under chapter three.
Here are some animations and a link to learn more:
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The
graph below shows position (not distance traveled) as a function of
time. Click here or on the
picture to see a linear motion animation.
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Ln(x)
animation In the animation y = ln(x) (in
purple) and y = 1/x (in
blue) are graphed in the same coordinate
system. Coordinated blue animated points are moving along the graph of
each. There is a vertical blue line segment extending from the point (1,0)
to the point (1,1). There is an animated vertical line segment coordinated
with the two animated points (i.e., same value for x). The area between
the x-axis, the vertical blue line segment, the graph of y = 1/x, and the
animated vertical line segment represents the absolute value of ln(x) where x is
the x value along the animated line segment. The value of ln(x) will be
negative where the animated line segment is red
and positive where the animated line segment is green.
I will go over this in class. Quicktime
animation
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JUST FOR FUN
| DANCING
CIRCLES
DANCE
LINE OF CIRCLES
Dancing Circles is
an animation consisting of eight small circles traveling around the four
limacons pictured below, two circles traveling on each limacon.
The centers of the circles follow the paths given parametrically at the
right. The centers always remain in a straight line. Dance
Line of Circles is the same animation with
the addition of the rotating straight line through the centers of the
circles and also a small black circle at the origin that does not move
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