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Math 1312 – MFW class Test IV review
1. True or False
_____________ a) all functions that are differentiable at x = 2 ( have a derivative at x = 2 ) will be continuous at x = 2.
______________ b) If f(x) is not continuous at x = 3, then f (x) can not have a derivative at x = 3.
______________ c) The inflection points can be found by solving the equation f ^{// } ( x ) = 0.
2. Use the first derivative of f (x) = x^{3} - 12x + 2 to find
a) the critical values of f(x). ( possible max, min ).
b) Use the 2^{nd} derivative to find the inflection points.
c) Is f(x) increasing or decreasing at x = 2 (Show work )
d) on what intervals if f(x) decreasing ? _______________
e) find the y-intercept(s). ___________________
f) Sketch the graph of f(x)
4. If C(x) = _____________________ represented the cost in producing x items, then find
( Please note that in using the function – the results may not make real world sense – it’s the method I am interested in)
Assume that C(x) = 3x^{2} – 2x + 200
a) the cost in producing 10 items
b) the cost in producing the 11^{th} item .
c) marginal cost -
d) give me two ideas that the value in part 4c represents (increasing – decreasing, predicts ? )
5. Find the area under the curve y = 4x – 4 from x = 3 to x = 5.
6. Find the value of the following indefinite integrals.
= _______________ dx = ______________
7. Find the value of the definite integrals
dx = _____________ = _____________
8. Find f (x) if
a) b)
f(x) = _________________ f(x) = ______________
9. Factorials:
Simplify – write down the formulas and then give me the final value.
1) b) P( 5000, 2) = _______________ c) C( 3000, 2998) = __________
10. Which is larger
a) P( n, r ) or C(n,r ) b) 0! or 1!
11. Multiplication Principle
a) How many two digit numbers can be created using { 1, 2, 3, 4, 5 } so that the number is an even number
1) if repetitions are allowed ____________
What is the probability that the number created at random will be an even number ? _________
2) if repetitions are not allowed ____________
What is the prob. that the number created will be an even number ? __________
b) A company is to select three different representatives to send to three different conventions during the year. Forty-two
people volunteer to attend. How many different groups of three can be selected ?
_________
c) Three movies are to shown at a local theater. The movies will be selected in the following format;
The first movie will chosen from a group of six G-rated movies. The second movie will be selected from a group of five
PG rated movies. The last movie selected will come from any one of seven unrated movies.
How many different groups of movies can be shown ? _______________
11. Simple permutations and Combinations
a) A company has two openings in its accounting department. A total of twenty applicants are interviewed for the job.
Eight of the applicants are female.
How many different ways can the company fill the two opening if anybody can be chosen ? __________
How many ways if at least one must be female ? _____________
What is the probability that exactly one is a female ? __________________
11b) A family drives to the local shop to pick up a fresh Christmas tree. The store has 20 available trees – four of these have
a minor but visible defect. Once there they decide to pick up an **additional **two trees. If the trees are selected at
random,
a) how many different groups of trees could be selected ? ____________
b) how many groups of trees with only one defective tree ? _________________
c) what is the probability that the group selected has only one bad tree ?
12.
a) A box contains four red balls, 1 blue ball, and 5 yellow balls. You draw a ball record the outcome and replace the ball.
Suppose that three balls are drawn in this manner.
What is the probability of drawing
1) all blue balls ? ______________ 2) exactly 2 red and 1 yellow ? ______________
b) If the balls are not replaced, what is the probability of drawing
1) all blue balls ? ______________ 2) exactly 2 red and 1 yellow ? ______________
13. A Bernoulli Experiment consists of how many outcomes ? __________
What are they called ?
If a Bernoulli Experiment is repeated n times ( n-independent trials), then what do we call the experiment ?
14. What is the formula that we called the Binomial formula – ( this means give me the formula !)
15. Simplify C( 40, 2) ( 0.2)^{2}(0.8)^{38} = ____________
16. Every time a ship leaves the harbor during a storm, there is a 2 % that the ship will sink. Last week twenty ships left
the harbor during a storm. What is the probability that
a) exactly 7 of the ships will sink ? ___________ b) at least one will sink ? __________
IF an experiment can be classified an a binomial experiment, we can find the expected value µ = E( X ) by
µ = np and the variance ^{2} by ^{2} = npq.
If the experiment above (#16) is a binomial experiment,
define the random variable x, define success, find success probability, and finish by finding
the expected value and the variance.
17. Find the expected value of the random variable that has the following probability distribution.
X=x P(X=x)
======================
-2 1/7
0 2/7
1 4/7
18. Find the variance and standard deviation of X in the problem above.
19. A child has 26 blocks. Each of the blocks corresponds to one letter of the alphabet.
Selects three blocks and holds them together in his hands.
a) How many three letter “words” are possible ? __________ ( Assume that any of the 26 can be selected )
b) How many different groups of three blocks can be selected ? ___________
20. A class consists of 10 people. Three different individuals are selected. If the class consists of 4 male and three female.
What is the probability that the group consists of two male and 1 female member ?
21. On three different days a person will be chosen at random to lead the group. What is the probability that the selection will
consists of exactly two male selections ?
4. Find P ( 2000, 2 ) = _____________________ P( n, n ) = _______________
C( 2000, 1999) = _____________________ 0 ! = ______________ |