MATH 180 December 2017
• Q1 (a) • Q1 (b) • Q1 (c) • Q2 (a) • Q2 (b) • Q2 (c) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 • Q7 • Q8 (a) • Q8 (b) • Q9 (a) • Q9 (b) • Q9 (c) • Q10 (a) • Q10 (b) • Q10 (c) • Q11 (a) • Q11 (b) • Q11 (c) • Q11 (d) • Q11 (e) • Q12 (a) • Q12 (b) • Q13 (a) • Q13 (b) • Q13 (c) • Q13 (d) •
Question 11 (c)
is an example of a Gamma distribution, a function that may be used to describe certain probabilities.
(c) The derivative of is
Identify all the intervals where is increasing, and all the intervals where is decreasing.
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Recall that a function is increasing if its derivative is positive, and decreasing if its derivative is negative.
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By the Hint, we need to look at the sign of . Since the fraction and the exponential function are positive, the sign of is the same as the sign of
For , that is for , we have that . So . In other words, is increasing on the interval .
Similarly, for we have and hence . So and is decreasing on the interval .
Answer: is and .