Welcome to Wk04 Relations A relation is a correspondence between two sets of information Relations An identifier is paired to
some information about the subject identified Relations In a table: Subject Alice Ben Carlos Delphine
Height 58 53 6 57 Relations These can also be written as ordered pairs
Usually the identifier is first and the information about that subject is second Relations The set or list of identifiers is called the domain Relations The set or list of
information is called the range Relations These ordered pairs form a set Relations The pairs are paired inside parentheses:
S = {(Alice,58),(Ben,53), (Carlos,6), (Delphine,57)} Relations Another graphical way to display a relation: Relations If there is only one piece of
information for each identifier, the relation is called binary Relations If more information is available, it is called n-ary Relations Subject | Height |
Weight Alice 58 170 Ben 53 120 Carlos 6 200
Delphine 57 250 Questions? Functions The term, "function" was coined by Gottfried Wilhelm Leibniz in 1673
Functions Leibnitz observed that in many real-world processes there are inputs and outputs Functions In a factory, the inputs are the components
Functions The output is the final product Functions In formulas, some variables are input variables The thing you are calculating with the formula is the output
Functions Functions IN-CLASS PROBLEMS In the formula: A = r2
What is the input variable? What is the output variable? Functions IN-CLASS PROBLEMS A =
r2 The size of r the radius is the input, it completely determines A the area of the circle (the output) Functions The input variable is ALWAYS put on the x-axis
(horizontal axis) No reason its a TRADITION! Functions IN-CLASS PROBLEMS Which variable goes on the horizontal axis in A = r2
Functions What the output turns out to be depends on what the input is The output is frequently called the dependent variable Functions The input is called the
independent variable because its value comes from an outside (sometimes uncontrollable) source Functions When you have a certain input, you want to be able to forecast exactly what
the output will be Functions You dont want several possible outputs Functions We call these mathematical models deterministic
Functions A special type of mathematical formula was designed for input-outputs: functions Functions For a formula to be a function, every x can have only one y
Functions Example: The set: {(1,3)(2,4)(7,25)} is a function because: every x-value (1,2,7) has only one y-value: 1 3 2 4 7 25
Functions IN-CLASS PROBLEMS Is this set of points a function?
{(0,-2)(5,0)(13,4.2)(8.1,6)} Functions IN-CLASS PROBLEMS Is this set of points a function? {(0,-2)(5,0)(13,4.2)(8.1,6)} YES! Because every x-value has only one y-value:
0 -2 5 0 13 4.2 8.1 6
Functions IN-CLASS PROBLEMS Is this set of points a function? {(0,-2)(5,0)(0,4.2)(8.1,6)}
Functions IN-CLASS PROBLEMS Is this set of points a function? {(0,-2)(5,0)(0,4.2)(8.1,6)} NO! Because when x = 0, the y-value could be -2 or 4.2:
0 -2 or 4.2 5 0 8.1 6
Functions Nikolai Lobachevsky gave the modern "formal" definition of a function as a relation in which every first element has a unique second element (1830-ish)
Functions When you graph a function, none of the points fall on a vertical line a function not a function Functions
To test whether a formula is a function, you graph it and do the vertical line test If it passes the VLT, it is a function Functions IN-CLASS PROBLEMS
Is it a function? Functions IN-CLASS PROBLEMS Is it a function? Functions Naturally, there is a special symbol for functions
Where you used to write y = 8x + 13 A function is written: f (x) = 8x + 13 read as: f of x Functions Alexis Claude Clairaut introduced the familiar notation "f (x)" in 1734
Functions f (x) shows that the formula uses x as the input and that the output depends on it Functions
When you evaluate a function, you need to substitute the given value for x in the formula Functions f (3) means evaluate the function for x=3
Functions IN-CLASS PROBLEMS f (x) = 3x + 7 find f (4) find
f (x+1) find f (-x) Functions IN-CLASS PROBLEMS
