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Learning:
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A Cloud Chamber of the Mind

March 2006 Mathematics Notebook
Introduction      
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An Example of a "Learning Process" Journal
Wednesday March 08, 2006
Learning Log Number 18
5:10 am Ballina NSW Australia

5:10 am Ballina NSW Australia

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

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

I have finished problems 1 - 22, but there are still a large number to complete.

One of the "obvious" facts that I have realized is the periodicity of the various expressions involving the trigonometric functions. I was aware of this for the functions themselves, but hadn't realized the obviousness of this for any expression involving these functions.

I am pleased with the use of Mathematica to quickly give me the appropriate graph.


23. . In words, what is the value of the secant of 2x as the angle approaches 0? Answer 1 as the secant is 1/cos and the cosine approaches 1 as the angle approaches 0. This is confirmed with Mathematica.

24. . In words, what is the value of cos x as the angle approaches 3p? This is equivalent to when the angle approaches p ( -180 degrees), which is -1. Mathematica agrees.

25. . In words, what is the value of sin x as the angle approaches 5p/6 (150 degrees)?

Answer is -sin 30 degrees (BUT I don't know this value!)

. Mathematica says 1/2.

I am missing some very basic understanding here. What is the sine of 30 degrees? I have two problems. I now see that the sine in the second quadrant is still positive. Good. Thus the sine of 150 degrees is the same as the sine of 30 degrees. But why is the sine of 30 degrees 1/2? Okay, I now have it.

26. . In words, what is the value of cos as the angle approaches 5p/3? This is the same as an angle of 330 degrees. The answer is + one half the square root of 3. Mathematica says 1/2. I have the proper sign but the wrong ratio (again!).

I am still having difficulty with my basic trig ratios. This time it is the value of the angle. An angle of 5p/3 is nto the same as an angle of 330 degrees, but is the same as an angle of 300 degrees.

27. . In words, what is the value of the tangent as the angle approaches 3p/4? Answer -1. Mathematica agrees.

28. . In words, what is the value of the secant as the angle approaches 7p/6? This would be in the third quadrant and thus the sign will be -. The value is the same as the secant of p/6, which is 2/square root of 3. Mathematica agrees.

37. . This is equivalent to the function x-1except at one point. Therefore the limit is -2. Mathematica confirms this.

38. . This is equivalent to the function (2x - 3) except at one point. Therefore the limit is -5. Mathematica agrees.

39. .

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

Reflection:

Difficulties: