There will be two mid term exams. Each will be during a regular class period. You should bring a scientific calculator but it does not have to be a graphing calculator and it MUST NOT be a calculator with Symbolic Algebra Capability such as the TI 92. You may also bring a single 8.5 by 11 inch "cheat sheet" on which you have written formulas , hints etc for use during the exam.
Brief , clear explanations of your work are required on the exams. Computations without explanations will not get credit , not even on multiple choice questions.
Check this page from time to time for exam locations
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Tuesday February 27 |
Ch 5,6,7 of Finite ; Ch 12 of Calc |
Logan Hall Room 17 |
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Tuesday April 3 |
Chapters 7 and 11 of Calculus (except 7.5) |
Logan Hall Room 17 |
REMEMBER TO GO TO LOGAN HALL (NOT DRL) FOR MIDTERM #2 !
REMEMBER TO MAKE AND BRING YOUR CHEAT SHEET !
The exam begins promptly at noon. Please arrive on time to maximize your own working time.
Here are the AQs we have had since Midterm #1 You should be able to do all of them with ease.
Most answers are provided as well. To make it easier to post , mathematical symbols have
mostly been written in"Maple-ese". If you don't understand any of them you could try typing
them directly into a Maple worksheet and Maple should reply with a nice symbolic form.
AQ #10 Let f(x) := sqrt(cos(x));
1. Find the Taylor polynomial of degree two for f(x) at x=0.
2. Use that polynomial to estimate Int( sqrt(cos(x)),x=-1..1);
Ans. 1. p(x) = 1- (x^2/4)
2. Int( sqrt(cos(x)),x=-1..1) is approximately Int( 1- (x^2/4),x=-1..1)=1.833333
AQ #11
1.Use Newton' s Method to estimate the positive zero of f(x) =x2 -29 What are
you finding when you do this? Take x0=5 and compute x1and x2
2. Begin with x0=5 again and find x1and x2 according to the new formula
xn+1 = (1/2) { xn + (29 / xn)} Show that x1and x2 are exactly as in 1. above
3. Prove the equivalence of 1. and 2. Modify the rule to approximate sqrt(40) ;
and then sqrt(a). This method of approximating square roots is very old and
famous. It was known to the Babylonians. In modern times it was often called
the Mechanic's Rule.
Ans. For 1. and 2. x1 = 5.4 and x2 = 5.38518
AQ #12 1. Find a rational number whose decimal expansion is 0.441 441 441 441 etc
2. Find the sum of the inifinite series: Sum ((2/3)^k ,k=0..infinity);
3. Find the sum of the inifinite series: Sum ((2/3^k) ,k=0..infinity);
Ans. 1. 49 /111 2. 3 3. 3
AQ #13 Determine whether te given series converges or diverges. Use the integral test or the
comparison test as appropriate/convenient.
1. Sum ( k / (k^2 +1 ) . k=1..infinity);
2. Sum ( 1/ (k^2 +1 ) . k=1..infinity);
3. Sum ( 1 / (2^k + k ) . k=1..infinity);
4. Use the comparison test to prove that Sum ( 1/ (k^2 +k ) . k=1..infinity); converges. Better still,
figure out the sum by using the fact that 1/ (k^2 +k ) = (1/k) - 1/(k+1). Write out a few terms
according to 1/ (k^2 +k ) = (1/k) - 1/(k+1) and deduvce what will remain after an infinite number of
cancellations
Ans. 1. Diverges (Integral Test) 2. Converges ( Compare with (1/k^2) , which is a convergent p-series)
3. Converges (compare with convergent geometric series (1/2^k) ). 4. Sum =1
AQ #14 Find the Taylor series for each given function , centered at a=0. Do this by manipulation of
known series and not by resorting to the definition.
