Applied Math 40S Slides June 1, 2007

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Applications of sequences and series, review video on YouTube, introduction to fractals.

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Applied Math 40S Slides June 1, 2007

  1. 1. A good resource for learning your way around the calculator or to review what we've learned in class ... Working with Sequences on the TI-83+ or 84+
  2. 2. A small forest of 4000 trees is under a new forestry plan. Each year 20% of the trees will be harvested and 1000 new trees are planted. (a) Will the forest ever disappear? (b) Will the forest size ever stabilize? If so, how many years and with how many trees?
  3. 3. Introduction to today's class by Mr. Green on YouTube ... a summary of almost everything in this unit ... Sequences and Series on YouTube
  4. 4. The Bouncing Ball A ball is dropped from one metre, and the height is recorded after each bounce. A 'Super Bouncer' sold locally is guaranteed to bounce to 90 percent of its drop height if it is dropped onto concrete from a height of less than two metres. 1. How high does the ball bounce on its eighth bounce? 2. How many times does the ball bounce before it rises to less than half of its original drop height? 3. How many times does the ball bounce before it stops bouncing? 4. How far has the ball travelled as it reaches the top of its 10th bounce. 5. Construct a graph that shows the bounce height versus bounce number.
  5. 5. The Bouncing Ball A ball is dropped from one metre, and the height is recorded after each bounce. A 'Super Bouncer' sold locally is guaranteed to bounce to 90 percent of its drop height if it is dropped onto concrete from a height of less than two metres. 1. How high does the ball bounce on its eighth bounce?
  6. 6. The Bouncing Ball A ball is dropped from one metre, and the height is recorded after each bounce. A 'Super Bouncer' sold locally is guaranteed to bounce to 90 percent of its drop height if it is dropped onto concrete from a height of less than two metres. 2. How many times does the ball bounce before it rises to less than half of its original drop height?
  7. 7. The Bouncing Ball A ball is dropped from one metre, and the height is recorded after each bounce. A 'Super Bouncer' sold locally is guaranteed to bounce to 90 percent of its drop height if it is dropped onto concrete from a height of less than two metres. 3. How many times does the ball bounce before it stops bouncing?
  8. 8. The Bouncing Ball A ball is dropped from one metre, and the height is recorded after each bounce. A 'Super Bouncer' sold locally is guaranteed to bounce to 90 percent of its drop height if it is dropped onto concrete from a height of less than two metres. 4. How far has the ball travelled as it reaches the top of its 10th bounce.
  9. 9. The Bouncing Ball A ball is dropped from one metre, and the height is recorded after each bounce. A 'Super Bouncer' sold locally is guaranteed to bounce to 90 percent of its drop height if it is dropped onto concrete from a height of less than two metres. 5. Construct a graph that shows the bounce height versus bounce number.
  10. 10. What is a Fractal? Fractal Zoom on YouTube
  11. 11. The Koch Snowflake This fractal -- the Koch Snowflake -- was developed by in 1904 by Helge von Koch, a Swedish mathematician. The fractal is started by drawing an equilateral triangle. Each side of the triangle is trisected, and the middle section forms the base of a new equilateral triangle outside the original one. The process is then continued. The diagram below shows three generations of the Koch Snowflake.
  12. 12. The Sierpinski Triangle Waclaw Sierpinski, a Polish mathematician, developed another fractal known as the Sierpinski Triangle. This fractal also starts with an equilateral triangle. To draw the fractal, you find the midpoint of each side of the original triangle, and then draw three segments joining the midpoints. There are now four triangles inside the original triangle. The middle triangle is not shaded, and the process is continued with the other three shaded triangles, as shown in the diagram below.
  13. 13. Draw a Fractal Use pencil and paper (metric graph paper if possible) to draw the fractal described below. • Draw a square with 8-cm sides in the middle of the paper. • Position the paper horizontally (in landscape format). Extend the diagram to the left and right by drawing a square on each side of the original square -- touching the original square. The sides of the new squares should be half as long as the side lengths of the original square. • Repeat the previous step three times. Your fractal should now have five generations, including the original square.
  14. 14. Question: Will the fractal ever be too large for this page? Explain.
  15. 15. The Rectangle ... Draw a rectangle that measures 12 cm by 8 cm, and shade the inside of the rectangle. Construct the midpoints of each side of the rectangle, and then draw a quadrilateral by joining these points. Shade the quadrilateral white. Now continue the process by finding the midpoints of the quadrilateral, drawing the rectangle, and shading it the same colour as the first rectangle. Draw six generations. (The initial rectangle is the first generation.)
  16. 16. The Square ... Create a fractal that begins with a large square 20 cm on each side. Each pattern requires that the square be divided into four equally sized squares, that the bottom-left square be shaded, and the process continues in the upper-right square. Repeat the process four times.
  17. 17. The Circle-Square ... Create a fractal where a square is inscribed in a circle. The diameter of the original circle is 16 cm. Shade the area between the circle and the square. Inscribe a circle inside the resulting square, and then inscribe a new square inside that circle, and shade the area between the new circle and square. Repeat this process three times.
  18. 18. HOMEWORK Page 282 question number 3 only. DO NOT do question 4. Let's see how many of your classmates read these slides; we'll find out on Monday. ;-)

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