In this lab you will look at two problems that are at the heart of calculus. Each of these experiments illustrates a core calculus concept. You should perform each experiment taking notes and pictures. You will use these to write up your results.
You are expected to use a word processor to produce the laboratory. Graphing software should be used to draw your graphs and illustrations. You can also include pictures you have taken. Equations should be written using “equation editor” software. In short, the laboratory should have a professional look and feel to it. It should be of publishable quality.
You report should be printed on 8.5 x 11 inch paper and include a title page (format will be discussed in class). Each page should be numbered. You can work in groups of 3 on this laboratory. If you do this, you must include a page right after the title page and before the report that includes a list of the contributions of each member of the group has made.
Suppose you start 10 feet away from a wall and walk 5 feet toward the wall and stop. Now walk
2.5 feet toward the wall and stop. Keep going each time walking half the distance of your previous walk toward the wall.
1.
Where are you after three walks?
2.
Where are you after 2, 3, 4, 5, 10 walks?
3.
Create a function where n is the number of the walk and f(n) is the distance from the wall.
4.
Graph this function.
5.
Using your modeling skills find a model for this function.
6.
If you walk forever, were will you end up? For this one write a paragraph defending your location.
7.
If instead of walking one half as far as the previous walk, walk one third. That is start 9 feet away from the wall and walk 3 feet, then 1 foot, then 1/3 of a foot, etc. Where do you end up this time? Again write a paragraph.
8.
Discuss you experiment in relation Zeno’s Paradox called Achilles and Tortoise.
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Here you are going to find the circumference and area of a circle by approximating it with polygons.
1.
Start by drawing a circle with radius 3” on a sheet of paper. (You should include your drawings in laboratory report. You should be able to get two per page.)
2.
Divide the circle into 3 equal parts.
3.
Now connect adjacent points on the circumference to form 3 triangles as shown below.
You need to find the area of these isosceles triangles and the length of the bases (red lines).
4.
In a table keep track of the following: a.
The number of triangles. b.
The sum of the lengths of the bases. This is your approximation for the circumference. Label this column, C. c.
The sum of the areas of the triangles. This is your approximation for the area of the circle. Label this column , A. d.
In a column divide your approximation for the circumference by 2*r. This value should be 6 since r is the radius of your circle is 3. Label this column P1 e.
In a column divide your approximation for the area by r
2
or 9. Label this column
P2.
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5.
Repeat this process for n = 4 … 15 recording your results in the correct columns.
6.
Create the two functions described below. You should the graph for each of these functions separately. a.
C(n) which associates n to the corresponding approximation of the circumference. b.
A(n) which associates n to the corresponding approximation of the area.
7.
For the two functions created in step 6 find a model for each function.
8.
If we were to continue this experiment --- let n grow larger without bound then what values do C and A will approach. Write a paragraph for each variable explaining your reasoning.
9.
Then examine the P1 and P2 columns of your table. Write a paragraph on what you if n is allowed to grow larger without bound.
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