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CALCULUS I EXAM II NOTES AND LINKS |
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There is online support for the Larson et al Calculus textbook.
They have free online support material for Chapters P, 1,
2, 3 at http://hmco.tdlc.com/public/calc7esample/
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Printable worksheets for graphical exercises can be found at mathgraphs.com.
Lots more can be found by going to
www.tdlc.com. |
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For a nice set of review modules go to
Visual
Calculus: Derivatives. |
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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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You can compute derivatives using Derive
or Maple on your computer to check your
answers. You can also use the derivatives feature on the Vanderbilt
Toolkit. Here are two
other sites that offer many of the same tools including tools to compute
derivatives: Dr. Huang's site
and QuickMath. 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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Harvey Mudd College has some online
tutorials relating to the limit
definition of the derivative, product
rule, quotient
rule, tangent
line problem, and the chain
rule that might be helpful. |
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Here are some online
solutions to typical calculus problems. |
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This
animation is intended as a demonstration of the definition of
derivative applied to the function f(x) = 4 - x2 at the point
(-1,3). The green triangle represents taking the limit from the
right and the length of the green tangent line segment corresponds to the
changing value of the difference quotient. The red triangle
represents taking the limit from the left and the length of the red
tangent line segment corresponds to the changing value of the difference
quotient. Both the red and green tangent line segments are approaching a limiting length of 2.
Quicktime animation |
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Click here to see
some sample problems. |
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Related Rate Examples |
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Here is a
Maple Worksheet with
differentiation (and integration) examples. |
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Click on the picture on
the left to see an animation.
The green line is the tangent to the graph
of
y = 4 - x2 at the point (1,3). |
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| 2.1 You need to be able to
approximate the slope of a line tangent to the graph of a given function
at a given point by calculating the slope of the line through two points
on the graph close to the given point. For example, given the
points (1.23, 6.41), (1.24, 6.48), (1.25, 6.55), (1.26, 6.64), (1.27, 6.75) on
the graph of a function f, approximate the slope of the line tangent to the
graph of f at (1.25, 6.55). These data points do not
relate to the graph at the right. In the graph at the right the
function is graphed in black, the tangent line at point B is given in red,
the line through points A and C is in blue, the line through points A and
B is in magenta (purple), and the line through points B and C is in
green. Which line (blue, purple, or green) appears to have a slope
which most closely approximates the slope of the red tangent line? |
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| 2.1 (cont) Click this Animation
to see secant lines approaching the tangent line. Be able to apply the
definition of derivative of a function to derive a function's derivative by
computing the appropriate limit. Be able to approximate the derivative of a function at a point given
only data points on the graph of the function. Be able to show how to approximate the
derivative of f at c by computing
Be able to find the
equation of the line tangent to the graph of a given function at a given point.
You should know what
happens in the relationship between a line tangent to a curve at a point and
the curve itself when you zoom in on the point. What is the visual nature
of the graph of any function differentiable at a point when you zoom in on the
point? Here is a nice java
applet graphically demonstrating the definition of the derivative and its
relation to the tangent to a graph. SOS
Mathematics has some very nice support material for
this chapter. Here
is a link to their material on the definition of the derivative as well as
average and instantaneous velocity and the geometrical concept of the derivative
(with another tangent line animation). Know that differentiability
implies continuity. Here
is another applet relating to the definition of the derivative and its use
in finding the slope of a line tangent to the graph of a function at a point on
the graph.
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Click on the picture
below to see an animation of the tangent line moving across the graph.
Click here to see an animation with both the tangent line and the normal line moving across
the graph. QT Version
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2.1 (cont) Here is another applet demonstrating
the secant
line approaching the tangent line and yet another
java applet for the same thing. Here is one more applet demonstrating
approximating the slope of a tangent line by using a secant
"chord": Group
of Applets links to a group of applets by MathinSite. Click on Differentiation
1. Here is a nice animation showing the graph of a
function and depicting the ever
changing tangent to the graph at a point running along the graph (similar to
my animation above) and another
demonstration of the secant
line approaching the tangent line.
2.2 Be able to compute average velocity and
instantaneous velocity given a position function (pp113-14: examples 9 and 10).
Cynthia Lanius has a helpful site titled Slope
as Rate of Change. Here are more links to discussions of average
and instantaneous velocity and average and
instantaneous rate of change in general. Here is a terrific presentation
on speed from Calculus Quest. Here is an applet
on instantaneous speed that nicely relates it to the derivative of the
position function.
2.2 - 2.4 Be able to apply the differentiation
formulas in these sections to compute derivatives. Look at the
following: Drill work with worked out solutions for the Product
Rule. Drill work with worked out solutions for the Product
Rule with emphasis on trigonometric functions. Drill work with worked
out solutions for the Quotient
Rule. Drill work with worked out solutions for the Quotient
Rule with emphasis on trigonometric functions. Drill work with worked
out solutions for the Chain
Rule. Langara College has some nice material
on derivatives including links to other helpful sites. SOS has helpful
material on applying
differentiation formulas, derivatives
of trig functions, and the
chain rule.
Here is a link to free
online software powered by Mathematica that will not only compute
derivatives but will also show every step involved in using the differentiation
formulas discussed in sections 2.2 to 2.4.
2.5 Be able to find dy/dx implicitly for a
relation between x and y defined implicitly, and be able to find the slope of
the tangent line and the slope of the normal line at a particular point on the
graph of the relation. Look at the
following: Drill work with worked out examples for finding dy/dx
implicitly. Drill work with worked out examples for finding dy/dx
implicitly and the equation of the tangent line at a point. Here is
the SOS material on implicit
differentiation.
2.6 Related Rates--Study the homework problems
and the problems done in class. Some additional examples can be found at applications
of the derivative in Sparknotes. Use the blue drop-down toolbar at the
left to find examples of related rate problems with solutions and suggestions on
how to solve such problems. You will get a lot of "garbage"
advertising at this site. More help can be found in the chain
rule applications for the UBC Math100 course. Here is a nice
related rate applet (from Robert Lozicki courtesy of the US Naval Academy)
relating the changing
distance between two ships traveling along paths that are perpendicular to
each other. Here is a tutorial
on related rates with examples. Elijah Loeffel found this expanding
sphere with displayed changing volume and radius demonstration found by clicking click
here after
following the preceding link. He also found this
related rate material on the sphere. Here is my own expanding
sphere animation and another version.
You will need to download
the free DPGraph Viewer to view it.
Here is a link to the free Hotmath solution to
problem
number 25 on page 155.
Related Rate Examples
Pictures for problems 3, 9, 12,
13, 14, and 15 on Exam II. (These pictures relate to an old exam.)
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