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.
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.
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.
P(0.3 < x < 0.5) = 0.284
C. Find the expected value E(t) and interpret what it means.
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.
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.
a0 = 9442 a1 = (9442)(1.2)
a2 = (9442)(1.2)2 a3 = (9442)(1.2)3
Hopefully the pattern is apparent. The general term is
an = (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.
a16 = (9442)(1.2)16 = 174568
We can expect that the benefit package will cost $174,568 if trends continue.
Determine the convergence or divergence of the following series.
We take the limit using L'Hopital's rule:
Since the limit is not zero, the series diverges.
This is a geometric series with r = 1/3, hence by the geometric series test the series converges.
Explain why the series converges. Then approximate the following sum using 5 terms. Then estimate the maximum error of your approximation.
This is a p-series with p = 8 > 1, hence by the p-series test the series converges.
We compute the maximum error by finding