If f (x) = x2 - 2x +7 evaluate each of the following: f (-5) f (x+4) f (-x) Functions The legal values for a function are called:
Domain - the xs Range - the ys Functions IN-CLASS PROBLEMS What is the domain for this function: {(0,-2)(5,0)(13,4.2) (8.1,6)}
Functions IN-CLASS PROBLEMS What is the domain for this function: {(0,-2)(5,0)(13,4.2) (8.1,6)} Domain = 0,5,13,8.1
Functions IN-CLASS PROBLEMS What is the range for this function: {(0,-2)(5,0)(13,4.2) (8.1,6)} Functions IN-CLASS PROBLEMS
What is the range for this function: {(0,-2)(5,0)(13,4.2) (8.1,6)} Range = -2,0,4.2,6 Questions? Inverse Functions
Undo a function Inverse Functions A function (x) has an inverse -1 (x) Inverse Functions (x) -1(x)
Inverse Functions The domain of is equal to the range of -1 and viceversa Inverse Functions The xs become the ys and vice versa Inverse Functions
IN-CLASS PROBLEMS Is {(2,5), (7,3)} the inverse of the function: {(5,2), (3,7)}? Inverse Functions IN-CLASS PROBLEMS Given the function :
(x) = {(7,1), (0,5), (6,2), ( 8,3)} is the inverse also a function? Inverse Functions Finding the inverse for formulas 1) 2) 3)
4) Replace "(x)" with "y" Interchange "x" and "y" Solve for "y" Replace "y" in with -1 Inverse Functions Example: (x) = 8x 9
find -1 (x) Inverse Functions Example: (x) = 8x 9 y = 8x 9 x = 8y 9 y = (x + 9)/8 -1 (x) = (x + 9)/8
Inverse Functions IN-CLASS PROBLEMS Find the inverse of the function: (x) = 9x Inverse Functions IN-CLASS PROBLEMS
Find the inverse of the function: (x) = 4x + 12 Inverse Functions Because the domain of is equal to the range of -1 and vice-versa, an inverse will rotate a function:
Inverse Functions The function is always rotated around the 1:1 y=x line Inverse Functions For a function to have an inverse that is also a function, the original
function must pass a horizontal line test Inverse Functions IN-CLASS PROBLEMS Does it have an inverse? Questions?
Sequences 1 2 3 Whats the next item in this sequence? Sequences 1 3 5
7 Whats the next item in this sequence? Sequences Sequences are ordered lists Sequences They dont have to be
numbers Sequences Sequences Sequences IN-CLASS PROBLEMS
Sequence or not? Sequences Each member of a sequence is called a term Sequences Each term is designated by a subscript: a1, a2, a3, ..., an, ...
The 2nd term in the sequence is a2 The nth term is a Sequences IN-CLASS PROBLEMS What is a17 in the sequence: 1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Sequences IN-CLASS PROBLEMS What is a5 in the following sequence: @ $ # % ^ & * ( ~ )
Sequences An infinite sequence goes on forever: a1, a2, a3, ... A finite sequence stops at some point: a1, a2, a3, ..., a12 Sequences IN-CLASS PROBLEMS
Which is the infinite series: 1 2 3 5 8 13 21 2 4 6 8 10 Sequences Numerical sequences are often formed by a formula that uses one term to calculate the next term in
the sequence Sequences You evaluate a sequence by plugging in the requested numbers for each term Sequences In an explicit formula each term in the sequence will be
based on the subscript (or index) k: ak = k/(k+1) a1 = 1/(1+1) = 1/2 a2 = 2/(2+1) = 2/3 ... Sequences An alternating sequence: ak = (-1)k
a1 = (-1)1 = -1 a2 = (-1)2 = 1 a3 = (-1)3 = -1 a4 = (-1)4 = 1 ... Sequences IN-CLASS PROBLEMS What is a3 if
n = 3 and an = 3 / n ? Sequences IN-CLASS PROBLEMS What is a3 if n = 3 and an = 3 / n ? Plug in n=3! Sequences
IN-CLASS PROBLEMS What is a3 if n = 3 and an = 3 / n ? an = 3 / n a3 = 3 / 3 a3 = 1 Sequences IN-CLASS PROBLEMS
What is an if n=2 a1 = 4 an = an-1 + 1 Sequences IN-CLASS PROBLEMS What is an if
n=2 a1 = 4 an = an-1 + 1 Plug in n = 2 Sequences IN-CLASS PROBLEMS What is an if n=2
a1 = 4 an = an-1 + 1 a2 = a2-1 + 1 a2 = a1 + 1 What is a1? Sequences IN-CLASS PROBLEMS What is an if
n=2 a1 = 4 an = an-1 + 1 a2 = a2-1 + 1 a2 = a1 + 1 a2 = 4 + 1 a2 = 5 Sequences These are called recursion
formulas - the formula calculates each term based on the previous term Sequences (You have to be given a starting term) Questions?
Arithmetic Sequences 1 2 3 Whats the next item in this sequence? 1 3
Arithmetic Sequences 5 7 Whats the next item in this sequence?
