Download Jan 22, 2004 - Angelo State University

Name _______________________________________ Math 1312 - Quiz #1 - Jan 22, 2004
1. Write in symbols
“ 4 is an element of the set A” → _________________
“ A is not a subset of B” → _____________
2. Complete the blank.
a) The set U is called the ____________________ set because it contains all objects under consideration for the given
experiment.
b) A is said to be a __________________ of B provided that for every x in A, x is also in B.
c) the set of _____________________ consists entirely of all whole numbers and their opposites.
d) Every real number can be classified as being one of two types either _________________ or ________________.
3. List all of the subsets of
A = { a, b, c } → ___________________________________________________________________
4. Why is the following “set” not a well-defined set ?
Let A = { x | x is a smart student in class } →
5. List all the values of each set. Use the proper notation → { 1, 2, 3 }
U = { x | x is a whole number < 5 } , A = { x | x is negative }, B = { x | x 2 = 4 }
a) A = ___________________________
b) B = ___________________________
c) U = ____________________________
6. True or False.
______________a) { 1, 3, 1 } = { 3, 1 }
_____________ c)
A  A for any set A.
_______________ b)
  { 1, 2 }
_______________ d)
{2 }  { 1, 2 }
Name ___________________________________________ Math 1312- Short Quiz #2 – Jan. 27, 2004
1.
Let A and B be two given sets, if A ∩ B = , we say that A and B are _________________ sets.
2.
Let A be a given set. The ___________________ of A written A ’ is the set that contains the set of all elements that
are in the universal set U but are not in A.
3. True or False.
___________ a. The union of sets A and B consists of all elements that are in A, in B, or in both A and B.
___________ b. A x B = { (a, b ) | a  A and b  B }
4. Find A x B if A = { 1, 2 } and B = { a, b, c }.
A x B = { _______________________________________________________________ }
5. Complete the properties.
6.
a) ( A / ) / = ____________
b)  / = _____________
c) ( A  A / ) = ____________
d) ( A / ∩ B / ) / = ______________
Let U = { x | x is a natural number less than 5 }, A = { x | x > 2 }, B = { x | x 2 = 4 }. Find
a) A  B = _____________
b) A ∩ B = ____________
c) B / = ______________
7.
Shade the following sets.
a) A ∩ B
8.
b) A/
A class consists of 30 students. Twenty are known to be female, 15 are known to write with a pencil, and six of the
female students write with a pencil. How many are neither female nor they write with a pencil ?
Name ___________________________________________ Math 1312 – QZ #3, Jan. 29, 2004
1. If x + 2/3 + 1/5 = 1, then x = ________________
2.
The teacher gives five exams in class. Each is worth the same and the class grade will consist only on these five
exams. Write the value of each exam in terms of the whole grade as a fraction.
3.
A class consists of 20 students , 12 are male and 8 are female. A student is selected at random. What is the
likelihood that the student selected is male- as a fraction.
4.
An experiment has four individual outcomes ( say a roll of a four sided die).
The set S = { s1, s2, s3, s4 } is called the
________________
The set is said to have ________________ probability if each of the outcomes {s1}, {s2}, {s3}, {s4} is equally likely
to occur. Those events just listed are called
_____________________ events.
5.
If an experiment S = { s1, s2, s3, s4, s5, s6, s7, s8, s9, s10 } consists of outcomes that are all equally likely to occur,
then likelihood that s1 will occur is
_____________
6. How many different events are possible from the sample space S = {s1, s2 } ? ___________
7. In probability we do not say sets are disjoint, we say the sets are _________________
Name ____________________________________________ Math 1312 – Short Quiz #4– Feb. 3, 2003
1. An event that can never happen is called an _______________________ event.
We call the sample space S a certain event because the probability that it will happen is equal to ______ ( a number )
When we say two events A and B are mutually exclusive, we mean that P ( A ∩ B ) = __________.
When all sample points are equally likely to occur ( elementary events ) , we say the sample space has
___________________ probability.
2.
(HINT: equal is not an answer)
Suppose a 12-sided die is rolled. If the faces on the die are labeled; 1, 1, 1, 2, 3, 3, 4, 4, 4, 4, 5, 6 then
a) Write a sample space for this experiment
S=
b) does your experiment have uniform probability ?
3.
A class consists of 40 students. Twenty-five of the students came early to class, 30 of the students live on campus
38 of the students either came early or live on campus. How many do
a) both ? _____________________
b) do neither ?
c) If a student is selected at random, what is the probability that the student is early ? ____________
d) if the student is known to have come early, what is the probability that the student lives on campus ? ______
Name __________________________________________ Math 1312 – Quiz #5 – Feb. 5, 2004
1.
If E is an event so that P(E ) = 1, we call E a certain event and E = __________
2.
True or False.
If E = { s1, s2, s3 } and S = { s1, s2, s3, s4 }
_________
a) , then P(E) = ¾
_________
b) If F = { s4 }, then F is an elementary event.
3. Shade the se that corresponds to
a) A/  B/
4.
b)
A–B
Given the following Venn diagram find the indicated probabilities.
a) P ( A ) = _____________
0.20
b) P( A B ) = ______________
c) P( A / ) = _____________
5.
Complete the formula:
for any two sets A and B, n( A  B ) = ______________________________
0.1
0.5
Name _______________________________________ Math 1312 – quiz #6, Feb. 10, 2004
1.
Complete the formulas.
a) P ( A  B ) = ______________________________________
b) P ( A/ ) = __________________
c) P( A | B ) = _______________
2. If A and B are mutually exclusive, then A  B = ____________
and P ( A | B ) = ____________
If A and B are independent, then P ( A ∩ B ) = _____________
3. Let P(A ) = 0.4, P ( B ) = 0. 8, and P (A  B ) = 0. 9 . Find
a) P ( A ∩ B ) . ____________
b) P( A | B ) . ________________
c) P (B | A ). _____________
d) Are A and B mutually exclusive ? why or why not ? ________________________________________
e) Are A and B independent ? Why or Why not ? ______________________________________________
4. Draw a picture of two sets ( Venn Diagram) that illustrate two sets that are definitely independent.
5.
A loaded coin is tossed five times. What is the probability that all five tosses are heads ? ( Assume a head occurs
an average of 2 out 7 seven times)
__________
Name ______________________________________________ Math 1312 – Quiz #7 – Feb. 12, 2004
1. Complete the formulas. Do not include extra – parts.
a) If A and B are mutually exclusive then P ( A  B ) = _______________________________ in simplest form.
b) For any two nonempty sets,
P ( A ∩ B ) = P ( A ) • P ( B ). True or false. → _________________
c) If A and B are independent, then P ( A ∩ B ) = ______________
2.
There are two choices for chairman of the board. The probability that the first member interviewed will be
selected is 0. 4, while the second one has a probability of 0. 3. The remaining probability would be attributed to
continue to find a new chairman. What is the probability that one or the other member will be selected ?
_____________
3.
Every year a member of the local union is selected to represent them. Based on previous results, a woman has a
probability of 0.4 of being selected. Assuming that a woman is running for the position in each of the next three
years, what is the probability that
a) no woman is selected during any of the three years ? ______________________
b) a woman is selected in at least one of the years ? _________________
4.
Given that P( A ) = 0. 4 , P ( B ) = 0.3, with P( A  B ) = 0. 7
are A and B mutually exclusive ? → _________________
5.
Let P(A ) = 0. 4 and P(B ) = 0.6 give me a numerical ( number ) reason as to when A and B are independent ?
Why ? →
6. Assume that P(E) = 0.2 and P( F ) = 0. 6
a) If E and F are mutually exclusive, then P( E | F ) =____________
b) If E and F are independent, then P( E | F ) = _______________
Name _______________________________________ Math 1312 – Quiz #8 – Feb. 19, 2004
1. A = { 1, 2,3 }
B = { 1, 4, 5 }
A  B = _________
A x B = _________________________
n ( A x B x A ) = _______________________
2. Set of Whole numbers = _____________________
3. Set of positive integers = ___________________________
4. Which term describes P( A ∩ B ) = P (A ) P( B ) ? _______________________
5. When is P( A ∩ B ) = P(A ) P(B | A ) ? ______________________
6. What does a sample space look like for an experiment with five distinct simple outcomes ?
7. True or false
_____________ n( A  B ) = P(A ) + P( B) - P ( A ∩ B )
______________ n(S ) = 1
8. Use the tree diagram to find
a)
b)
c)
d)
Name ________________ _______________________ Math 1312 – Quiz # 9 , Feb. 24, 2004
No Calculators
1. Find 7 ! __________________
2.
3.
4.
0 ! = ____________
1 ! = _____________
n ! • ( n+1) = ______________
(n+1) ! / n ! = ___________
2000 ! / 1999 ! = ______________
5. A job consists of three different tasks. Task one can be done in any one of four ways. Task two can be done in any one of
three ways , and Task three can be done in any one of five ways. If all three tasks are to be completed, then how many
different ways are there for the job to be done.
6.
A die is rolled, a coin is flipped, and a card is selected from a standard deck of cards. If the three results are recorded,
then how many different outcomes are possible ?
Name ____________________________________ Math 1312 – Quiz #10
1) Which of these best satisfies the statement: “ an arrangement of objects in which order does not matter “
permutations
combinations → _________________
2) A three digit number is to be created by using any of the digits { 2, 4, 6, 7, 8} . How many three digit numbers are
possible if
a) no repetitions are allowed ? → _________________
b) repetitions are allowed ? → _________________
3)
A club is voting to select a President, a VP, and a Treasurer. If they are selected in that order and there are 12
members wishing to serve in any of the three positions, then how many different ways can the three be selected ?
_______________
4) A jury of 12 is to be selected from a group of 20 → is this a combination or a permutation problem ?
_______________
5) Find
a) P ( 32, 1) = ____________
C( 100, 2) = ________________
6) A six sided die is rolled three times.
a) how many sequences are possible ? _______________
b) how many with exactly one six ? _______________
c) What is the probability that in the three rolls exactly one of the rolls is a six ? ____________
Additional Properties of Permutations and Combinations
Formulas for
P(n,r) = ________________ = ________________
C(n,r) = _________________ = _______________ = _________________
Properties
1) 0 ! = _________
1 ! = ___________
2) P(n,1) = __________
C(n, 1) = ______________
3) P(n,n) = __________
4) C(n,n) = ____________
5) C(n,r) = C(n,s) provided that _______________
Name _____________________________________________ Math 1312 – Quiz #11 – March 2, 2004
1.
A department consists of 10 men and 6 women. A group of four is selected to attend a demonstration of a new
product. The selection is done at random.
a) How many different groups are possible ? ________________________
b) How many with exactly three men ? _____________________
c) How many if no man attends ? ________________
d) What is the probability that no man attended if everybody was eligible to go ? _________________
2.
A group of three individuals walk into a theater to watch a movie. How many different seating arrangements are
possible if there are 20 rows with 15 chairs in each row ?
___________
3. A student takes a five problem multiple choice quiz (three choices per problem). Since the brain bug has been
eating his memory he forgot to study and must guess randomly. What is the probability that all of the questions
are answered correctly ?
___________
What is the probability that he answers at least one correctly ? _____________
4. A four-sided die is rolled, a pair of coins is tossed, a number between 0-11 is chosen ( 1 through 10).
How many different outcomes are possible? ________________
5. You want to create all possible three digit numbers using { 1, 2,3,4, 5 } without repetitions.
that the number you create will be an even number ?
6.
What is the probability
A class of 10 students take a quiz. The quiz is graded and then the group that fails must stay in class – the others
leave. How many different groups could possibly stay after class if you know that at least one failed ?
_________
Name ____________________________________ Math 1312- Quiz #12, March 4, 2004
1. Write down the 3x2 zero matrix.
2. Write down the the 3x3 identity matrix.
3.
Find the product of
 3 0


b) 0 0 0 •  2 4 =


 5 7
1 0  3 
a) 
 •  =
0 1    2 
4. Perform the given operation – write in simplest form. If not possible, then write NP.
2
 
a)  1
 
 3 
b)
-
2  1
0 1  •


1
 2  3 =
b) -3
2
5 
 
3
4 =
 
5. If the initial matrix I = [ 0.4
0.6] and the Transition matrix T is represented by
0.3 0.7
0.6 0.4


What does the product IT represent ?
6.
A Bernoulli experiment has two types of outcomes
a _______________________ or a _______________
7.
A student is selected to write up a problem on the board. You want to find the probability that the student is you.
Is this a Bernoulli experiment? Is yes, then identify the success and the success probability if the class consists of
students.
Name _________________________________________ Math 1312 – quiz #13, March 9, 2004
1.
Write down the Binomial formula.
2. A four sided die is rolled five times. The die is fair and is labeled with 1, 2, 3, 4 (one per face-side)
The die is rolled five times. What is the probability
a) none are four’s ? ________________
b) at least one is a four ? ______________
c) exactly two are four’s ? ______________
3. A card is drawn from a standard deck of 52. The card is replaced and a second card is drawn (after shuffling).
This pattern continues. A total of four cards are selected. What is the probability that
exactly three of the four selections were aces ? ______________
4. Four cards are selected from a standard deck of cards and kept together in your hand ( four-card hand). What is
the probability that three of the four are aces ?
Name ________________________________________ Math 1312 – Quiz #14, March 11, 2004
1.
A local weatherman ( although it could have been a woman – this case it turned out to be a man) predicted a 30 %
chance of rain. During the year there have been 60 such days with the exact same conditions.
a) How many days do you expect rain to have fallen ? ____________
a) what is the probability that rain will fall on 20 out of the 60 days ? ____________
2. Given the data 10, 20, 10, 20 what is
a) the arithmetic mean ? _____________
median ? ______________
3. Given the data
-1 with frequency 12
0 with frequency 64
and 2 with frequency 24
What is the average squared-deviation ? __________
4. The probability that a student will answer question #5 correctly is 0.8 There are 30 such students ( same type)
all provided an answer that is independent of what the other students answered. What is the probability that
out of the 30 students exactly 20 will answer the question correctly ?
_______________________________
5.
What is the probability that in the problem above at least one of the students will answer the question correctly ?
_________________
6. How many exams are expected to take in this class (excluding the final exam ) after spring break ? _______
7.
Evaluate the Binomial formula with n = 100, x = 12, and p = 3 % . ___________________
( answer to the nearest hundredth )
Name ___________________________ Quiz #15
Name _____________________________________ Math 1312 – Quiz #16, April 1, 2004
1.
2.
Write down the binomial formula.
P ( x successes ) =
nCx
• ___________
A loaded coin is tossed. It is known that a head is three times more likely to occur than a tails.
Let the r.v. X represent the number of heads in four tosses of this coin.
a) What are all the possible values of X: ______________________________
b) What is the probability distribution of X ?
X=x
P(X = x)
3. Given the data; 2, 3, 5, 2 . Find the arithmetic mean and the variance. Assume that this is the entire population.
mean = _________
variance = __________
4. If you were trying to find the average height of the students at ASU (all 6000 students),
describe how you would do it.
5. Given the following r.v. and its probability distribution.
X=x
P(X =x )
==============================
-2
1/10
---------------------------------------------- ---0
2/10
--------------------------------------------------1
3/10
----------------------------------------------------2
4/10
-----------------------------------------------------
Find the expected value of X. ___________
Find its variance. ______________
Draw a histograph of X and its probability distribution.
Name ___________________________________________ Math 1312 – Quiz #17
1. If an experiment is a binomial experiment, we can easily find its expected value and its variance by
a) E(X) =
b) Variance =
2.
Find the expected value and variance of each of the following cases.
a) a coin is tossed twice; if a head comes up you get $2, if two heads come up , you get $ 5. How much should you
pay to play this game in order for the game to be fair ( E(X) = 0 ) ( get = “win” = )
b) The prob. of a crash on a rainy day is 0.001. Four thousands cars were driven on this day. What is the
expected value of accidents ? What is the variance ?
c) A man buys a $100,000 policy for $350 per year. The chance that this person will die is 0.002
Find the expected value of this policy in terms of the insurance company.
3. Find the variance of X .
X=x
P( X = x )
======================
0
0.2
------------------------------------------------1
0.3
------------------------------------------------2
0.4
-------------------------------------------------3
0.1
---------------------------------------------------
4.
What is the total area under a normal curve ? _______________
5.
If the mean is 42 and the variance is 16, then where are the inflection points ?
6. Each student has a 7 in 10 chances of answering all questions correctly. A class consists of 30 students. How many
students are expected to answer all questions correctly ?
Name _______________________________________
Math 1312 – Short Quiz #18 April 8, 2004
1.
A normal curve is given with mean = 25 and variance = 9. Find the exact area to the right of 25.
2.
A standard normal curve always has a mean of ____________ and a standard deviation of _______
3.
Find the inflection points of the normal curve in #1 above. ___________ and _____________
4.
5.
Find the area to the right of 0.45 under a standard normal curve. _________________
A normal curve with mean = 100 and variancle = 64 is given. Find the area to the right of 90 under this curve.
6. You are going approximate the P ( X > 20 ) for a binomial experiment ( binomial r.v.) with n = 100 and p = 0.3
