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CALCULUS II EXAM I NOTES AND LINKS |
| If it is not already on your hard drive, you will need to download
the free DPGraph Viewer to view some of
the pictures linked to on this web site. |
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QuickTime free download. |
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THE
INTEGRATOR is a free function integrator. You need to read
their instructions on entering input to use it effectively. It is from
Wolfram Research and powered by Mathematica. |
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Quickmath This is a free
online source for solving math problems powered by Mathematica. |
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You can also compute
integrals using Derive on your computer in class to check your answers
or you can use the antiderivative feature on the Vanderbilt
Toolkit. The Vanderbilt Toolkit includes some specialized utilities
that relate directly to chapter 6. Regrettably, this free online
software has not been available for quite a while now. |
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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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Traces
Animes A really nice set of 2D and 3D WIMS graphing tools by XIAO Gang
which will produce pictures and graphs that can then be saved and printed or
used on a web site. Traces Animes
sample animation |
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Printable worksheets for graphical exercises can be found at mathgraphs.com. |
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Visual Calculus
has lots of great stuff. |
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Textbook support site |
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Here is a
Maple
Worksheet with integration (and differentiation) examples. |
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Here is a
Maple
Worksheet on differentiating and integrating logarithmic and exponential
functions. |
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Here is a
preview
of the area applications in Calculus II including
area
between two curves. |
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Extra
Credit |
5.6 You will need to be able to define and
differentiate the inverse trigonometric functions. Here
is a presentation of the inverse tangent, inverse sine, and inverse cosine
from the UBC online calculus course.
  
y = arcsin(x)
y = arctan(x)
y = arcsec(x)
5.7 You will need to be able to use the
inverse trigonometric functions in integration. You will need to complete
the square in computing some of the integrals in this section.
5.8 You will need to be able to define the
hyperbolic functions and their inverses and use them in differentiation and
integration.
  
y = arcsinh(x)
y = arccosh(x)
y = arctanh(x)
6.1-6.3 We will review solving a few differential
equations and their applications.
Extra Credit
| Hopefully in Calculus I you learned 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 blue 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. |
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| Try using the method of separation
of variables to solve the following: dy/dx = y(cos(x)),
y(0) = 1 The
solution can be found in the examples.
The graph of the solution is shown on top at 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. The lower graph on the right shows the
solutions for y(0) = -1, -0.4, 0.4, and 1 along with the direction
field. Click on the graph to see an enlargement.
In addition to the graphs shown at the
right you can look at various solutions corresponding to an initial
condition of the form y(0) = c by following this link to a DPGraph
of a surface and the plane y = c. The curve of intersection of
the surface and the plane when c = 1 is the graph of the solution to the
initial value problem above. You can use the scrollbar and activate
c to look at solutions for various values of c. You can also use the
z-slice feature.
Here is another java applet DE
solution grapher that also draws the direction field.
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Here are more
examples of solutions to first order
separable differential equations from my Differential Equations course web
site.
7.1 You will need to be able to find the
area of a region between two curves. Here is the SOS Math introduction
to the area
between two curves and an applet demonstrating area
between two curves. Here are some drill
problems on area between two curves. This is an animation
reviewing approximating the area under part of a sine wave.
7.2 You will need to be able to compute
volumes using the disk method. You should also be able to compute the
volume of a solid whose cross section area function is known (See Page 461 and
these two Quicktime movies by Bruce Simmons: The
Concept, An
Example). Another example of this would be a solid whose bottom is bounded by a
circle and whose cross sections are squares. Here is a java
applet illustrating such a problem.
Here is the surface formed by revolving the
graph of y = (x3 - 2x2 - 3x + 10)/10 about the x-axis over
the interval [-2,2].
This DPGraph picture is of the Surface
of Revolution formed by revolving the graph of y = 1 +
(x+4)(x-4)((x-1)/20 about the x-axis over the interval [-3,3].
This DPGraph picture is of the Surface of Revolution 2
formed by revolving the graph of y = 3 + (2 + 8sin(x))/(1.2x + 1) about
the x-axis over the interval [0,18].
7.3 You will need to be able to compute
volumes using the shell method.
Here
is the HMC tutorial
on finding volumes and the UBC
tutorial on finding volumes.
7.4 You will need to be able to compute arc length and
the area of a surface of
revolution. Arc
Length1 This utility will compute the arc length for a function f(x)
over an interval [a,b]. Arc
Length2 This is a nice applet demonstrating the approximation of arc
length using line segments. Solid
of Revolution 1 This utility will draw the surface formed by rotating
the graph of y=f(x) over [a,b] about the x-axis or y-axis. Solid
of Revolution 2 This utility will draw the solid formed by rotating
the intersection of the graphs of y=f(x) and y=g(x) over [a,b] about the x-axis
or y-axis. Another nice arc length applet is found in the HMC
arclength tutorial. Below are two pictures demonstrating the
approximation of arc length along the graph of the parabola y = x2
from x = -2 to x = 2. The figure on the left uses 4 line segments and the
figure on the right uses 8.
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7.5 You will need to be able to compute the work
done when the force being applied is variable. Examples of this would be
the work done in compressing or stretching a spring (using Hooke's Law),
emptying the liquid out of a tank, and lifting a chain. Look at this animation
(see picture on the left) and see if you can figure out how much work is done against gravity in
lifting the chain and the weight attached to the end of it up to the
ceiling. The chain weighs one pound per foot. After the "oops" the weight comes loose and falls
back to the floor. How fast is the weight traveling (neglecting air
resistance) when it hits the floor? Here is the UBC
tutorial on work.
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Spring Stretching Example
A force of 100 pounds will stretch a spring 2
feet beyond its equilibrium length of 5 feet. Find the work done in
stretching the spring from a length of 5 feet to a length of 8 feet.
Click here to see an animation
and click here to see an
animation with scales.
100 = 2k so the spring constant is 50. The work
done would be
How much additional work would be done in
stretching the spring two more feet (assuming we are still within the
elastic limits of the spring and Hooke's Law holds)?
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7.6 You will need to be able to find the moments
and center of mass of a planar lamina with uniform density. You will need to be able to find the centroid of a
plane region. Center
of Mass This utility will compute the center of mass of a plate with
uniform density whose boundaries are y=f(x) (on top), y=g(x) (on the bottom),
x=a, and x=b.
7.7 You will need to be able to find the total
fluid force on one side of a vertical submerged surface. We will also look
at how to find the total fluid force on one side of a submerged flat surface
that is not vertical.
| Extra
Credit Here is a
picture of
the vase and the data for the volume and surface area extra credit problem
presented in class.
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