Week+02+Aug+25-29

toc

August 25th 2008
The two major concepts of calculus learned so far, the Derivative and the Definite Integral of a function, both involve the idea of a limit. From precalculus and analysis courses we experienced a conceptual introduction to limits but today we learned the calculus definition of a limit. the two images below are the slides used during class. media type="custom" key="1710239"

August 26th 2008
Today we spent the majority of the class period going over homewrok problems. Members of the class had experienced difficulty finding values of delta for a given value of epsilon and for finding delta in terms of epsilon algebraically. Several homework problems asked us to both of these things in parts c) and d) of each problem and, depending on the type of function presented, various issues arose with the algebra. The key to solving for delta in terms of epsilon is remembering i) how to solve a double - sided inequality, ii) inverses of functions and iii) characteristics of the inverse functions, particularly the inverse sine function used in #19. media type="custom" key="1724489"

August 27th 2008
Review for the test was the priority today, Several students had questions from Tuesday's homework so we spent some time addressing those issues, Part d) of Problem R2 on page 26 presented a table showing distances ran after t seconds during a 200 meter race. The questions asked were: Estimate the instantaneous velocity in m/sec when t = 2, t = 18 and t = 24. When t = 2, the instantaneous velocity is approximately (13 - 0)/(4 - 0) = 2.25 m/sec.  When t = 18, the instantaneous velocity is approximately (138 - 103)/(4 - 0) = 8.25 m/sec When t = 24, the instantaneous velocity is approximately (200 - 154)/(4 - 0) = 11.5 m/sec  For which time intervals did the velocity stay relatively constant? The velocity remains relatively constant from t = 8 to t = 16. Resources for further review include the concepts test on page 27 and the chapter test on pages 28 and 29.
 * t(s) || d(m) || t(s) || d(m) ||
 * 0 || 0 || 14 || 89 ||
 * 2 || 7 || 16 || 103 ||
 * 4 || 13 || 18 || 119 ||
 * 6 || 33 || 20 || 138 ||
 * 8 || 47 || 22 || 154 ||
 * 10 || 61 || 24 || 176 ||
 * 12 || 75 || 26 || 200 ||
 * t(s) || d(m) || Estimate of velocity || t(s) || d(m) || Estimate of velocity ||
 * 0 || 0 || 3.5 || 14 || 89 || 7 ||
 * 2 || 7 || 2.25. || 16 || 103 || 5 ||
 * 4 || 13 || 6.5 || 18 || 119 || 8.25 ||
 * 6 || 33 || 8.5 || 20 || 138 || 8.75 ||
 * 8 || 47 || 7 || 22 || 154 || 9.5 ||
 * 10 || 61 || 7 || 24 || 176 || 11.5 ||
 * 12 || 75 || 7 || 26 || 200 || 12 ||

August 28th 2008.
Test #1 covering sections 1.1 - 1.4 and 2.2

August 29th 2008
Having been introduced to the definition of the limit of a function at a particular point, the next step was to investigate limits of different combinations of functions with a view to develop and understand the limit theorems. During the class period, our investigation took us through the sum, difference, product and quotient of two functions and we discussed the relationship between the limits of the two functions at a particular x value and the limit of the combined function at that same x value. We found that as long as the limits of the functions f and g existed at x = c, then the limit of the sum, difference and product f functions f and g existed at x = c. Examples of such limits, and many others are shown in the slides below.

media type="custom" key="1824867"