What should you select as the mean and the variance of the normal curve used to find the approximation ?
_______________
____________
Name _________________________________________ Math 1312 – Quiz #19, April 13, 2004
1.
A binomial r.v. has values n = 100 and success probability p = ¼.
If we wanted to find P( X > 30 ) by using a normal curve to approximate, then what would
you use as the
a) mean ( µ ) of the normal curve ? µ = _________
b) standard deviation (  ) ?  = ___________
Find P ( X > 30 by using a very rough estimate ) . Area to the left of 30 under the normal curve. __________
Find P( X > 30 ) by using a better estimate. Area to the left of 30.5 ( Why are we using 30.5 ? )
2.
True or False.
___________ a) a relation is always a function
____________ b) the equation ax + by = c represents a line
____________ c ) x = - b/2a represents the vertex of a parabola
____________ d ) f(x) = - 2x2 + 2 is a parabola that open downward
3.
Sketch the graph of
a) 2x – y = 4 by first finding the x intercept_________ and the y-intercept ________ and the slope _________
b) f(x) = - x2 + 4x by finding the vertex ____________
the y-intercept ________ and the x-intercepts ________
Name __________________________________________ Math 1312 – Quiz #20 – April 15, 2004
1.
Sketch the graph of
a) y = 4x
b)
c) g(x) = 2x – 3
e)
d) h(x) = | 2x – 6 |
y = x2 – 2
f) f(x) =
6. What is the domain of the function
a)
b)
7. What is the range of the function
a)
8. What is the x-intercept of
9. What is the y-intercept of
f(x) = log 4 x
b)
x2  4
x2
Name _____________________________ Math 1312 – Quiz #21 – April 20, 2004
1.
If f(x) =
a)
x
, then find
x2
f( 2 ) = ___________
b) f( h ) = __________
c) lim f(x) = _______
x→3
x2
2.
d) lim f(x) = __________
x→ -2
e) lim f(x) = _____
x → 2-
if x ≥ 2
If f(x) =
, then find
x – 1 if x < 2
a)
f( - 2 ) = ___________
c) lim f(x) = _______
x→ 3
3.
b) f( h ) = __________ , if h is a an odd prime number
d) lim f(x) = __________
x→ 2
Use the given graph of the function f(x) to find the values.
e) lim f(x) = _____
x → 2-
f(x)
g(x)
a) f( - 4 ) = __________
b) f( 0 ) = ____________
c) f ( 4 ) = __________
d) lim f(x) = _________
x→ 4
e) lim g(x) = _______
x→-2
f) lim g(x) = ________
x→-2+
Name ____________________________________ Math 1312 – Quiz # 22 – April 22, 2004
1.
Find the slope of the line
2x – 3y = 12. → m = ____________
2.
Let f(x) = x3 – 2x + 2 be given.
a) Find f(1) = __________
b) Find f( 4 ) = ____________
c) Give me two points that f(x) = x3 – 2x +2 passes through. --- I would hope (and will grade it as such ) that you
will select the two most obvious points available.
______________________
d) Take the two points that you selected above and find the slope of the line that connects (passes through) them.
You can have lots of answers – but only one will be graded correctly. ______________
This is called the slope of the secant line passing through your selected points.
3.
Find
lim f(x)
x→c
If f(x) =
2x  4
and
4 x 2  16
a) c = 1 → lim f(x) = ______
x→1
b) c = 2 →
lim
c) c = h →
e) What is f( 0 ) ? ___________
d) c = “∞” →
f) What is f( 2) ? __________
Name __________________________________________ Math 1312 – Quiz #23 – April 27, 2004
1.
We say a function is ______________________ if it has no breaks or gaps
2.
A function is continuous at x = c if
1) _____________ exists
3. Show that f(x) =
2) ________________ exists
3) __________________
x2
is continuous at x = 1.
x2  4
1)
2)
3)
4. What are the points of discontinuity (what x’s ) of each of the following functions
a) y =
x2
→ _____________________
x2  4
x–2
x
→ _______________
3
if x ≥ 3
2 if x ≠ 3
c) y =
d) f(x) =
x2
if x < 3
x
___________________
e)
b) y =
3x
if x > 1
x2 + 2
if x ≤ 1
f(x) =
if x = 3
_______________________
______________________
5. Given f(x) = x 2 – 2x + 1
find f(3) = ____________
f( 5) = __________
Find the slope of the line that connects ( 3, f(3) ) and ( 5, f(5) ) --- the average rate of change – Arc.
m = _____________________
Answers
Quiz # 1
1. 4  A, A
 B
2.
Universal set, subset, integers, rational or irrational
3. { a }, { b }, { c } , { a, b }, { a, c } , { b, c }, {a, b, c }, 
4. “smart” is not well-defined
5.
A = { } or 
B={2}
U = { 0, 1, 2, 3, 4 }
6. a) True
b) True
c) True
d) false
=================================================================================
Quiz #2
1. disjoint sets
2. complement of A, do not say “ A prime or the prime of A”
3. a) True
b) true
4. A x B = { (1, a), (1, b), (1, c), (2, a), (2,b), (2,c) }
b)  / = U
5. ( A / ) / = A
c) ( A  A / ) = U
6. U = { 1, 2, 3, 4 } , A = { 3, 4 } , B = { 2 }
a) A  B = { 2, 3, 4 }
b) A ∩ B =  or { }
d) ( A/ ∩ B/ ) / = A  B
c) B / = { 1, 3, 4 }
7. a) A ∩ B = shade the intersection of A and B – what they have in common.
b) A/ = shade the outside of A
8. Use a Venn Diagram: n( F ) = # of female students = 20 , n( P ) = # of students that use a pencil = 15
n ( F ∩ P ) = 6 → this gives you the intersection
F
P
14
6
9
answer: 1 student is neither female nor writes with a pencil.
===========================================================================
Quiz #3
1. If x + 2/3 + 1/5 = 1, then x = ________________ ans. x = 2/15
2.
The teacher gives five exams in class. Each is worth the same and the class grade will consist only on these five
exams. Write the value of each exam in terms of the whole grade as a fraction.
ans. five exams = 100 % → each exam is 20 % or 1/5 of the grade
3.
A class consists of 20 students , 12 are male and 8 are female. A student is selected at random. What is the
likelihood that the student selected is male- as a fraction.
ans. 12/20 or 6/10 or 3/5
4.
An experiment has four individual outcomes ( say a roll of a four sided die).
The set S = { s1, s2, s3, s4 } is called the
________________ ans. sample space
The set is said to have ________________ probability if each of the outcomes {s1}, {s2}, {s3}, {s4} is equally likely
to occur. Those events just listed are called
_____________________ events.
ans. uniform probability ,
5.
elementary events
If an experiment S = { s1, s2, s3, s4, s5, s6, s7, s8, s9, s10 } consists of outcomes that are all equally likely to occur,
then likelihood that s1 will occur is
_____________ ans. 1/10
6. How many different events are possible from the sample space S = {s1, s2 } ? ___________ ans. 2 2 = 4
7. In probability we do not say sets are disjoint, we say the sets are _________________ ans. mutually exclusive
Quiz #4– Feb. 3, 2003
1. An event that can never happen is called an _______________________ event .
ans. impossible
We call the sample space S a certain event because the probability that it will happen is equal to ______ ( a number )
answer:( 1 )
When we say two events A and B are mutually exclusive, we mean that P ( A ∩ B ) = __________. ans. ( 0 )
When all sample points are equally likely to occur ( elementary events ) , we say the sample space has
___________________ probability.
(HINT: equal is not an answer)
ans. (uniform)
2.
Suppose a 12-sided die is rolled. If the faces on the die are labeled; 1, 1, 1, 2, 3, 3, 4, 4, 4, 4, 5, 6 then
a) Write a sample space for this experiment
S=
possible ans. S = { 1, 2, 3, 4, 5, 6 }
b) does your experiment have uniform probability ?
mine does not → the probability distribution would be
P(1) = 3/12, P(2) = 1/12 , P( 3) = 2/12, P(4) = 4/12, P(5) = 1/12, P(6) = 1/12
3.
A class consists of 40 students. Twenty-five of the students came early to class, 30 of the students live on campus
38 of the students either came early or live on campus. How many do
a) both ? _____________________
ans. Let E = set of students that came early and C = set of students that live on campus
n( E) = 25, n(C) = 30 , n( E  C ) = 38 → so n( E ∩ C) = ?
38 = 25 + 30 - ? → n( E ∩ C ) = 17
b) do neither ?
ans.
total – n( E  C ) = 40 – 38 = 2
c) If a student is selected at random, what is the probability that the student is early ? ____________
ans. 25/40
d) if the student is known to have come early, what is the probability that the student lives on campus ? ______
ans 17/25
Name __________________________________________ Math 1312 – Quiz #5 – Feb. 5, 2004
1.
If E is an event so that P(E ) = 1, we call E a certain event and E = __________
ans. E = S
2.
True or False.
If E = { s1, s2, s3 } and S = { s1, s2, s3, s4 }
_________
a) , then P(E) = ¾
_________
b) If F = { s4 }, then F is an elementary event.
ans. a ) false → it is only true if S has uniform probability
b) true → F contains a single sample point
3. Shade the se that corresponds to
a) A/  B/
b)
ans. you should shade everything outside
A∩B
4.
A–B
ans. you should shade everything that inside of A
except for the part that can be included in B
Given the following Venn diagram find the indicated probabilities.
a) P ( A ) = _____________ ans. 0.3
0.20
b) P( A B ) = ______________ ans. 0.8
c) P( A / ) = _____________ ans. 0.7
5.
Complete the formula:
for any two sets A and B, n( A  B ) = ______________________________
ans.
n( A  B ) = n(A ) + n(B) – n( A∩ B )
0.1
0.5
Name _______________________________________ Math 1312 – quiz #6, Feb. 10, 2004
1.
Complete the formulas.
a) P ( A  B ) = ______________________________________ ans. P( A  B ) = P(A ) + P( B ) – P( A ∩ B )
b) P ( A/ ) = __________________ ans. P( A / ) = 1 – P( A )
c) P( A | B ) = _______________ ans. P ( A | B ) = P( A ∩ B ) / P( B )
2. If A and B are mutually exclusive, then n(A  B) = ____________ ans. n( A  B ) = n(A ) + n(B) – n( A ∩ B )
and P ( A | B ) = ____________ ans. P( A ∩ B ) =0
If A and B are independent, then P ( A ∩ B ) = _____________ ans. P( A ∩ B ) = P(A ) P(B )
3. Let P(A ) = 0.4, P ( B ) = 0. 8, and P (A  B ) = 0. 9 . Find
a) P ( A ∩ B ) . ____________ ans. P( A  B ) = P(A ) + P( B ) – P(A ∩ B) →
0.9 = 0.4 + 0.8 – P( A ∩B) → P( A ∩ B ) = 0.3
b) P( A | B ) . ________________
ans. P( A| B ) = P( A ∩ B ) / P( B )
= 0.3 / 0.8 = 3/8
c) P (B | A ). _____________
ans. P( B | A ) = P( B ∩ A ) / P(A )
= 0.3/0.4 = 3/4
d) Are A and B mutually exclusive ? why or why not ? ________________________________________
ans. NO – P( A ∩ B ) ≠ 0
e) Are A and B independent ? Why or Why not ? ______________________________________________
NO - P ( A) • P( B ) = 0.4 • 0.8 = 0.32 ≠ P( A ∩ B )
4. Draw a picture of two sets ( Venn Diagram) that illustrate two sets that are definitely independent.
ans. actually need two sets that intersect with very specific values such as
P ( A ) = 0. 4, P( B) =0.5, and P( A ∩ B ) = 0.2 → if you label the values in all four regions correctly, you have
independent events.
5.
A loaded coin is tossed five times. What is the probability that all five tosses are heads ? ( Assume a head occurs
an average of 2 out 7 seven times) → 2/7 • 2/7 • 2/7 • 2/7 • 2/7 = ( 2/7)5
__________
Quiz #7 – Feb. 12, 2004
1. Complete the formulas. Do not include extra – parts.
a) If A and B are mutually exclusive then P ( A  B ) = _______________________________ in simplest form.
ans. P( A  B ) = P(A ) + P(B)
b) For any two nonempty sets, P ( A ∩ B ) = P ( A ) • P ( B ). True or false. → _________________
ans. false → only true if A and B are independent
c) If A and B are independent, then P ( A ∩ B ) = ______________
ans. P( A ∩ B ) = P(A) • P(B )
2.
There are two choices for chairman of the board. The probability that the first member interviewed will be
selected is 0. 4, while the second one has a probability of 0. 3. The remaining probability would be attributed to
continue to find a new chairman. What is the probability that one or the other member will be selected ?
_____________
ans.
3.
P ( one or the other ) = P( one) + P(other ), mutually exclusive since they both can not be chairman
= 0.4 + 0.3 = 0.7
Every year a member of the local union is selected to represent them. Based on previous results, a woman has a
probability of 0.4 of being selected. Assuming that a woman is running for the position in each of the next three
years, what is the probability that
a) no woman is selected during any of the three years ? ______________________
ans. P( W/ ∩ W/ ∩ W/ ) = P( W/ ) • P( W/ ) • P(W/) = (0.6) (0.6)(0.6) = _________
b) a woman is selected in at least one of the years ? _________________
ans. 1 - P( no woman) = 1 – (0.6)3 = _______
4.
Given that P( A ) = 0. 4 , P ( B ) = 0.3, with P( A  B ) = 0. 7
are A and B mutually exclusive ? → _________________
ans. yes - P ( A  B ) = P(A) + P( B ) – P(A ∩ B ) but → P( A ∩ B ) = 0 → mutually exclusive
5.
Let P(A ) = 0. 4 and P(B ) = 0.6 give me a numerical ( number ) reason as to when A and B are independent ?
Why ? → ans. independent if P ( A ∩ B ) = P(A) • P(B) .
So, A and B are independent if (0.4) (0.6) = 0.24 = P( A ∩ B )
6. Assume that P(E) = 0.2 and P( F ) = 0. 6
a) If E and F are mutually exclusive, then P( E | F ) =____________ ans. P( E | F ) = 0
b) If E and F are independent, then P( E | F ) = _______________ ans. P ( E | F ) = P(E ).
Quiz #8 – later
Quiz #9 Name ________________ _______________________ Math 1312 – Quiz # 9 , Feb. 24, 2004
No Calculators
1. Find 7 ! __________________
2.
Ans. 7 ! = 7 •6•5•4•3•2•1 = __________
0 ! = ____________
3.
1 ! = _____________ ans. 1 for both of them
n ! • ( n+1) = ______________
4.
(n+1) ! / n ! = ___________
ans. (n+1) !
ans. (n+1)
2000 ! / 1999 ! = ______________ ans. 2000
5. A job consists of three different tasks. Task one can be done in any one of four ways. Task two can be done in any one of
three ways , and Task three can be done in any one of five ways. If all three tasks are to be completed, then how many
different ways are there for the job to be done.
ans. (4)(3)(5) = 60
6.
A die is rolled, a coin is flipped, and a card is selected from a standard deck of cards. If the three results are recorded,
then how many different outcomes are possible ?
ans. (6) (2)( 52) = 624
Quiz #10
1) Which of these best satisfies the statement: “ an arrangement of objects in which order does not matter “
permutations
combinations → ans. combination
2) A three digit number is to be created by using any of the digits { 2, 4, 6, 7, 8} . How many three digit numbers are
possible if
a) no repetitions are allowed ? → _________________ ans. 5 •4•3 = 60
b) repetitions are allowed ? → _________________ ans. 5 •5•5 = 125
3)
A club is voting to select a President, a VP, and a Treasurer. If they are selected in that order and there are 12
members wishing to serve in any of the three positions, then how many different ways can the three be selected ?
_______________ ans. 12 •11 •10 = 1320
4) A jury of 12 is to be selected from a group of 20 → is this a combination or a permutation problem ?
_______________ ans. a combination problem
5) Find
a) P ( 32, 1) = ____________
ans. 32
C( 100, 2) = ________________
ans. 100 • 99 /2= 9900/2 = 4950
6) A six sided die is rolled three times.
a) how many sequences are possible ? _______________ ans. 6 •6•6 = 216
b) how many with exactly one six ? _______________ 3( 1•5•5) = 75 ( you have to multiply by 3 because
you do not know where the six – in which of the three spots
c) What is the probability that in the three rolls exactly one of the rolls is a six ? ____________ans. 75/216
Name ____________________________________ Math 1312 – Quiz #10
1) Which of these best satisfies the statement: “ an arrangement of objects in which order does not matter “
permutations
combinations → _________________ ans. combinations
2) A three digit number is to be created by using any of the digits { 2, 4, 6, 7, 8} . How many three digit numbers are
possible if
a) no repetitions are allowed ? → _________________ ans. 5 • 4 • 3 = 60
b) repetitions are allowed ? → _________________ ans.: 5•5 • 5 = 125
3)
A club is voting to select a President, a VP, and a Treasurer. If they are selected in that order and there are 12
members wishing to serve in any of the three positions, then how many different ways can the three be selected ?
_______________ ans. 12 •11• 10 = 1320
4) A jury of 12 is to be selected from a group of 20 → is this a combination or a permutation problem ?
_______________ ans.: combination – order does not matter
5) Find
a) P ( 32, 1) = ____________ ans. 32
C( 100, 2) = ________________ ans. 100 •99 /2 = 9900/2 = 4950
6) A six sided die is rolled three times.
a) how many sequences are possible ? _______________ ans.: 6 • 6 • 6
b) how many with exactly one six ? _______________ ans.: ( 3 )(1 • 5 • 5 ) = 75
c) What is the probability that in the three rolls exactly one of the rolls is a six ? ____________ ans.: 75/216
Name _____________________________________________ Math 1312 – Quiz #11 – March 2, 2004
1.
A department consists of 10 men and 6 women. A group of four is selected to attend a demonstration of a new
product. The selection is done at random.
a) How many different groups are possible ? ________________________ ans.: C(16, 4) = ____
b) How many with exactly three men ? _____________________ ans.: C(10,3) • C(6,1) = ________
c) How many if no man attends ? ________________ ans.: C(10,0) • C(6,4) = _________
d) What is the probability that no man attended if everybody was eligible to go ? ____________
ans.: C(10,0)•C(6,4) / C(16,4)
2.
A group of three individuals walk into a theater to watch a movie. How many different seating arrangements are
possible if there are 20 rows with 15 chairs in each row ?
___________
ans.:
total of 20 • 15 chairs = 300 chairs → three people → 300 • 299 • 298 = ___________
3. A student takes a five problem multiple choice quiz (three choices per problem). Since the brain bug has been
eating his memory he forgot to study and must guess randomly. What is the probability that all of the questions
are answered correctly ?
___________
ans.:
E = event all questions are correct → P ( E ) = (1 • 1 • 1 • 1 •1 ) / (3 • 3 • 3 •3 •3 ) = 1 / 35 =
What is the probability that he answers at least one correctly ? _____________
ans.:
1 -
25
= _________
35
4. A four-sided die is rolled, a pair of coins is tossed, a number between 0-11 is chosen ( 1 through 10).
How many different outcomes are possible? ________________
ans.: 4 • 4 • 10 = 160
5. You want to create all possible three digit numbers using { 1, 2,3,4, 5 } without repetitions.
that the number you create will be an even number ?
What is the probability
ans.: 4•3•2 = 24 , n(S) = 5 • 4• 3• = 60 → P ( even # ) = 24 / 60
6.
A class of 10 students take a quiz. The quiz is graded and then the group that fails must stay in class – the others
leave. How many different groups could possibly stay after class if you know that at least one failed ?
_________
ans.: 2 10 - 1, you are looking for all possible subsets ( 2n ) but excluding the empty set.
Name ____________________________________ Math 1312- Quiz #12, March 4, 2004
1. Write down the 3x2 zero matrix.
0 0 


ans. 0 0


0 0
1 0 0


2. Write down the the 3x3 identity matrix. ans.: 0 1 0


0 0 1
3.
Find the product of
1 0  3 
3
a) 
• 
= ans. 



0 1    2 
 2
 3 0


b) 0 0 0 •  2 4 = ans. 0 0


 5 7
4. Perform the given operation – write in simplest form. If not possible, then write NP.
2
 
a)  1
 
 3 
-
1
 2  3 =
b) -3
2
 5  = ans.
 
6
 15


ans. NOT POSSIBLE – NP
b)
2  1
0 1  •


3
2
 4  = ans.  4 
 
 
5. If the initial matrix I = [ 0.4
0.6] and the Transition matrix T is represented by
0.3 0.7
0.6 0.4


What does the product IT represent ?
ans. The probabilities in the next (2nd) iteration
6.
A Bernoulli experiment has two types of outcomes
ans.: a success and a failure
7.
A student is selected to write up a problem on the board. You want to find the probability that the student is you.
Is this a Bernoulli experiment? Is yes, then identify the success and the success probability if the class consists of
30 students. Bernoulli Exp. : Look at one student only
ans.: yes, it is a Bernoulli Experiment → s = success = the event that you are selected → p = 1/20 if done at random
Math 1312 – quiz #13, March 9, 2004
1.
Write down the Binomial formula.
ans.:
C(n,x) px qn-x
2. A four sided die is rolled five times. The die is fair and is labeled with 1, 2, 3, 4 (one per face-side)
The die is rolled five times. What is the probability
a) none are four’s ? ________________
s= event a four comes up
p= ¼→q=¾
→ P ( x = 0 ) = C ( 5, 4 ) (1/4)0 ( ¾)5
b) at least one is a four ? ______________
ans.: P ( x ≥ 1 ) = P ( x = 1 ) + P(x=2) + P(x=3) + P(x=4)+P(x=5)
= 1 – P(x=0 ) =
= 1 - C ( 5, 4 ) (1/4)0 ( ¾)5
c) exactly two are four’s ? ______________
ans.: P ( x = 2 ) = C (5, 2) (1/4) 2 (3/4)3
3. A card is drawn from a standard deck of 52. The card is replaced and a second card is drawn (after shuffling).
This pattern continues. A total of four cards are selected. What is the probability that
exactly three of the four selections were aces ? ______________
s = event an ace is drawn → p = 4/52 = 1/13 → q = 12/13
n = 4, x = 3
ans.: C(4, 3) ( 4/52)3 ( 48/52)1
4. Four cards are selected from a standard deck of cards and kept together in your hand ( four-card hand). What is
the probability that three of the four are aces ?
ans.:
not a binomial experiment → P ( three are aces and one is not )
=
C (4,3)  C (48,1)
C (52,4)
Name _____________________________________ Math 1312 – Quiz #16, April 1, 2004
1.
2.
Write down the binomial formula.
P ( x successes ) =
nCx
• ___________
answer: nCx • px qn-x
A loaded coin is tossed. It is known that a head is three times more likely to occur than a tails.
Let the r.v. X represent the number of heads in four tosses of this coin.
a) What are all the possible values of X: ______________________________ ans. : 0, 1, 2, 3, 4
b) What is the probability distribution of X ?
X=x
P(X = x)
0
C(4,0)(3/4)o(1/4)4 =
1
C(4,0)(3/4)1(1/4)3 =
2
C(4,0)(3/4)2(1/4)2 =
3
C(4,0)(3/4)3(1/4)1 =
4
C(4,0)(3/4)4(1/4)0 =
3. Given the data; 2, 3, 5, 2 . Find the arithmetic mean and the variance. Assume that this is the entire population.
mean = _________
variance = __________
x = (2 + 3 + 5 2 ) / 4 = 12 /4 = 3
variance = (average squared deviation) =
(2  3) 2  (3  3) 2  (5  3) 2  (2  3) 2
= 6/4 = 3/2
4
4. If you were trying to find the average height of the students at ASU (all 6000 students),
describe how you would do it.
take a sample of the students (random – about 50 to 100 ) – find their ht. , average their heights.
5. Given the following r.v. and its probability distribution.
X=x
P(X =x )
xp
x2p
=======================================
-2
1/10
-2/10
4/10
---------------------------------------------- -------------------0
2/10
0
0
--------------------------------------------------------------------1
3/10
3/10
3/10
--------------------------------------------------------------------2
4/10
8/10
16/10
----------------------------------------------------- ---------------
 xp = 9/10
x
2
p = 23/10
Find the expected value of X. ___________ ans.: = 9/10
Find its variance. ______________ ans.: 23/10 – (9/10)2
Draw a histograph of X and its probability distribution.
Name ___________________________________________ Math 1312 – Quiz #17
1. If an experiment is a binomial experiment, we can easily find its expected value and its variance by
a) E(X) = ans.: E(X) = np
b) Variance = ans.: Variance = npq
2.
Find the expected value and variance of each of the following cases.
(win = you get your money back plus the amount shown)
a) a coin is tossed twice; if a head comes up you win $2, if two heads come up , you get $ 5. How much should you
pay to play this game in order for the game to be fair ( E(X) = 0 )
ans.: three types of outcomes : + 2, + 5, - x ( the amount that it costs to play the game )
E(X) = 2( 2/4) + 5 (1/4 ) - x ( ¼ ) = 17/4 - x/4. If the game is to be fair, then E(x) = 0 → 17/4 - x/4 = 0
→ 17/4 = x/4 → x = 17. It costs $17 to play the game. --- not very attractive.
b) The prob. of a crash on a rainy day is 0.001. Four thousands cars were driven on this day. What is the
expected value of accidents (expected number of accidents) ? What is the variance ?
E(X) = np = 4000 ( 0.001) = 4 → variance = npq = 4000 ( .001) ( 0.999) = 4 ( 0.999) = 3.996
c) A man buys a $100,000 policy for $350 per year. The chance that this person will die is 0.002
Find the expected value of this policy in terms of the insurance company.
lives, dies → 350 ( .998) - ( 100000-350)(.002) = ______________
3. Find the variance of X .
X=x
P( X = x )
xp
x2p
========================================
0
0.2
0
0
--------------------------------------------------------------------1
0.3
0.3
0.3
--------------------------------------------------------------------2
0.4
0.8
1.6
--------------------------------------------------------------------3
0.1
0.3
0.9
----------------------------------------------------------------------
 xp = 1.4
 xp = 2.8 → variance = ( 2.8) - (1.4)
2
= _________
4.
What is the total area under a normal curve ? _______________ ans.: one unit
5.
