Math 195 - Chapter 4

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Section 4.1 – Maximum and Minimum Values
- Read pp. 277 – 283
- online resources:
  1. Extreme Value Theorem - http://www.orst.edu/instruct/mth251/cq/Stage4/Lesson/EVT.html
    This site discusses the extreme value theorem and illustrates it with three examples.
Section 4.2 – The Mean Value Theorem
- Note: this section is not part of the syllabus. Here are some good references for Rolle's Theorem and the Mean Value Theorem (MVT)
- online resources:
  1. Mean Value Theorem - http://www.frederickps.k12.ok.us/fhs/nicks_assign/mean-theorem.htm
    This site provides a link to 10 different sites that describe these theorems.
Section 4.3 – How Derivatives Affect the Shape of a Graph
- Read pp. 294 – 302
- online resources:
  1. What does f ^'(x) say about f(x)? - http://www.dean.usma.edu/math/research/mathtech/java/FandFPrimeProject/fandfprimepage.htm
    This site provides a java applet that illustrates the relationship between a function and its derivative.
  2. A First and Second Derivatives Applet - http://www.slu.edu/classes/maymk/Applets/Derivatives2.html
    This site provides a java-based applet that ties together the graph of the function with the graphs of the first and second derivatives.
Section 4.4 – Indeterminate Forms and L'Hospital's Rule
- Note: this section is not part of the syllabus. Here is a good reference for L'Hospital's Rule
- online resources:
  1. L'Hopital's Rule - http://www.ies.co.jp/math/java/calc/lopi/lopi.html
    This site provides a java applet that demonstrates l'Hopital's rule.
Section 4.5 – Summary of Curve Sketching
- Read pp. 314 – 321
- online resources:
  1. Graphing of Functions Using First and Second Derivatives - http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/graphingdirectory/Graphing.html
    This site provides problems that illustrate detailed graphing of functions of one variable using the first and second derivatives.
  2. Curve Sketching - http://hverrill.net/pages~helena/math006/week7.html
    This site provides problems that illustrate the use of asymtopes, concavity and second derivative test to sketch a graph of a function.
Section 4.7 – Optimization Problems
- Read pp. 329 – 334
- online resources:
  1. Max-Min Problems - http://www.ma.utexas.edu/users/kawasaki/mathPages.dir/calculus.dir/max-min.dir/max-min.html
    This site provides 5 max-min problems, their solutions and applets that demonstrate the solution.
  2. Area of a Rectangle Inscribed in a Semicircle - http://archives.math.utk.edu/visual.calculus/3/applications.1/index.html
    This site provides a java applet illustrating the solution to finding the maximum area of a rectangle which can be inscribed in a semicircle.
  3. The Longest Ladder - http://archives.math.utk.edu/visual.calculus/3/applications.2/index.html
    This site provides a java applet illustrating the solution to finding the longest ladder which can be carried through a hallway with a corner.
  4. The Fastest Route - http://archives.math.utk.edu/visual.calculus/3/applications.3/index.html
    This site provides a java applet illustrating the solution to finding the shortest time to move through the desert to a town.
  5. Shortest Ladder over a Fence - http://archives.math.utk.edu/visual.calculus/3/applications.4/index.html
    This site provides a java applet illustrating the solution to finding the shortest ladder that can lean over a fence against a building.
Section 4.9 – Newton's Method
- Note: this section is not part of the syllabus. Here are some good references for Newton's Method
- online resources:
  1. Newton's Method I - http://math.furman.edu/~dcs/java/newton.html
    This site provides a java applet that illustrates using Newton's method to approximate solutions to the equation.
  2. Newton's Method II - http://www.math.umn.edu/~garrett/qy/Newton.html
    This site provides a java applet that illustrates using Newton's method to approximate solutions to the equation.
  3. Newton's Method III - http://www.dartmouth.edu/~rewn/newton.html
    This site provides a java applet that illustrates using Newton's method to approximate solutions to a user defined equation.
  4. Pathological Example for Newton's Method - http://www.math.umn.edu/~garrett/qy/BadNewton.html
    This site provides a java applet that illustrates a pathological possibility using Newton's method to approximate solutions to the equation.
  5. Newton's Basins - http://aleph0.clarku.edu/~djoyce/newton/newton.html
    This site provides a detailed discussion of the fractals created when trying using Newton's method.
  6. Newton's Method and Fractals - http://www.math.hawaii.edu/206L/newton.html
    This site provides a java applet that shows five iterations of using Newton's method to find the cube root of 1.
  7. Newton's Fractal - http://www.geocities.com/CapeCanaveral/Hangar/7959/newtonapplet.html
    This site provides a java applet that shows the fractal created when trying using Newton's method to find the cube root of 1.
Section 4.10 – Antiderivatives
- Read pp. 351 – 356
- online resources:
  1. Antidifferentiation - http://mss.math.vanderbilt.edu/cgi-bin/MSSAgent/~pscrooke/MSS/antiderivative.def
    This site provides a tool that computes the antiderivative of the user defined function.
  2. Direction Fields and Euler's Method - http://www.math.ualberta.ca/~ewoolgar/java/Euler/Euler.html
    This site provides a java applet that displays the direction field for the differential equation dy/dx = ax^m + bxⁿ.
  3. The First-Order Differential Equations - http://www.ies.co.jp/math/java/calc/DiffEqu/DiffEqu.html
    This site provides a tjava applet that displays the direction field for the differential equation dy/dx = f(x, y).

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