1 4 Arithmetic Sequences 7 10 Whats the next item in this
sequence? Arithmetic Sequences These are called arithmetic sequences air-rith-MEH-tic not ah-RITH-meh-tic
Arithmetic Sequences Each term in the sequence (after the first) differs from the preceding one by a constant amount (positive or negative)
Arithmetic Sequences start with a1 increase each time by "d" Arithmetic Sequences IN-CLASS PROBLEMS
start with a1 increase each time by "d" 3 7 11 15 19 What is a1? What is d? Arithmetic Sequences start with a1
increase each time by "d" General term of an arithmetic sequence: an = a1 + (n-1)d Arithmetic Sequences IN-CLASS PROBLEMS an = a1 + (n-1)d
If a1 = 0 and d = 3 what are the first four terms of the arithmetic sequence? Just plug in a1 and d and n= 1,2,3,4 Arithmetic Sequences IN-CLASS PROBLEMS an = a1 + (n-1)d
an = 0 + (n-1)3 a1 = 0 + (1-1)3 = ? a2 = 0 + (2-1)3 = ? a3 = 0 + (3-1)3 = ? a4 = 0 + (4-1)3 = ? Arithmetic Sequences You can calculate the
10,000th term in an arithmetic sequence using the formula without having to list up the 9,999 that come before it! Arithmetic Sequences IN-CLASS PROBLEMS If a1 = 5 and d = 2 what
are the first four terms of the arithmetic sequence? Arithmetic Sequences IN-CLASS PROBLEMS If a1 = 5 and d = 2 what are the first four terms of the arithmetic sequence? an = a1 + (n-1)d
Arithmetic Sequences IN-CLASS PROBLEMS If a1 = 5 and d = 2 what are the first four terms of the arithmetic sequence? an = 5 + (n-1)2 n=?
Arithmetic Sequences IN-CLASS PROBLEMS If a1 = 5 and d = 2 what are the first four terms of the arithmetic sequence? a1 = 5 + (1-1)2 = a2 = 5 + (2-1)2 = a3 = 5 + (3-1)2 = a4 = 5 + (4-1)2 =
Questions? Geometric Sequences What is the next term in this sequence: 1
2 4 8 Geometric Sequences How about this one?
1 3 9 27 Geometric Sequences
Geometric sequences each term in the sequence (after the first) is a common multiple (positive or negative) of the previous term Geometric Sequences
For a geometric sequence, you need: the starting value a1 the multiple r Geometric Sequences IN-CLASS PROBLEMS For the sequence: 1 4 16 64
What is a1? What is r? Geometric Sequences General term of a geometric series: an = a1r n-1
Geometric Sequences IN-CLASS PROBLEMS If a1 = 1 and r = 2, what are the first four terms of the geometric sequence? Geometric Sequences IN-CLASS PROBLEMS
If a1 = 1 and r = 2, what are the first four terms of the geometric sequence? an = a1r n-1 Geometric Sequences IN-CLASS PROBLEMS If a1 = 1 and r = 2, what are the first four terms of
the geometric sequence? an = 1(2 n-1) What is n? Geometric Sequences IN-CLASS PROBLEMS If a1 = 1 and r = 2, what are the first four terms of the geometric sequence?
a1 = 1(2 1-1) = ? a2 = 1(2 2-1) = ? a3 = 1(2 3-1) = ? a4 = 1(2 4-1) = ? Geometric Sequences IN-CLASS PROBLEMS If a1 = 1 and r = 2, what are the first four terms of
the geometric sequence? a1 = 1(2 1-1) = 1(1) = 1 a2 = 1(2 2-1) = 1(2) = 2 a3 = 1(2 3-1) = 1(4) = 4 a4 = 1(2 4-1) = 1(8) = 8 Geometric Sequences IN-CLASS PROBLEMS If a1 = 2 and r = 2, what
are the first four terms of the geometric sequence? Geometric Sequences IN-CLASS PROBLEMS If a1 = 2 and r = 2, what are the first four terms of the geometric sequence? an = a1r n-1
Geometric Sequences IN-CLASS PROBLEMS If a1 = 2 and r = 2, what are the first four terms of the geometric sequence? an = 2(2 n-1) What is n?
Geometric Sequences IN-CLASS PROBLEMS If a1 = 2 and r = 2, what are the first four terms of the geometric sequence? a1 = 2(2 1-1) = ? a2 = 2(2 2-1) = ? a3 = 2(2 3-1) = ? a4 = 2(2 4-1) = ?