If the mean is 42 and the variance is 16, then where are the inflection points ? ans.: at x = 46 and x = 38
6. Each student has a 7 in 10 chances of answering all questions correctly. A class consists of 30 students. How many
students are expected to answer all questions correctly ?
E(X) = np = 30 ( 7/10) = 21
Math 1312 – Short Quiz #18 April 8, 2004
Name _______________________________________
1.
A normal curve is given with mean = 25 and variance = 9. Find the exact area to the right of 25.
ans.: since 25 is the mean: half the area under the curve is to the right and half to the left. The total area is 1 unit
so: the area to the right of 25 is 0.5
2.
A standard normal curve always has a mean of ____________ and a standard deviation of _______
ans.: mean = 0 and a standard dev. = 1
3.
Find the inflection points of the normal curve in #1 above. ___________ and _____________
answer: inflection points at: 22 and 25
4.
Find the area to the right of 0.45 under a standard normal curve. _________________
z = 0.45 →
5.
05
--------------------------------------0.4
1736
Area to the right: 0.5000-0.1736
= 0. 3264
A normal curve with mean = 100 and variance = 64 is given. Find the area to the right of 90 under this curve.
90  100
 10 / 8  1.25 →
answer: z =
8
0
1
2
1.2
3
4
5
0.3944
Area to the right: 0.5000 + 0.3944 = 0.8944
6. You are going approximate the P ( X > 20 ) for a binomial experiment ( binomial r.v.) with n = 100 and p = 0.3
What should you select as the mean and the variance of the normal curve used to find the approximation ?
_______________
answer: np = ( 100 ) ( 0.3 )
= 30
____________
variance: npq = (100)(0.3)(.7) = 21
Name _________________________________________ Math 1312 – Quiz #19, April 13, 2004
1.
A binomial r.v. has values n = 100 and success probability p = ¼.
If we wanted to find P( X > 30 ) by using a normal curve to approximate, then what would
you use as the
a) mean ( µ ) of the normal curve ? µ = _________ answer: np = (100 ) ( ¼ ) = 25
b) standard deviation (  ) ?  = ___________ answer:
npq  (100)(1 / 4)(3 / 4) =
10 3
 4.33
4
Find P ( X > 30 by using a very rough estimate ) . Area to the right of 30 under the normal curve. __________
answer: z =
30  25
= 1.15 (round to nearest hundredth) →0.3749 → Area = 0.5000-0.3749=0.1251
4.33
P(X > 30) ~ 0.1251
Find P( X > 30 ) by using a better estimate. Area to the right of 30.5 ( Why are we using 30.5 ? )
answer: z =
30.5  25
= 1.27 → Area = 0.5000 – 0.3980 = 0.1020
4.33
Compare these two values: 0.1251 -vs – 0.1020 → In terms of probability , there is a difference
( small )
2.
True or False.
___________ a) a relation is always a function ---------- ans.: false – but a function is always a relation
____________ b) the equation ax + by = c represents a line ------ ans.: yes
____________ c ) x = - b/2a represents the vertex of a parabola ------ answer: no, V (x,y) need a y-coordinate
____________ d ) f(x) = - 2x2 + 2 is a parabola that open downward ----- answer: true
3.
Sketch the graph of
a) 2x – y = 4 by first finding the x intercept_________ and the y-intercept ________ and the slope _________
answer: x-int = 2
y-intercept = -4
slope = 2
b) f(x) = - x2 + 4x by finding the vertex ____________ the y-intercept ________ and the x-intercepts ________
x = -b/2a=2, y=4 → V(2,4)
y-int= 0
x-int=0 and 4
Graph: parabola with vertex at V(2,4) opens
downward crossing the
x-axis at 0 and 4
the y-axis at 0
Name __________________________________________ Math 1312 – Quiz #20 – April 15, 2004
1.
Sketch the graph of
a) y = 4x
b)
f(x) = log 4 x
c) g(x) = 2x – 3
d) h(x) = | 2x – 6 |
- 6
| 3/2
x=3
- -3
e)
y = x2 – 2
f) f(x) =
x2  4
x2
2
| -2
- -2
vertex at V(0, -2)
6. What is the domain of the function
a) f(x) = log 4 (x+2) → (x+2) > 0 → x > -2
b)
f(x) =
x2  4
→ all real numbers ≠ -2
x2
7. What is the range of the function
a) y = 4x → all real numbers y, y > 0
8. What is the x-intercept of y = x2 – 4 ?→
b) y = x2 – 4 → all real numbers y, y ≥ - 4
x-intercepts: 2 and – 2
9. What is the y-intercept of h(x) = | 2x - 6 | → y-intercept = 6 ( make sure to correct this in your paper – I gave the
grader the wrong answer.
Name _____________________________ Math 1312 – Quiz #21 – April 20, 2004
1.
If f(x) =
a)
x
, then find
x2
f( 2 ) = ___________
b) f( h ) = __________
ans.: 2/4 = ½
ans.: f(h) =
c) lim f(x) = _______
x→3
ans.: 3/5
x2
2.
d) lim f(x) = __________
x→ -2
ans.: undefined
h
h2
e) lim f(x) = _____
x → 2ans.: 1/2
if x ≥ 2
If f(x) =
, then find
x – 1 if x < 2
a)
f( - 2 ) = ___________
b) f( h ) = __________ , if h is a an odd prime number
answer: (-2) – 1 = - 3
c) lim f(x) = _______
x→ 3
answer: 32 = 9
3.
answer: f(h) = h2
d) lim f(x) = __________
x→ 2
DNE since left and rt limits ≠
Use the given graph of the function f(x) to find the values.
e) lim f(x) = _____
x → 2answer: (2) -1 = 1
f(x)
g(x)
a) f( - 4 ) = __________
answer: 0
d) lim f(x) = _________
x→ 4
answer: 3
b) f( 0 ) = ____________
c) f ( 4 ) = __________
answer: 3
answer: undefined
e) lim g(x) = _______
x→-2
f) lim g(x) = ________
x→-2+
answer: DNE
answer: -3
Name ____________________________________ Math 1312 – Quiz # 22 – April 22, 2004
1.
Find the slope of the line
2x – 3y = 12. → m = ____________ answer:
2.
m = 2/3
Let f(x) = x3 – 2x + 2 be given.
a) Find f(1) = __________ answer: f(1) = (1) 3 - 2(1) + 2 = 1
b) Find f( 4 ) = ____________ answer: f(4) = 4 3 - 2 (4) +2 = 58
c) Give me two points that f(x) = x3 – 2x +2 passes through. --- I would hope (and will grade it as such ) that you
will select the two most obvious points available.
______________________
ans.: (1, 1) and (4, 58 ) --- you do not need additional points , the points you found in parts a and b above
should be enough !!
d) Take the two points that you selected above and find the slope of the line that connects (passes through) them.
You can have lots of answers – but only one will be graded correctly. ______________
This is called the slope of the secant line passing through your selected points.
answer: m =
3.
Find
f (4)  f (1) 58  1 57


