Learning:
The Journey of a Lifetime
A Cloud Chamber of the Mind

March 2006 Mathematics Notebook

An Example of a "Learning Process" Journal

5:30 am Ballina NSW Australia

This is a continuation of my notes for "Calculus".

Continuation of Exercises For Section 1.3 [p. 64 - 66]

39. . This limit has the form of 0/0 when using direct substitution. The numerator can be factored to the form and then (x+2) can be cancelled. Direct substitution into gives 12. Mathematica agrees.

40. . This limit has the form of 0/0 when using direct substitution. The numerator can be factored to the form and then (x+1) can be cancelled. Direct substitution into gives a value of 3.

41. . This limit may be reduced to . Direct substitution gives1/10. Mathematica agrees.

42. . This limit may be reduced to . Direct substitution gives -1/4. Mathematica agrees.

43. .This limit has the form 0/0 when using direct substitution. Both numerator and denominator can be factored and the term (x-1) cancelled, leaving (x+2)/(x+1). Direct substitution gives 3/2. Mathematica agrees.

44. . This limit has the form 0/0 when using direct substitution. Rationalizing the numerator gives which evaluates to . Mathematica agrees.

Exercises 45 - 56 also involve rationalizing the numerator.

57. . This is equivalent to , which is 1/5.

58. . This is equivalent to which is 3(0), which is 0. Mathematica agrees.

59. . This involves being familiar with various trigonometric relationships. This expression is equivalent to , which is the same as , which is 0.

60. Easy.

61. Easy.

62. 0. Easy. It is relatively easy to do the algebra but I have absolutely no idea what the overall graph looks like. This is where Mathematica becomes useful as one can see the behavior of the function over the entire domain, not just at a single point.

Here are the results using Mathematica (Web, Mathematica). 7:20 am

Reflection: This was a great session. Difficulties: None.