Forum: A Tale of Two Integrals

We have been working with Auntie Derivative and her Family of Functions, also known as the indefinite integral. We have also worked with the definite integral. Which image below illustrates which type of integral?

Did you pick the correct one? The indefinite integral is a family of functions with a given derivative and the definite integral is the area under a curve. Incredibly, they are related to each other, as we will soon discover!

What method have we used to approximate the definite integral (starts with a "t")? This method uses a strategy developed by Archimedes, considered by many to have been the greatest applied mathematician of antiquity. His method for finding areas under curves laid the groundwork for the invention of calculus by Newton and Leibniz two thousand years later. Read this recent New York Times article by Steven Strogatz,  a professor of applied mathematics at Cornell University. Professor Strogatz discusses Archimedes method of exhaustion and many of the ideas we have covered this year: Zeno's Paradox of Motion, infinity, linear approximations, and the underlying basis for calculus - the concept of a limit. Post a short comment on one "take-away" from the article.

Forum Question: Quantifying Humanity

In January, 2006 the cover story of BusinessWeek was about numbers. According to the article, "the world is moving into a new age of numbers," where the “mathematical modeling of humanity promises to be one of the great undertakings of the 21st century.”

This article is another sign of the need for a more quantitatively-literate society. As more of us live, work, and play online, the more our digital profiles are open to analysis and susceptible to exploitation. People have been using mathematics to extract hidden patterns from data for centuries. However the increased volume of data in modern times has elevated the knowledge of math and computer technology to a 21st century life skill. In order to be smart consumers and informed citizens we need the ability to sort through data to identify patterns and establish relationships. Giving over that authority to the software behind our computers does not relieve us from the responsibility to be critical thinkers. Companies like Google and Yahoo are charting these oceans of data navigated by quants and math geeks who are becoming the new elite of the global economy. In the information age of the 21st century the best job in the world is that of data miner.

This week's Forum Question is to read, or listen to (a podcast is available) the BusinessWeek cover story. Of particular interest to you will be the last page which contains the overviews, "How Much Math Do We Need to Know?" and “How Math Transforms Industries”. Write a short summary of the article and then share your “take-aways” from the story.

Forum Question: See Math

Photo from www.allposters.com

The Greek philosopher Heraclitus stated, Nothing is permanent except change. Evolution is based on the idea that the survival of a species depends on its ability to adapt to changes in the environment. Social change is advocated as a necessary means to creating a better world. It has been suggested that the key to happiness is the recognition and acceptance of the impermanence of life.

Calculus is the mathematics of change. You can train your eyes to look at the world through mathematical lens and see math in everything, everywhere. Our life experiences teach us that change can be sudden and discontinuous, or a seamlessly smooth continuum. The Forum Question this week is to capture change through the lens of calculus and the medium of photography. You are to take a real world photo to illustrate each of the following functions:
- continuous and differentiable.
- continuous, but not differentiable.
- not continuous and not differentiable.
- an invertible function.
- a function that is not one-to-one.

The Forum Question requires setting up a Flickr account. Flickr is an online photo management and sharing application. Flickr enables users to upload photos; organize and sort photos with tags, annotate photos with “hot spot” notes; add comments and captions; and share photos with friends and family. Your five photos are to be uploaded to your Your photostream in Flickr.

With each photo you are to include the following:
1. A title.
2. A description of the photo.
3. A “hot spot” somewhere on the photo, highlighting why you chose it to illustrate the type of function. A “hot spot” is a note that you can place directly onto the photo. Here is an example and  instructions.
4. The following three tags:
(1) Our class tag crmscalc2010.
(2) Your nom de plume.
(3) The tag seemath.

Note: Flickr uses the same tagging system as Delicious in that tags are to be separated by a space. DO NOT USE A COMMA to separate tags. Flickr will look on the comma as part of the tag.

You cannot use screen shots, clip art, or copy pictures off the internet. These must be real world photographs that you have taken. However, you may stage or "set up the shot" in any way you like. Be creative and have fun with this.

Forum Question: Choices

We all make choices. With those choices come consequences, some good and some not so good. As we learn and grow, we become more mindful of the consequences of our choices.

Today is Virtual Wednesday. You have the choice of joining our face-to-face class, or attending a virtual classroom of your own. The topic is the Derivative of the Inverse of a Function. You will find the objective of the lesson in Exploration 4.5A posted under Student Handouts on our CRMS Calculus website. After working through this Exploration, answer the following seven questions in your Forum Post:

1. State the Derivative of the Inverse of a Function property in words.

2. Using the Equation Editor (right sidebar), state this property in symbols.

3. The volume of a sphere is a function of its radius:

Is volume an invertible function?