Geometric Sequences IN-CLASS PROBLEMS If a1 = 2 and r = 2, what are the first four terms of the geometric sequence? a1 = 2(2 1-1) = 2(1) = 2 a2 = 2(2 2-1) = 2(2) = 4 a3 = 2(2 3-1) = 2(4) = 8
a4 = 2(2 4-1) = 2(8) = 16 Geometric Sequences IN-CLASS PROBLEMS Consider a string of amplifiers each with a voltage gain A If a signal V is applied to the first amplifier, show the
sequence that describes the successive output voltages of each stage: Geometric Sequences IN-CLASS PROBLEMS Consider a string of amplifiers each with a
voltage gain A If a signal V is applied to the first amplifier, show the sequence that describes the successive output voltages of each stage: Sequences IN-CLASS PROBLEMS
Which is geometric? 3 6 9 12 1248 7 10 13 16 2 3 4.5 6.75 2468 Sequences IN-CLASS PROBLEMS
Identify which are arithmetic and geometric series: 1, 5, 25, 125, ... 2, 5, 8, 11, ... 19, 26, 33, 40, ... Sequences IN-CLASS PROBLEMS
Identify which are arithmetic and geometric series: 1, 5, 25, 125, ... geometric 2, 5, 8, 11, ... arithmetic 19, 26, 33, 40, ... arithmetic Questions?
Sequences Limit of a sequence: if the terms of a sequence approach a specific number, we say the limit of the sequence exists and it converges to the number Sequences Example: height of a
bouncing ball Sequences Monotonic either increasing or decreasing only Sequences Non-decreasing each term is greater than or equal to
the preceding term Sequences Non-increasing each term is less than or equal to the preceding term Sequences Sequences
Bounded all terms are less than or equal to some finite number in magnitude Sequences http://math.feld.cvut.cz/mt/txta/1/gifa1/pc3aa1ba.gif http://upload.wikimedia.org/wikipedia/commons/thumb/f/f4/
Sequences Alternating Sequence: Each term: (-1)n an Sequences 5 -5 5 -5 5 -5 5 -5 Sequences IN-CLASS PROBLEMS
Is this an alternating sequence? Sequences Fibonacci Sequence a series of numbers in which each number is the sum of the two preceding numbers 0, 1, 1, 2, 3, 5, 8, ...
Sequences IN-CLASS PROBLEMS What is the next number in the Fibonacci sequence? 112358 Sequences
(Amend 2005) Sequences The Fibonacci sequence first appears in the book Liber Abaci (1202) Leonardo of Pisa (Fibonacci) (1175-1250)
Sequences Fibonacci considers the growth of an idealized rabbit population, assuming: Start with a single newly born pair of rabbits (m/f) Leonardo of Pisa
(Fibonacci) (1175-1250) Sequences Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair
of rabbits Leonardo of Pisa (Fibonacci) (1175-1250) Sequences Rabbits never die and a mating pair always produces one new pair (m/f)
every month from the second month on Leonardo of Pisa (Fibonacci) (1175-1250) Sequences IN-CLASS PROBLEMS How many pairs will there
be in one year? Leonardo of Pisa (Fibonacci) (1175-1250) Sequences How many pairs will there be in one year? Mont
h Pairs 0 1 2 3 4 5 6 7 8 9
0 1 1 2 3 5 8 1 3 2 1 3
4 1 0 5 5 1 1 8
9 12 144 Questions? Series If you combine the numbers in a sequence into a single
number, its called a series Series A sequence is a list of terms A series is a sum of terms Series IN-CLASS PROBLEMS
What is the sum of the first 10 counting numbers? Series IN-CLASS PROBLEMS What is the sum of the first 10 counting numbers? 1+2+3+4+5+6+7+8+9+10
Series IN-CLASS PROBLEMS Heres a trick: 1 2 3 4 5
+ + + + + 10 = 11 9 = 11 8 = 11
7 = 11 6 = 11 So the answer is 5 * 11 = Series Its math class so theres always a new symbol!
called summation means add em up! Series We could have written: What is the sum of the first 10 counting numbers? as: 1 2 310
Series We usually use with a formula: 10 n n=1 This means add up the ns where n goes from 1 to 10
Series We usually use with a formula: 10 upper limit n n=1
lower limit This means add up the ns where n goes from 1 to 10 Series Calculating a series: 5
k=2+3+4+5 k=2 = 14 Series IN-CLASS PROBLEMS 4
In: (n + 1) n=1 what is the lower limit? what is the upper limit? Series IN-CLASS PROBLEMS 4
What is (n + 1) n=1 Series IN-CLASS PROBLEMS 4 What is (n + 1)
n=1 Just start adding up the (n + 1)s Plugging in n=1, n=2, n=3, n=4 Series IN-CLASS PROBLEMS
4 What is (n + 1) n=1 =1+1 +2+1 +3+1 +4+1 =2+3+4+5=?
Series No, all series do not add up to 14 (just a coincidence) Series IN-CLASS PROBLEMS
What is the sum of the first five terms in the arithmetic sequence 1, 4, 7? Series IN-CLASS PROBLEMS What is the sum of the first five terms in the arithmetic sequence 1, 4, 7?