= 19
4 1
4 1
3
lim f(x)
x→c
If f(x) =
2x  4
and
4 x 2  16
a) c = 1 → lim f(x) = ______ ans.:
2(1)  4
2

 1/6
4(1)  16  12
x→1
b) c = 2 →
lim
c) c = h →
d) c = “∞” →
ans.: lim
ans.: lim
1
2x  4
 1/8
 lim
2x  4
(2 x  4)( 2 x  4)
2x  4
2h  4
1
 2

2
4 x  16 4h  16 2h  4
ans.: lim f(x) = 0
x→∞
e) What is f( 0 ) ? ___________
f(0) = -4/-16 = ¼ ( this is the y-intercept)
f) What is f( 2) ? __________
f( 2) is undefined
Name __________________________________________ Math 1312 – Quiz #23 – April 27, 2004
1.
We say a function is ______________________ if it has no breaks or gaps
2.
A function is continuous at x = c if
1) _____________ exists
2) ________________ exists
f(c) exists
3. Show that f(x) =
1)
f(x) =
2) lim
answer: continuous
lim f(x) exists
x→c
3) __________________
lim f(x) = f(c)
x→c
x2
is continuous at x = 1.
x2  4
1 2
3

 1
1 4  3
x2
= -1
x2  4
3) answer from #1 = answer from #2 :. continuous at x=1
x→1
4. What are the points of discontinuity (what x’s ) of each of the following functions
a) y =
x2
→ _____________________
x2  4
discontinuous at x = 2, -2
x–2
x
→ _______________
3
discontinuous nowhere: continuous
if x ≥ 3
2 if x ≠ 3
c) y =
d) f(x) =
x2
if x < 3
x
___________________
discontinuous at x = 3
e)
b) y =
3x
if x > 1
x2 + 2
if x ≤ 1
f(x) =
if x = 3
_______________________
discontinuous at x = 3
continuous everywhere.
5. Given f(x) = x 2 – 2x + 1
find f(3) = ____________
f( 5) = __________
Find the slope of the line that connects ( 3, f(3) ) and ( 5, f(5) ) --- the average rate of change – Arc.
f(3) = 9 – 6 +1 = 4, f(5) = 25 – 10 + 1 = 16
m = Arc =
f (5)  f (3) 16  4

6
53
2