Math 117 Practice Midterm 2 Please work out each of the given problems. Credit will be based on the steps that you show towards the final answer. Show your work.
Problem 1 The following table gives the probability distribution for the number of kittens in a litter.
Determine E(x) and s. Explain the meaning of each.
We expand the table as follows
The expected value is 3.65. This means that the overall average number of kittens in a litter is 3.65 kittens. The variance is 1.40. We take the square root to get the standard deviation of 1.18.
Problem 2 You have experimented with the days for germination for a lupin and know that the distribution follows the parabolic distribution f(t) = kt(1  t) for 0 < t < 1 where t is measured in months and k is a positive constant. A. Find k. To find k we set the integral equal to zero.
Hence k = 6.
B. Find the probability that a lupine seed will germinate between 0.3 and 0.5 months. We find
Hence P(0.3 < x < 0.5) = 0.284
C. Find the expected value E(t) and interpret what it means. We find
Hence E(t) = 0.5 D. Find the standard deviation. We calculate the integral
Now calculate the variance V = 0.3  0.25 = 0.05 Take a square root to get the standard deviation of 0.22.
Problem 3 Currently, the benefit package costs the college $9442 and is expected to increase 20% next year. Assume the 20% increase becomes a trend over the next several years. A. Write an expression for the cost of the benefit package after n years. Notice that a 20% increase means that the next year will have 1.2 times the current year, since you have 100% of your expense from last year plus an additional 20%. Look at the first few terms. a_{0} = 9442 a_{1} = (9442)(1.2) a_{2} = (9442)(1.2)^{2} a_{3} = (9442)(1.2)^{3} Hopefully the pattern is apparent. The general term is a_{n} = (9442)(1.2)^{n}
B. Compute the cost of benefits in the year 2020. (This year is 2004). The year 2020 corresponds with n = 2020  2004 = 16. a_{16} = (9442)(1.2)^{16} = 174568 We can expect that the benefit package will cost $174,568 if trends continue.
Problem 4 Determine the convergence or divergence of the following series. A. We take the limit using L'Hopital's rule:
Since the limit is not zero, the series diverges. B.
This is a geometric series with r = 1/3, hence by the geometric series test the series converges.
C.
S
Problem 5 Explain why the series converges. Then approximate the following sum using 5 terms. Then estimate the maximum error of your approximation.
This is a pseries with p = 8 > 1, hence by the pseries test the series converges. We find
We compute the maximum error by finding