an = a1 + (n-1)d What is a1? What is d? What is n? Series IN-CLASS PROBLEMS What is the sum of the first five terms in the arithmetic
sequence 1, 4, 7? a1 = 1 + (1-1)3 = a2 = 1 + (2-1)3 = a3 = 1 + (3-1)3 = a4 = 1 + (4-1)3 = a5 = 1 + (5-1)3 = Series IN-CLASS PROBLEMS
What is the sum of the first five terms in the arithmetic sequence 1, 4, 7? a1 = 1 + (1-1)3 = 1 a2 = 1 + (2-1)3 = 4 a3 = 1 + (3-1)3 = 7 a4 = 1 + (4-1)3 = 10 a5 = 1 + (5-1)3 = 13 Series
IN-CLASS PROBLEMS So the sum is: 1+4+7+10+13 = 14*2 + 7 = 35 Series Sum of the first n terms of an arithmetic series: Sn = (2a1 + (n-1)d)
For ours: S5 = (2 + 4(3)) = 35 Series IN-CLASS PROBLEMS What is the sum of the first five terms in the geometric sequence 1, 4, 16?
Series IN-CLASS PROBLEMS What is the sum of the first five terms in the geometric sequence 1, 4, 16? an = a1r n-1 What is a1? What is r?
Series IN-CLASS PROBLEMS What is the sum of the first five terms in the geometric sequence 1, 4, 16? an = 1(4 n-1) What is n? Series
IN-CLASS PROBLEMS What is the sum of the first five terms in the geometric sequence 1, 4, 16? a1 = 1(4 1-1) = ? a2 = 1(4 2-1) = ? a3 = 1(4 3-1) = ? a4 = 1(4 4-1) = ? a = 1(4 5-1) = ?
Series IN-CLASS PROBLEMS What is the sum of the first five terms in the geometric sequence 1, 4, 16? a1 = 1(4 1-1) = 1 a2 = 1(4 2-1) = 4 a3 = 1(4 3-1) = 16
a4 = 1(4 4-1) = 64 a = 1(4 5-1) = 256 Series IN-CLASS PROBLEMS What is the sum of the first five terms in the geometric sequence 1, 4, 16? 1 + 4 + 16 + 64 + 256
= 341 Series Sum of the first n terms of a geometric series: Sn = For ours: S5 = = = = 341 Series
Another new symbol: which is the product operator (multiply em all together) Series IN-CLASS PROBLEMS
3 What is n=1 n? Series IN-CLASS PROBLEMS
3 What is n=1 2n ? Series Another new symbol:
n! which is the factorial operator (the product of the integers from 1 to n) Series n! = n(n-1)(n-2)...321 Series
IN-CLASS PROBLEMS What is 5! ? Questions? Series Convergence Not in your book Series Convergence
Not in your book but important for IT Series Convergence Given a sequence of numbers {an}, the sum of the terms of this sequence: a1 + a2 + a3 + ... + an + ... is called an infinite series
Series Convergence The sequence Sn defined by S1 = a1 S2 = a1 + a2 Sn = a1 + a2 + a3 + ... + an n = ak k=1
is the sequence of partial Series Convergence The number Sn is the nth partial sum Series Convergence If the sequence of partial sums converge to a limit L,
then we can say that the series converges and its sum is L Series Convergence If the sequence of partial sums of the series does not converge, then the series diverges
Series Convergence IN-CLASS PROBLEMS How well is y = Approximated by: 1 + x + x 2 + x3 + ? Series Convergence IN-CLASS PROBLEMS
How well is y = Approximated by: 1 + x + x 2 + x3 + ? Plug in x=5 Series Convergence IN-CLASS PROBLEMS How well is y =
Approximated by: 1 + x + x 2 + x3 + ? Plug in x=5 Is the series a good estimator of the original function? Series Convergence If not, the series is said to diverge
Series Convergence IN-CLASS PROBLEMS How well is y = Approximated by: 1 + x + x 2 + x3 + ? Plug in x=1/2 Series Convergence
IN-CLASS PROBLEMS How well is y = Approximated by: 1 + x + x 2 + x3 + ? Plug in x=1/2 Is the series a good estimator of the original function?
Series Convergence If the series is a good estimator of the original function, the series is said to converge Series Convergence To what extent does the original function equal the series?
Series Convergence To what extent does the original function equal the series? A good estimator converges Series Convergence To what extent does the
original function equal the series? A good estimator converges A bad estimator diverges Series Convergence Range of convergence Where is the series a good estimator of the original?