4. Use the Equation Editor (right sidebar) to write the inverse function for the radius of a sphere as a function of its volume, r(V).

5. Using the online graphing tool (right sidebar) or another graphing tool, plot the following four graphs:
(1) The volume function, V(r);
(2) Choose any point on V(r) and graph the tangent line through this point.
(3) The inverse function r(V);
(4) Graph the tangent line through the “mirror” point on the graph of the inverse function.
* Note: The graphing window should clearly display all four graphs.

6. Show that the slopes of these tangents are reciprocals.

7. Briefly explain your choice of either coming to class or attending your virtual classroom.

Forum Question: www.edu-cusp

Do you remember the definition of cusp? Today we are at the cusp of a momentous change in teaching and learning on a global scale. Your generation has grown up connected to the world. The toys and tools of the Internet are an integral part of your daily life. According to our Student Internet Use Questionaire you spend an average of 1-2 hours online each day. Marc Prensky called you digital natives in his essay, Digital Natives, Digital Immigrants. You think and process information differently than those of previous generations. You are used to getting information really fast and like to multi-task. You prefer graphics and games over text and tasks. You are impatient; you crave instant gratification and thrive on frequent rewards. Prensky suggests that your brains are physically different because of the digital input you receive growing up. What does this all mean for you as you try to navigate through the information wilderness that lies ahead?

Read through the following questions, and then choose one on which to write a thoughtful reflection.

1. In the introduction to our weblog it was suggested that because of Web 2.0, “Anyone can learn anything, from anyone, anywhere, at anytime”. Do you agree or disagree with this statement? If you agree, then what role do schools and formal education play in your learning? If you disagree, then what gets in the way of learning?

2. Mark Twain said, “Never let formal education get in the way of your learning.” What did he mean by that? Is technology blurring the boundary between formal and informal learning?

3. How do schools need to change to better serve the needs of you digital natives?

4. The video Did You Know presented a provocative view of the world of the 21st century changing at an exponential rate.The video states that , “… schools are preparing students for jobs that don’t yet exist…with technologies that haven’t been invented … in order to solve problems we don’t even know are problems yet …” using technology that quickly becomes obsolete. If all this is true, then what should schools be teaching?

5. Describe the essential skills, knowledge and expertise you think you need to master in order to succeed in work and life of the 21st Century.

Forum Question: The Product Rule

Delicious is a social bookmarking service that allows users to tag, save, manage and share web pages from a centralized source. Delicious greatly improves how people discover, remember and share on the Internet (excerpted from the Delicious website).

This Forum Question requires setting up a Delicious account. Once you create an account, take some time to learn how to use the features. ... Ready to try it out?

The next derivative rule you will learn is the Product Rule for finding the derivative of a product of two functions. Search the Internet for a site which explains the Product Rule well enough for you to understand and apply. Bookmark this site on Delicious. So that your classmates can easily find this bookmark, you want to assign a tag which will be recognized by our class. The tag you will use for this bookmark (and all future bookmarks for this course) is crmscalc2010. Add a second tag of your nom de plume, and a third tag of Product Rule. Now the bookmark can be easily shared with anyone in our class. Look for your bookmark in our Delicious library on the left sidebar of our blog.

Once you have bookmarked the site, read Section 4.2 of our textbook on the Product Rule. Write a blog post comparing the explanations given on your bookmarked site and in the textbook. From which source did you get a better understanding of the rule? Include in your post what you liked and disliked about each explanation. End your post by stating the Product Rule and giving an example.

After posting to our blog, take this short quiz to check your understanding of the Product Rule.

Forum Question: Mind Map of Semester 1

A mind map is a diagram showing the relationships among concepts. It is a graphical tool for organizing and representing information. You can read more about mind maps in our Delicious library on the left sidebar.

Your first Forum Question is to create and post a mind map of what you learned in the first semester of this course. Include all the important topics and procedures that highlight your understanding of what was covered in the first three chapters of our textbook. On the right sidebar you will find a free online tool to create your mind map. Here is an example of a mind map that I created using this tool. The three required labels for your post are: Your nom de plume, the title of the forum question (Mind Map of Semester 1), and the type of post (Forum Post). Look here for an explanation of labels.

Have fun mapping your knowledge of Calculus!