The bulleted links below relate to an Exam III from a few years ago.

	A quiz problem  will be to find the dimensions of the rectangle of maximum area that can be inscribed in a semicircle of radius 2.
	Click here to view the take-home problems on this exam and an alternate quiz problem.
	Click here for some figures and an animation for Exam III.

Hopefully Helpful Links

	Here is a Maple Worksheet supporting problems 4, 5, 7, and 10 on Exam III, Term 2067.
	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/
	Printable worksheets for graphical exercises can be found at mathgraphs.com.  Lots more can be found by going to www.tdlc.com.
	Hotmath  You can look at free solutions to problems in exercise sets from a wide variety of mathematics textbooks including Calculus by Larson, Hostetler, Edwards, sixth edition.  Many of these exercises are identical to those in the seventh edition.  They now have chapters 1 - 6 and 85% of chapters 7 -12 completed for the seventh edition.  Only a few solutions are still free.
	You can compute derivatives using Maple on your computer to check your answers.  Here are two sites that offer many 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.
	Here is another link to free online software powered by Mathematica that will compute derivatives and also show every step involved in using the differentiation formulas.
	For graphing you may find it helpful to make use of this graphing applet that will draw the graphs of a function and its first and second derivatives, each in a different color.
	UC Davis has a site with a lot of problems and worked out solutions.

3.1  For a given function be able to find the absolute maximum and minimum values over a closed interval.  Also be able to find relative extreme values (extrema) if any over an interval.  Here are some drill problems with solutions for finding max, min on a closed interval.

Lawn Sprinkler

Here is an example of the lawn sprinkler problem found in the exercises (#61) 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?  For more information on the "calculus of lawn sprinklers" see the article "Design of an Oscillating Sprinkler" by Bart Braden in Mathematics Magazine.  You can view the article at matharticles.com.

3.2  You will need to understand the meaning of Rolle's Theorem and the Mean Value Theorem (MVT) and understand the proof of the MVT.  Be able to apply Rolle's Theorem and the MVT in problems like the exercises in section 3.2:  11-24, 39-46.  Here is a tutorial on the MVT, Rolle's Thm, and the First Derivative Test and here is some drill on the MVT.  The site containing many of the tutorials and drills on this page also contains a TI-85 MVT illustration.  Here is a different tutorial on Rolle's Theorem and the MVT from Harvey Mudd College.

Here is some other nice stuff on Rolle's Theorem and the Mean Value Theorem from Karl's Calculus Tutor and here is a nice interactive Mean Value Theorem applet.  Here is another Mean Value Theorem applet.  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.  Quicktime animation for the cubic function Click here to see a PowerPoint presentation to prepare you for part of one of your Exam III questions.

3.3-3.5  For a given polynomial function be able to find all x-intercepts, the y-intercept, relative maximum and relative minimum points, inflection points, and state where the graph of the function is increasing and decreasing, concave up and concave down.  Here is some drill on the First Derivative Test and some drill on inflection points and intervals where the graph of a function is concave up, concave down.  Here is an applet relating to second derivatives and concavity.  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.

Karl's Calculus Tutor discusses relative max and relative min points on his page titled Hilltops and Valley Floors.  He discusses the significance of second derivatives in Squigglies.

For a rational function be able to find equations of all asymptotes (vertical, horizontal, slant, all types), relative maximum and relative minimum points, inflection points, and intercepts.  Also be able to find the equation of any polynomial function that the graph of the rational function is asymptotic to.  Be able to compute limits at infinity.  Check out Karl's help on graphing.  

You do not need to show the use of synthetic division in factoring a polynomial the way I did on the first classroom example.  You may use your computer and/or calculator.  If you are interested, here is a nice review of synthetic division from Purplemath.

Click here to see some worked out examples and some examples to practice on.

3.7  Optimization Problems:  Be able to do applied maximum and minimum problems similar to those done in class or assigned for homework.  Here is another link to the take-home problems for this exam.  Here is another link to some worked out examples of optimization problems with accompanying graphics.

Below are some online examples.

Example 1:  Find the maximum area of a rectangle inscribed in a semicircle.

Example 2:  (Tougher one)  Find the longest ladder which can be carried through a hallway with a corner.

Example 3:  Find the shortest time to move through the desert to a road and then along the road to a town.

Example 4:  Find the shortest ladder that can lean over a fence and against a building on the other side of the fence.

Example 5:  A volume presentation by Cynthia Lanius.

Here are some other useful max-min examples.

Here is another example.  This one involves maximizing the volume of a box with a square base and no top using a fixed amount of material.  I seem to have lost this one but it was located somewhere in the Mathematics Help Central Home.  There is lots of neat stuff here including graph paper you can print out.

Here is an example involving maximizing the cross section area of a gutter.  The English is not entirely clear but it does have a Java applet.

This is another example involving maximizing area.

3.8  Newton's Method:  Be able to write the iteration formula for approximating a zero of a given function using Newton's Method and show the result of each iteration leading to the final approximation (accurate to a prescribed number of significant digits).  Here is a tutorial on Newton's Method and drill problems on Newton's Method.  You can also implement Newton's Method using the Vanderbilt Toolkit or check out the following Newton's Method Demo.  Here is another nice demo of Newton's Method.  And if this is not enough, here is yet another demo of Newton's Method and also an example showing when Newton's Method runs into problems.

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.  Quicktime Version On my TI in class I stored the value 0.5 in x and then entered the command x+cos(x)/sin(x) > x   and repeatedly hit the enter key to generate x1, x2, . . . Click here to see a Maple generated picture showing five iterations of Newton's Method applied to find a zero of the function given below.

Here is one more Newton's Method Applet.  Here is another one--very nice.  My Powerpoint Presentation

3.9  Differentials:  You will not be tested directly on this section on this test but in preparation for the next unit you might benefit by looking at a graphic similar to the TI demonstration I did in class.  Here is an animation of the same thing looking at the graph of y = x2 starting out with a delta x value of 1/2 at x = 1/4 as in the picture at the right and animating as delta x approaches 0.  Click here or on the picture to see the animation.  Click here to see a pair of animations, one zooming, one not zooming.

Quicktime Version Quicktime Version with Zoom

Here are some links to SOS Math Calculus tutorials:  Mean Value Theorem,  Increasing-Decreasing Functions,  Local (relative) Max and Min,  Global (absolute) Extrema Over An Interval,  Concavity And Inflection Points,  Newton's Method.

Here is a Maple Worksheet with differentiation (and integration) examples.

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        Lane Vosbury, Mathematics, Seminole State College   email:  vosburyl@seminolestate.edu

        This page was last updated on 08/21/14          Copyright 2002          webstats