Series Convergence The formulas in calculators and computers are not lookup tables The formulas are calculated using series approximations
Series Convergence Companies look for: accurate approximations approximations that converge quickly (after only a few iterations) Series Convergence The Squeeze Theorem: a function gets squeezed
between the other functions Series Convergence Series Convergence Divergence & Convergence Tests If a series converges, then the
larger terms get closer and closer to zero If not then the series diverges Series Convergence IN-CLASS PROBLEMS Alternating series: 1-1+1-1+1-
Does it converge? Series Convergence IN-CLASS PROBLEMS The series: 1/1 + 1/4 + 1/9 + Does it converge? Series Convergence
Series convergence is actually hard to test and requires calculus Questions? Matrices Matrix (plural matrices) - a rectangular table of elements (or entries),
numbers or abstract quantities Arranged in rows and columns Matrices Matrices in everyday life Matrices An n-ary relation is a
matrix: Subject Weight Alice Ben Carlos | Height |
58 53 6 170 120 200 Matrices
The term "matrix" for arrangements of numbers was introduced in 1850 by James Joseph Sylvester Matrices Horizontal lines in a matrix are called rows Vertical lines are called columns
Matrices A matrix with m rows and n columns is called an m-by-n matrix (written m n) m and n are called its dimensions Matrices The dimensions of a matrix
are always given with the number of rows first, then the number of columns: rxc Matrices Matrices are usually given capital letter names like A
Matrices To designate a specific element in a matrix, you use small letters with subscripts r and c: a37 means the element in matrix A in row 3 column 7 Matrices
In Math class, matrices are shown by listing the elements inside square braces: [ ] Matrices a11 a12 a13 A = a21 a22 a23
a31 a32 a33 Matrices IN-CLASS PROBLEMS The Matrix Game Questions?
Matrix Calculations When information is entered into a computer, it is recorded in a field that behaves in the same way as a mathematical matrix Matrix Calculations When the computer adds,
subtracts, multiplies or divides the information in a field, it is performing the same calculations that we execute when using matrices Matrix Calculations In mathematics, a matrix is used to store individual
numbers or other pieces of data Matrix Calculations The amount of data that can be stored in the matrix depends upon its size The size is expressed as the number of rows and columns in the matrix
Matrix Calculations Although each of these matrices has six slots for data, they have different shapes and are different sized matrices 2 rows 3 columns
d a d d
e b e e f c f f
3 rows 2
columns e a c e e f
b d f f
Matrix Calculations If the number of rows is the same as the number of columns, it is a square matrix d e f
a d g d b e h
e c f i f
Matrix Calculations Two or more matrices may be added or subtracted only if they are the same size Matrix Calculations A 2 x 3 matrix can be added
to another 2 x 3 matrix but not to a 3 x 2 matrix Matrix Calculations To add two matrices, add the components in each of the slots:
d a c e f b d
f +
d e g e f f h
f =
dxc a+e c+g dxc doc
b+f d+h doc Matrix Calculations
To subtract matrices, subtract the second component from the first dxc d f dinf each component slotdoc a-e b-f
a b e f c d - g h = c-g d-h e f e f dxc doc Matrix Calculations
Example: d 5 7 e
f 6 8 f
- d 2
5 e f 1 4 f
= =
dxc 5-2 7-5 dxc doc
3 5 2 4 doc doc 6-1 8-4 doc
Matrix Calculations IN-CLASS PROBLEMS
Find: d 2 3 6 7 e f 4 1
f d f 1 4 9 + 3 2 4 e
f Matrix Calculations IN-CLASS PROBLEMS Find: d 2 1 -4 6
7 -2 e f d f 3 1 7 + 4 -5 6
e f Matrix Calculations
IN-CLASS PROBLEMS Find: d d 2 -1 3 - 5
-6 -2 4 -9 d d Matrix Calculations
There are two kinds of matrix multiplication Matrix Calculations There are two kinds of matrix multiplication Scalar multiplication multiplies a matrix by a constant
Matrix Calculations Example: 35 36 d f 35 36 5 6 3 7 8 = 37 38 35 36 e f d
f 15 18 = 21 24 e f
Matrix Calculations IN-CLASS PROBLEMS Find: 2 3 2 5 2
3 3 6 3
Matrix Calculations IN-CLASS PROBLEMS Find: 2 -5 2 -3 4 -2 2
Matrix Calculations IN-CLASS PROBLEMS
Find: 2 3 -1 2 -3 -4 4 6 -2 -5 1 7 2 3
Matrix Calculations Multiplying two matrices is more complicated than addition or subtraction or scalar multiplication Matrix Calculations Only certain matrices can be multiplied
Matrix Calculations Only certain matrices can be multiplied If the number of columns of the first matrix is the same as the number of rows of the second matrix, they can be multiplied Matrix Calculations
We call these the inner dimensions of the matrices 7x5 5x8 inner dimensions Matrix Calculations For example, a 3 x 2 matrix
can be multiplied by a 2 x 3 matrix since the inner dimensions of 2 agree: 3x2 2x3 Terms agree Matrix Calculations
However, a 3 x 2 matrix cannot be multiplied by another 3 x 2 matrix since the inner dimensions do not agree: 3x2 3x2 Terms do not agree
Matrix Calculations Factoid: the size of the solution matrix will be the outer dimensions: for the 7 x 5 times 5 x 8 matrix multiplication, the solution matrix will be a 7 x 8 matrix
Matrix Calculations Example: Multiply 2 3 2 3 1 2 5 7 x 3 4 6 8 2 3 2 3 Matrix Calculations Example: Multiply
2 3 2 3 1 2 5 7 x 3 4 6 8 2 3 2 3 First, check to see if the two matrices can be multiplied Are the inner dimensions the same?
