Download Math 141 Formulas

Math 141 Formulas
Slope Formula: m 
y y 2  y1

x x2  x1
Point-Slope Formula: y  y1  m( x  x1 )
C(x) = cx + F, the Total Cost function, gives the total cost for manufacturing x units at
a unit cost of c and fixed costs F.
R(x) = px, the Revenue function, gives the total revenue realized from manufacturing
and selling x units at price p.
P(x) = R(x) – C(x) = px – (cx + F) = px – cx – F, the Profit function, gives the total profit
realized from manufacturing and selling x units.
Row-Reduced Form of a Matrix
1. Each row consisting entirely of zeros lies below any other row having nonzero
entries.
2. The first nonzero entry in each row is 1 (called a leading 1).
3. In any two successive (nonzero) rows, the leading 1 in the lower row lies to the
right of the leading 1 in the upper row.
4. If a column contains a leading 1, then the other entries in that column are zeros.
A standard maximization problem is one in which
1) The objective function is to be maximized.
2) All the variables involved in the problem are nonnegative.
3) Each linear constraint may be written so that the expression involving the variables
is less than or equal to a nonnegative constant.
Simple Interest
I = Prt, where I is the interest, P is the principal, r is the interest rate, and t is the time in
years.
A = P(1 + rt), where A is the accumulated amount, P is the principal, r is the interest
rate, and t is the time in years.
Compound Interest
nt
r

A  P1   , where A is the accumulated amount, P is the principal, r is the annual
 n
interest rate, n is the number of compounding periods per year, and t is the time in
years.
A  Pe rt , where A is the accumulated amount, P is the principal, r is the annual interest
rate compounded continuously, and t is the time in years.
n
ref
r

 1    1 , where re is the effective rate of interest, r is the annual
n

interest rate, and n is the number of compounding periods per year.
De Morgan’s Laws
Let A and B be sets. Then, ( A  B) C  AC  B C
and
( A  B) C  A C  B C
n(A  B) = n(A) + n(B) – n(A  B)
n(A  B  C) = n(A) + n(B) + n(C) – n(A  B) – n(A  C) – n(B  C) + n(A  B  C)
The number of permutations of n distinct objects taken n at a time:
P(n, n) = n!
The number of permutations of n distinct objects taken r at a time:
n!
P(n, r) =
(n  r )!
Permutations of n Objects, Not All Distinct:
Given a set of n objects in which n1 objects are alike and of one kind, n 2 objects are
alike and of another kind,…, and, finally, nr objects are alike and of yet another kind so
that
n1  n2  ...  nr  n
then the number of permutations of these n objects taken n at a time is given by
n!
n1!n2 !...nr !
The number of combinations of n distinct objects taken r at a time is given by
n!
C(n, r) =
(where r  n)
r!(n  r )!
P(E  F) = P(E) + P(F) – P(E  F)
P( E C ) = 1 – P(E)
P( E ) 
Number of favorable outcomes in E n( E )

Number of possible outcomes in S
n( S )
Conditional Probability
Number of elements in A  B n( A  B)

Number of elements in A
n( A)
P( A  B)
P( B | A) 
P( A)
P( B | A) 
Product Rule:
P( A  B)  P( A)  P( B | A)
P( E  F  G )  P( E )  P( F | E )  P(G | E  F )
Two events A and B are independent if and only if P( A  B)  P( A)  P( B) .
If E1 , E 2 ,..., E n are independent events, then
P( E1  E2  ...  En )  P( E1 )  P( E2 )  ...  P( En )
Baye’s Theorem
P ( A | D) 
P( A)  P( D | A)
P( A)  P( D | A)  P( B)  P( D | B)  P(C )  P( D | C )
P( A | D) 
Pr oduct of probabilit ies along the lim b through A
Sum of products of the probabilit ies along each lim b ter min ating at D
The expected value of X, E(X), is given by E ( X )  x1 p1  x2 p2  ...  xn pn .
If P(E) is the probability of an event E occurring, then
P( E )
P( E )

1. The odds in favor of E occurring are
1  P( E ) P( E C )
2. The odds against E occurring are
1  P( E ) P( E c )

P( E )
P( E )
[P(E)  1]
[P(E)  0]
Probability of an Event (Given the Odds)
If the odds in favor of an event E occurring are a to b, then the probability of E occurring
a
is P( E ) 
.
ab
A binomial experiment has the following properties:
1. The number of trials in the experiment is fixed.
2. There are two outcomes of the experiment: “success” and “failure.”
The probability of success in each trial is the same.
The trials are independent of each other.
If X is a binomial random variable associated with a binomial experiment consisting of
n trials with probability of success p and probability of failure q, then the mean
(expected value), variance, and standard deviation of X are
 = E(X) = np
Var(X) = npq
 x  npq