1. f(x) = e-x 2. f(x) = xcos(x2) 3. f(x) = ln (1+x)
Find both partial derivatives df/dx and df/dy for each
4. z=f(x,y) = 3x5y2+7xy4+x+2 5. z=f(x,y) = x2 y3 -y5sin(x)+x4 - cos(y)
Ans. 1. 1-x+x^2/2! - x^3/3! + x^4/4! ...... ( By subs x-> -x in ex series)
2. x- x^5/ 2! + x^9 / 4! -x^13 / 6! ..... ( By subs x-> x2 in cos(x) series and then multiplying by x)
3. x - x^2/ 2 + x^3 / 3 - x^4 / 4 + x^5 / 5 ........ ( By integrating the series for 1/ (1+x) )
4. df/dx = 15 x4y2 + 7y4 +1 ; df/dy = 6x5y + 28xy3
5. df/dx = 2xy3 - y5cos(x) +4x3 ; df/dy = 3x2y2 -5y4 sin(x) + sin(y)
AQ #15 1. For the function z = f( x, y ) =2x2+y3 -x-12y+7 : (i) Find all critical points ( there are two of them)
and then (ii) Use the second derivative tst to try to determine whether z = f( x, y ) has a local max, a local min
or a saddle point at the critical points. If the test fails at one or both of the critical points , just say so.
2. Find the value of the double (iterated) integral : Int (Int ( xy , y=x..x^2),x=1..4 ) ;
Ans. 1. Local miin at ( 1/4 , 2 ) Saddle point at ( 1 /4 - 2 )
2. 309.375 = 309+(3/8)
AQ #16 1. Find the value of the double(iterated) integral: Int (Int( (x+y) ,x=(y/2)..1) , y=0..2);
Or , if you don't feel like doing it honestly , explain why
Int (Int( (x+y) ,x=(y/2)..1) , y=0..2)= Int (Int( (x+y) ,y=0..2*x) , x=0..1) , and the latter integral we've done
It's 4/3 .
2. Use the method of Lagrange Multipliers to maximize z = f( x, y ) = -2 x2 -2xy -(3/2) y2 +x+2y
Ans. 2 . Max occurs when x= (1/2) , y=2. The max is z= -4.
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Here are the nine AQs for the first midterm.
AQ#1
A survey of 1500 Freshmen at Pacifica Tech revealed that:
520 took Chemistry
335 took Physics
425 took Math
100 took both Math and Chemistry
180 took both Physics and Chemistry
150 took both Physics and Math
28 took all three
Use this information and a diagram below to figure out:
1. How many freshmaen took at least one of these courses? Ans. 878
2. How many freshmaen took exactly one of these courses? Ans 504
AQ#2
A package contains 100 bulbs , of which 10 are defective. A sample of 5 bulbs is selected at random.
(a) How many different samples are possible? Ans. C(100 , 5)
(b) How many of the samples contain 2 defective bulbs? Ans. C(90, 3) * C ( 10 , 2 )
(c) How many of the samples contain at least one defective bulb? Ans. C(100 , 5) - C(90 , 5)
AQ#3
Prove that C ( n+1 , k ) = C ( n , k ) + C ( n , k-1 )
There are ( at least ) two ways to proceed. Take your pick or try both. First , you can write out
the sum C ( n , k ) + C ( n , k-1 ) according to our formulas , then do some slightly fancy
common denominator addition of the fractions and show that the result is the left hand side
C ( n+1 , k ). Or you can reason it out like so. The left hand side starts with a set of n+1
elements and asks for the number of subsets of size k. Imagine that you have attached a label , a *
say , to exactly one of the n+1 elements in the set. We think of the one with the * as the (n+1)st
member of the set. Now there are exactly two kinds of subsets of size k which can be taken from our starter set of n+1 elements; those that contain the element with the * label and those that don't. Count each type. Explain why their number is what you say it is. When you do this you will be done.
Since this logical reasoning method enables you to entirely dodge the calculation , you will be held to a rather high standard of presentation, so make it as precise , concise and convincing as you can.
AQ#4
A lot of 50 manufactured items contains 20 defectives and 30 non-defectives. What is the probability that a sample of 10 items selected at random will correctly reflect the quality of the lot. In other words what is the probability that a random sample of 9 items will have 4 defectives and 5 non defectives
Ans [ C ( 20 , 4 ) * C (30 , 5) ] / C(50 , 9)
AQ#5
Suppose that we have a white urn containing two white balls and one red ball and we also have a red urn containing one white ball and three red balls. Our experiment consists of selecting one ball at random from the white urn and then (without replacing that ball) selecting a ball a random from the urn whose color matches that of the first ball chosen.
