Math 131, Lab 4, Summer 1999 - Christian Brothers University
Math 131 C & D. Lab 5 9/22/05
Part # 1. Useful limits.
Calculate [pic](h in radians) in two ways:
(1) By evaluating the function for values of h approaching 0.
[pic] [pic]
(2) By using the graph of the function. (Show the graph of the function and the window range you selected and explain how you obtained the result).
We could use similar methods for [pic], however recall that last week we estimated[pic].
Part # 2. Derivatives of y = sin x and y = cos x.
Note: The derivative of a function y = f(x) is defined as [pic]
a) Fill in the following table (with h = 0.001) and for cos x in radian mode. [Note: you can use the seq command; e.g. seq((sin(x+0.001) – sin(x))/0.001, x, 0, 2(, (/4) ]
[pic]
What do you observe? Explain clearly why you would suspect from these values that the derivative of
sin x is cos x.
b) Find the derivative of y = sin x (x in radians) algebraically.
[pic]
Use the limits in part # 1 to get the result. Explain your work.
(Note: sin(x + h) = sin x cos h + sin h cos x)
(c) repeat part (b) for the function y = cos x (x in radians)
(Note: cos(x + h) = cos x cos h - sin x sin h)
Part #3. Differentiability and Continuity.
Consider the graph of each function. Use a table to estimate[pic]or determine that it doesn’t exist, for each c. (See sample table below.) Confirm that your answer is reasonable by looking at the graph.
(a) [pic] c = 1 c = -3
(b) [pic] c = 1 c = 0
Based on your experimentation, can you establish some criteria so that you can look at a graph and determine where the derivative does and does not exist? Perhaps further experimentation would help.
Optional: Consider[pic] c = -1
What does your experimentation tell you about the relationship between differentiability and continuity?
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[pic]
c = 1
[pic]
Explain.
(Note: One way of filling in the table is by placing f(x) in y1 in your TI-89 calculator. Then from the home screen enter: (y1(1+h) – y1(1))/h | h =1. After you get one value, just edit a previous command for others.)
Lab report: 1 per group.
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[pic]
[pic]
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