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Homework 2 - determine how old the picture is and conclude...

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MATH 106 CALCULUS I FOR BIO. & SOC. SCI. FALL 2011 INSTRUCTOR: JOS ´ E MANUEL G ´ OMEZ Homework 2. Due date Wednesday September 14, 2011 Please show all your work. (1) (5 Points) A certain strain of bacteria that reproduces asexually triples its size every 45 days. If after 180 days there are 1620 bacteria, how many bacteria were there originally? (2) (5 Points) A picture supposedly painted by Vermeer (1632-1675) contains 99.5% of its carbon–14. It is known that carbon-14 has a half life of 5730 years. From this information, can you decide whether or not the picture is a fake? Please explain your answer. Hint: Let’s call X 0 the initial amount of Carbon–14 in the painting. We don’t know what X 0 is but that does not matter at the end. Use the given information to
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Unformatted text preview: determine how old the picture is and conclude if the picture is fake or not. (3) (5 Points) Determine the values of the sequence { a n } for n = 0 , 1 , 2 , 3 , 4 in the following sequences (a) a n = e cos( πn ) . (b) a n = n 2 n + 1 . (c) a n = √ n + 4 . (d) a n = sin( πn 2 ) . (e) The recursively defined sequence with a = 128 and a n +1 = √ a n . (4) (5 Points) Determine if the following sequences converge or diverge. In case they converge find their limit. (a) a n = sin( πn ) . (b) b n = sin(2 πn ) . (c) c n = e-2 n . 1 2 INSTRUCTOR: JOS ´ E MANUEL G ´ OMEZ (d) d n = n + 1 n 2 . (e) e n = n 2 + 1 n ....
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