(i) What is the probability that the second ball chosen is red? Ans 7 / 12
(ii)What is the probability that the second ball chosen is white? Ans 5 / 12
AQ#6
Here is the distribution for the random variable X.
X Pr(X)
1 (1/3)
6 (1/3)
11 (1/3)
1. Find E(X) = ux Ans 6
Ms. Jones doesn't know the probability distribution for X but decides to estimate ux by sampling.She makes two observations and averages them. The sample mean is X- = ( X1 + X2 ) / 2 of course.
2. Make a table giving the distribution for X- , the sample mean.
3. Calculate directly the expected value of X- , the sample mean , thus showing that X- is an unbiased
statistic for ux Ans 6
AQ#7
1.Suppose that in a large factory there is an average of two accidents per day and the time between accidents has an exponential p.d.f. ( with , necessarily , expected value of 0.5 days)
Find the probabilty that the time between two accidents will be more than half a day and less than one day.
Ans e^(-1) - e^(-2)
2. Suppose that X is a random variable of the continuous type whose distribution is uniform on the
interval [ 10 , 90 ]. Compute the mean and variance of X. ( Better still , compute the mean and
variance of a random variable of the continuous type whose distribution is uniform on the interval
[ a, b ] thereby computing the mean and variance of all possible continuous , uniformly distributed
random variables.
Ans. mean = 50 variance = 1600 / 3
AQ#8
1. Four fifths of the students at Middleborough Junior High School have at least one cavity. The
school authorities decide that all of the students are to receive dental exams. What is the probability
that:
(i) The first student to have a cavity is the third student examined? Ans (1/5)^2 * ( 4 / 5 )
(ii) The first student to have a cavity is the tenth student examined? Ans (1/5)^9 * ( 4 / 5 )
2. The monthly number of fire insurance claims filed against the Firebug Insurance Company is a
Poisson distributed random variable with mean = 10.
(i) What is the probability that in a given month no more than two claims are filed? Ans 61 * exp(-10) = .002769
(ii) What is the probability that in a given month at least three claims are filed? Ans1-.002769=0.997230
3. Professor Smith believes that 1% of the population is infected with a certain bacterium. She is about to test 200 people and will use a Bernoulli model to calculate the probability that exactly three
of the people examined are infected. What will she get? Professor Jones agrees with Professor Smith's assessment of the mean infection rate but believes that the distribution follows the Poisson
probability law. When he calculates the probability that exactly three of the 200 examined people are
infected what will he get?
Ans (Smith) C(200 , 3 ) * (0.01)^3 * (0.99)^197 = 0.181355
Ans. ( Jones) (4 / 3 ) * exp (-2) = 0.180447
4. Let X be a random variable. (We are completely infdfferent as to its distribution). Prove that
Var(X) can be calculated according to the (technicallu useful) rule:
Var(X) = E[X(X-1] + ux - ( ux ^2) where , of course , ux = E (X)
Hint: Recall that Var(X) = E(X^2) - ( ux ) ^2 and consider E [ X^2 - X + ux - ( ux ^2) ]
AQ#9
1. The Jump Start Company makes 12 volt car batteries. They know that the life of their batteries
is a normally distributed random variable and that the average life is 55 months. They know also
that the standard deviation of the battery life is 5 months. Jump start will replace any battery that
fails within four years of purchase.
What is the probability that a randomly chosen battery will fail during the guarantee period?
Ans. 0.0808
2. Laboratory mice are given an illness for which the natural recovery rate is (1/6). A new drug is
tested on 20 infected mice and 8 of them recover. In order to help judge the efficacy of the
drug the investigator wants to know the probabilty that 8 or more would have recovered without
the drug. Answer this this question by considering a binomial experiment and then approximating
the binomial by the appropriate normal lly distrubuted variable.
Ans 0.0062