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MERLOT II


    

Personal Collection Detail View


Calculus 101

by MERLOT Adminstrative Team
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Description:
These are resources that can be used for a college calculus course. The subject in this first course is differential calculus. Beginning with functions and limits, this course includes techniques and applications of differentiation, indefinite and definite integrals and applications of integration. Topics include functions, limits, continuity, differentiation of algebraic and trigonometric functions, mean value theorem, and applications of derivatives. A suggested textbook would be "Calculus" by Howard Anton, Sixth Edition, Vol. 1, Wiley
Course:
Beginning Calculus
Prerequisites:
Math placement exam; completion of high school algebra and trigonometry
Learning Outcomes:
1 To master algebraic topics introduced in precalculus and trigonometry. 2. To understand limits and investigate some of their basic properties. 3. To understand the basic relationship between tangent lines, rates of change, and the derivative. 4. To master the techniques of differentiation. 5. To become familiar with the standard applications of the derivative in physics, engineering, biology, chemistry, and economics. 6. To become familiar with parts of the theoretical framework that are appropriate at this level. 7. To understand the integral and its relation to the derivative. 8. To master techniques of integration for simple integrals. 9. To learn how to use a graphing calculator.
Pedagogy Approach:
Various
Assessment Methodology:
Problem completion

Authorized users only:


1.

Numerical Integration Rules


Added 09/30/2009
Limited functionality, but very clear depiction of integration and the error involved with different integration methods. This applet provides a visual representation of various techniques of numerical integration including left rectangle, right rectangle, midpoint, Simpson's, and trapezoidal. The absolute value of the error is reported for each requested partition. This makes it easy to evaluate the effectiveness of changing the number of subintervals, and makes it possible to compare the accuracy of the various methods of integration.
2.

Numerical integration simulation


Added 09/30/2009
This applet provides a visual representation of various techniques of numerical integration including right rectangle, midpoint, and trapezoidal. The visual representation of the partitions and the corresponding sum of their areas approximates the area under the curve. The exact value is also given and compared to the calculated value. Numerical integration methods in this applet include right rectangle, midpoint and trapezoid. The exact value is also provided. The difference and percent difference between the Riemann sum and the exact value are given. A pie graph of the difference allows the user to visually determine if the value of the numerical method has improved with increased rectangles or change of method. Twelve preset functions are provided.
3.

Riemann Sums


Added 09/30/2009
Numerical integration methods include left rectangle, right rectangle, midpoint, Simpsons, and trapezoidal. The user has the option of plotting the graph of the numeric antiderivative as well as plotting the graph of a formula that is the user’s guess of the antiderivative. The graphics are very distinct, with colors effectively used to coordinate the graph with its function. The number of partitions are quickly changed by entering a number for the subintervals or doubling the previous number selected. The function and the interval can easily be changed and the values recalculated. The values of the various methods are reported simultaneously. This applet is very effective in comparing the results of the numerical integration methods, both visually and numerically. Area that occurs above the x-axis is yellow and area that is below the x-axis is green.
4.

Mathematical Visualization Toolkit


Added 09/30/2009
This site consists of a collection of plotting and solving applets featuring a uniform user interface. This site was selected as the 2005 MERLOT Classics Award winner for the Mathematics discipline due to its value and effectiveness as a set of teaching/learning tools. Visualizing mathematical concepts, especially in three-dimensional space, can be quite difficult for students. These tools and applications enable students to see the concepts in action and to come a deeper understanding of the underlying mathematics. In addition, the collaboration between the faculty, students and Sun Microsystems staff who together designed and constructed these tools was quite unusual and impressive. The collaboration itself is an inspiring model.
5.

Numerical Integration Applet


Added 09/30/2009
This applet provides a visual representation of various techniques of numerical integration including left rectangle, right rectangle, midpoint, Simpson's, and trapezoidal. The graphics are very distinct, and the partitions are quickly changed by entering a number for the subintervals or by changing the integration method. The function and the interval can easily be changed and the values recalculated. The values of the various methods are reported simultaneously, and the one whose partitioning is shown is given in red font. This applet is very effective in comparing the results of the numerical integration methods, both visually and numerically.
6.

Gallery of Volume Visualizations


Added 09/30/2009
This is an extensive gallery of colorful visualizations for disk, washer, shell and cross-section methods of finding volumes of solids of revolution. It is useful for viewing the incremental approach in constructing three-dimensional solids.
7.

Partial Derivative Geometrically Gallery


Added 09/30/2009
This is a great collection of animations of partial derivatives and the slope of the tangent lines. A series of Web-based animations are used to illustrate the surface of each function, the path of the indicated partial derivative for a specified value of the variable and the value of the derivative at each point along the path. Students find it difficult if not impossible to visualize most three-dimensional surfaces without assistance; the Web-based animation gallery provides an excellent source of visual illustrations that allow students to connect their abstract mathematical computations with geometric representations
8.

Volumes


Added 09/30/2009
This is a collection of tutorials (Flash and Java) on various applications of integration including area between two curves, volumes of solids of revolution, arc length, average value of a function, work, and moments and centers of mass. This instruction comes in many forms: text, graphics, animations, visualizations, practice exercises with solutions, etc. In addition to Macromedia Flash and Java, the tutorials employ a number of computer programs including: LiveMath (MathView), Microcalc, JKGraph and MPP3D (a plotting program). Most of these programs, both stand-alone and plug-in types, are free downloads. The site provides numerous examples with detailed instructions for using the TI-85 and -86 graphing calculators. One of the topics (Area between two curves) also includes practice exercises with solutions.
9.

Visual Calculus: Applications of Integration


Added 09/30/2009
This site is a collection of tutorials (Flash and Java) on various applications of integration including area between two curves, volumes, arc length, work and centers of mass. The applets are extremely well-designed and implemented and provide step-by-step solutions for various application exercises. The use of color in the tutorials and the animations and visualizations are excellent and would certainly help students come to an understanding of the mathematical concepts. Some of the computer programs used to generate visualizations are available as downloads from the UTK Math Archives site; while these programs are a bit dated, they may still be of value to an interested user.
10.

Visual Calculus: Applications of Integration-Work


Added 09/30/2009
This site is a collection of tutorials (Flash and Java) on the concept of Work including Hooke's Law, pumping liquids and lifting a weight. The purpose of this site is to enable students to 1) develop their understanding of the concept of work as a product of force and distance; 2)exercise their mathematical intuition and verify it via appropriate calculations, 3) recognize and correct a common misconception concerning work, and 4) recognize and correct a common calculation error involving work.
11.

Animations (for freshman and sophomore calculus and differential equations)


Added 09/30/2009
This is a collection of over three dozen animations of standard calculus and differential equations topics that are enhanced by visualization; the main sections include limits, derivatives, Riemann sums, multivariable Calculus, numerical methods and Statistics. All files are uncompressed avi files; for most animations, MathCAD code is available. The site will enable students to gain a visual as well as conceptual understanding of selected mathematical concepts. As indicated by the author in his introduction, these animations provide a most effective means by which to introduce and illustrate dynamic mathematical concepts. The animation topics are well-chosen and cover a variety of subjects.
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