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CALCULUS I EXAM V NOTES AND LINKS |
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The
Natural Logarithmic Base |
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Applet for estimating
the number e from y = (1 + 1/x)x, e being the limit as x approaches
infinity. |
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Applet to better understand the derivative
of exp(x) and ln(x). |
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Here is some help in introducing the calculus approach
to the natural logarithmic function from the University of Oregon's Calculus
Quest. |
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Here is a very nice java
applet for computing derivatives in which you are shown each step in
applying differentiation formulas such as the product rule, quotient rule,
and chain rule. |
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Hotmath
You can look at solutions to problems in exercise sets
from a wide variety of mathematics textbooks including Calculus
by Larson, Hostetler, Edwards, sixth , seventh, and eighth editions. They have chapters 1 -
12 available. Only a few solutions are
still free (solutions to problems 15, 25, 35 in each section are free). |
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Printable worksheets for
graphical exercises can be found at mathgraphs.com. |
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UC Davis has a site with a lot of problems
and worked out solutions. |
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Here is a Maple Worksheet with
integration (and differentiation) examples.
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Examples |
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Here is another
Maple Worksheet on differentiating and
integrating logarithmic and exponential functions. |
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Extra Credit |
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5.1 You need to understand the definition of the
natural logarithmic function as given on page 322 of your text and know how to
approximate a natural logarithm using this definition. You need to be able
to differentiate composite functions involving the natural logarithmic function.
Here is Karl's derivation of the derivative
of the logarithmic function.
5.2 You need to be able to use the natural
logarithmic function in integration.
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f(x) = ln(x) |
5.3 You need to be able to tell whether a
function is one to one or not and be able to find its inverse if it is one to
one.
| 5.4 You need to understand the
definition of the natural exponential function as given on page 350 of your text. You need
to be able to differentiate and integrate functions involving the natural
exponential function including composite functions. Here is Karl's derivation of the
derivative
of the exponential function. Click here
for an animation comparing the graph of un(x) = (1 + x/n)n
as n goes from 1 to 100 with the graph of f(x) = ex. What is
the limit of (1 + x/n)n as n approaches infinity? |
f -1(x) = exp(x) |
5.5 You need to be able to differentiate and
integrate exponential and logarithmic functions involving bases other than e
including composite functions. Many worked out examples can be found at
the site ecalculus.
This site includes worked out detailed solutions to problems (by D. A. Kouba), and
animations (by Deej Heath) for calculus.
6.2 You need to be able to solve the differential
equations governing simple growth and decay including population growth,
radioactive decay, and Newton's Law of Cooling and use your solutions in
application problems. Here are some links for exponential
population growth and logistic population growth, radioactive
decay, and Newton's
Law of Cooling.
| 6.3 You need to be able to solve
some ordinary differential equations (ODES) by the method of separation of
variables. To have a better sense of what the solution to a first order
ODE is take a look at this First
Order DE Solution Grapher. The picture on the right shows a blow-up of a
portion of what the DE Solution Grapher will produce, in this case showing part
of the graph of the solution to dy/dx = 2x, y(0) = 0. The line segments
indicate the direction of tangents to the graph of any solution to dy/dx = 2x
that went through the left endpoint of the line segment. The Vanderbilt
DE Toolkit can solve a variety of differential equations. |
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Try using the method of separation
of variables to solve the following:
dy/dx = y(cos(x)),
y(0) = 1 Solution
The graph of the solution is shown on the
right. Click on the graph to see an animation of the direction field
vectors moving across the screen for increasing values of x along with an
animated solution point.
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Here is a nice
set of examples including applications involving separable differential
equations. The examples include an excellent one
on population growth. Here are more examples including Newton's
Law of Cooling and mixture examples from the same source, Joseph Mahaffy
from San Diego State University. The link will take you to his
Differential Equations site. When you get there you will need to click on Lectures
and then on Linear Differential Equations.
There are lots of other good DE things at this site too.
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