Matrix Calculations Example: Multiply 2 3 2 3 1 2 5 7 x 3 4 6 8 2 3 2 3 First, check to see if the two matrices can be
multiplied Since the inner dimensions are both 2, the matrices Matrix Calculations Example: Multiply 2 3 2 3 1 2 5 7 x 3 4 6 8
2 3 2 3 Next, we look at the outer dimensions to determine the expected dimensions of the solution matrix Matrix Calculations Example: Multiply 2 3 2 3 1 2 5 7
x 3 4 6 8 2 3 2 3 Since the outer dimensions of both are 2, we know that the solution matrix must be 2x2 Matrix Calculations To multiply the two
matrices, each row of the left matrix is multiplied by each column of the right matrix These individual products are then added to get the entries in the solution Matrix Calculations
Matrix Calculations Its easier done than said Matrix Calculations Its easier done than said but it IS tricky! Matrix Calculations Use the ROWS in the first matrix
And the COLUMNS in the second matrix Matrix Calculations The first row in the first matrix and the first column in the second matrix are combined to create the top left entry in the solution matrix
Matrix Calculations Example: Multiply 2 3 2 3 1 2 5 7 = x 3 4 6 8 2 3 2 3 Top left entry:
2 3 15+26 17+28 15+26 17+28
2 3 Matrix Calculations Example: Multiply 2 3 2 3 1 2 5 7 = x 3 4 6 8
2 3 2 3 Top right entry: 2 3
15+26 17+28 15+26 17+28 2 3 Matrix Calculations Example: Multiply 2 3 2 3 1 2 5 7 =
x 3 4 6 8 2 3 2 3 Bottom left entry:
2 3 15+26 17+28 35+46 17+28 2 3 Matrix Calculations Example: Multiply 2 3 2 3
1 2 5 7 = x 3 4 6 8 2 3 2 3 Bottom right entry: 2 3
15+26 17+28 35+46 37+48 2 3
Matrix Calculations Example: Multiply 2 3 2 3 1 2 5 7 = x 3 4 6 8 2 3 2 3 2 3 2
15+26 17+28 17 23 = 35+46 37+48 39 53 2 3 2 3 Find:
Matrix Calculations IN-CLASS PROBLEMS 2 3 2 2
3 1 5 3 2 6 x 4
2 3 1 9 3
Matrix Calculations IN-CLASS PROBLEMS Find: 2 3 2 3
2.8 9.1 x 4.2 3.1 4.7 2 3 2 3 Matrix Calculations IN-CLASS PROBLEMS
2 3 Find: 2 5 9 x 2 3
2 1 5 4 2
Questions? Systems of Equations A set of two equations
containing two variables ... A set of three equations containing three variables ... Are you seeing a pattern? Systems of Equations To solve a system, you
must have at least as many equations as you have variables Systems of Equations 2x + 5y = 38 x + 7y = 37 Lots of equations come in
groups! Systems of Equations If they give you values for x and y and ask if they are true values, plug them in to both equations both equations must be true for
the values Systems of Equations Are x = 4 and y = 6 true values for the system: 2x + 5y = 38 x + 7y = 37 Plug in x = 4 and y = 6
Systems of Equations Plug in x = 4 and y = 6 2(4) + 5(6) =? 38 (4) + 7(6) =? 37 Are the values x = 4 and y = 6 a true solution to this system?
