(0a) Please type your name at the top of your worksheet as a comment.
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(1) Define , , and .
(1a) Graph the three functions on the same set of axes, choosing scaling that shows the key features of the graphs.(5)
(1b) Explain which graph is the graph of , which is the graph of , and which is the graph of in a complete sentence. (0)
I did not receive a response to this question.
(2) The function is called the hyperbolic cosine function and can be written in the equivalent form .
Remember to divide by 2 in .
(2a) Use Sage to provide evidence that these two functions and are the same, perhaps by graphing them both on the same set of axes. (2)
(2b) What is an approximate value for , accurate to three decimal digits? Does this answer look right based on your graph? Explain, rescaling your graph as needed. (2)
(3a) Is it true that for all ? (5)
(3b) Explain your answer to (3a) in a complete sentence. (5)
Good analysis.
(4a) What is the limit of as approaches infinity? (5)
(4b) Is your answer in (4a) the same as if you use the definition of in terms of exponentials, as given in (2)? Briefly explain. (5)
(5a) Use Sage to find the derivative of . (5)
(5b) Is your answer to (5a) consistent with the definition ? Briefly explain. (0)
Is your answer to (5a) consistent with the definition as opposed to other possible definitions of sinh(x)?