Systems of Equations Suppose we wanted to find x and y for the equations: 2x + 5y = 38 x + 7y = 37
Systems of Equations Strategy: try to get one of the variables written in terms of the other variable: 2x + 5y = 38 x + 7y = 37 Look for a loner
Systems of Equations x + 7y = 37 Use algebra to solve for x: x = 37 7y Now, plug in this x into the other equation
Systems of Equations 2x + 5y = 38 x = 37 7y To get: 2(37-7y) + 5y = 38 Now, solve for y! Systems of Equations
2(37-7y) + 5y = 38 74 14y + 5y = 38 -9y = 38 74 = -36 y = -36/-9 y=4 Then we plug in this value for y and solve for x Systems of
Equations We already have: x = 37 7y y=4 So just plug in y = 4: x = 37 7(4) x = 37 28 x=9
Systems of Equations So, the true solution to the system: 2x + 5y = 38 x + 7y = 37 is x = 9, y = 4 or (x,y) = (9,4)
Systems of Equations To find the true solution to the system: 2x + 5y = 38 3x + 7y = 50 Solve one of the equations for either x or y
Systems of Equations 2x + 5y = 38 2x = 38 5y x = 38 5y 2 Then plug in this x into the other equation
Systems of Equations 3x + 7y = 50 x = 38 5y 2 Systems of Equations
3(38/2 - 5y/2) + 7y = 50 57 7.5y + 7y = 50 Solve for y: -7.5y + 7y = 50 57 -0.5y = -7 y = -7/-0.5 y = 14 Systems of Equations
You already have: x = 38 5y 2 and y = 14 so: x = 38 5(14) = 38-70 2 2 x = -32/2 = -16
Systems of Equations So the true solution to the system: 2x + 5y = 38 3x + 7y = 50 is x = -16, y = 14 or (x,y) = (-16,14)
Systems of Equations For more equations Same stuff, just more of it! Systems of Equations Solve the following system
of equations: x+z=3 x + 2y z = 1 2x y + z = 3 Systems of Equations x+z=3 is the easiest lets start
there: z=3x Plug that in! Systems of Equations z=3x x + 2y z = 1 2x y + z = 3
become: x + 2y (3 x) = 1 2x y + (3 x) = 3 Now its just two equations! Systems of Equations x + 2y 3 + x = 1
2x y + 3 x = 3 or 2x + 2y = 4 xy=0 Use x y = 0 x = y x=y Systems of
Equations 2x + 2y = 4 2x + 2x = 4 4x = 4 x=1 Systems of Equations So, now we have:
x=1 x=y z=3x So, (x,y,z) = (1,1,2) Systems of Equations Lots of business
applications use systems of equations Systems of Equations The income function is the total amount generated by selling the product: I(x) = (price per unit sold) *
x x is the quantity sold Systems of Equations The cost function is the total cost of producing the product: C(x) = fixed cost +
(cost per unit produced) * x x is the quantity produced Systems of Equations The profit function is the difference between the cost and the revenue functions: P(x) = I(x) C(x)
x is the quantity of the product Systems of Equations The x-solution for the two equations Cost and Income is the true solution of the system of the two
equations (the Cost and Income equations) Systems of Equations The solution to the system is called the break-even point
Systems of Equations IN-CLASS PROBLEMS Example: A widget-maker produces widgets Each widget sells for $300 They sell 20,000 widgets each year What is their annual
Systems of Equations IN-CLASS PROBLEMS I(x) = (price per unit sold) * x I = $300 (20,000) = $6,000,000 Systems of Equations
IN-CLASS PROBLEMS They have a fixed cost (rent, utilities, salaries, etc.) of $950,000 per year Each widget costs $10 to make They make 20,000 widgets each year
Systems of Equations IN-CLASS PROBLEMS C(x) = fixed cost + (cost per unit produced) * x C = $950,000 + $10(20,000) = $1,150,000
Systems of Equations IN-CLASS PROBLEMS What is their annual profit? Systems of Equations IN-CLASS PROBLEMS P(x) = I(x) C(x) P = $6,000,000 $1,150,000
= $4,850,000 Systems of Equations IN-CLASS PROBLEMS Lets buy Widget stock!!! Systems of Equations IN-CLASS PROBLEMS
Break-even point: How many widgets you need to make for the cost to equal the income (profit of $0) Systems of Equations IN-CLASS PROBLEMS Break-even point:
C(x) = I(x) fixed cost + (cost per unit) * x = (price per unit sold) * x Systems of Equations IN-CLASS PROBLEMS Break-even point: 950,000 + 10(x) = 300(x)
Algebra magic 950,000 = 300x - 10x 950,000 = 290x 950,000/290 = x So x = 3275.86 units Systems of Equations IN-CLASS PROBLEMS They will break even
when they make and sell 3276 units Since they make and sell a lot more, they are making a huge profit! Systems of Equations If you graph the income
and cost curves, the breakeven point is where the two curves cross Systems of Equations Actually this is true for all systems of equations the
true solution (x,y) is the point where the graph of the two curves cross Systems of Equations IN-CLASS PROBLEMS For this example, w is the number of wheelchairs
produced Revenue: R = $600w Cost: C = $500,000 + $400w Break-even point: Questions?
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