The Number Line, Absolute Value, Inequalities, and Properties of R

The Number Line

To draw a number line we draw a line with several dashes in it and ordered numbers below the line, both positive and negative.  The number corresponding to the point on the number line is called the coordinate of the number line. 

       

Absolute Values

Absolute value signs make the inside positive:

        |-3|  =  3        |2| = 2        |4 - 1|  =  |3|  =  3

We can write

We can read this as "Multiply by (-1) if x is negative, and leave it alone if x is positive."

Inequalities

Recall the four inequalities:

Symbol Meaning

< Less Than

> Greater Than

< Less Than or Equal to

> Greater Than or Equal to

When we graph an inequality on a number line we use "["  or "]" to include the point and "(" or ")" to not include the point.  For example, [1, 3)  means all, the points between 1 and 3 including 1 but not including 3.  We can write this on the number line as

   

Exercises

Graph the following on a number line

A. {x| x < 3}

B. {x| x > 2}C. {x| 3 < x < -5}D. {x| |x| > 4}

Properties of Addition and Multiplication

A. Commutative Propertyo Addition:  a + b = b + ao Multiplication:  ab = ba

B. Associative Propertyo Addition:  (a + b) + c = a + (b + c)o Multiplication:  (ab)c = a(bc)

C. Identity Propertyo Addition:  there is a 0 such that a + 0 = 0 + a = ao Multiplication:  there is a 1 such that a1 = 1a = a

D. Inverse Propertyo Addition:  for any a, there is a -a with a + -a = 0o Multiplication:  for any a not 0, there is a 1/a with a(1/a) = 1

E. Distributive Propertyo a(b + c) = ab + aco (a + b)c = ac + bc 

F. Trichotomy

If a and b are real numbers then one of the three must hold

i. a < bii. a > biii. a = b

B. Transitivity

 If a, b, and c are real numbers and 

        a  <  b    and    b  <  c then       a  <  c

Examples

        (2 + 3) + 4  =  2 + (3 + 4)  (Associative Property of Addition)

        (x - y)(x + y)  =  (x + y)(x - y) (Commutative Property of Multiplication)

                     1        (2 - y)             =  1  (Multiplicative inverse)                   2 - y

Exercises:  

Complete the following.  (If you hold your mouse on the yellow rectangle, you will see the solution.)

A. x - z  =  ________  (Commutative)                               

B. w (0)  =  _________  (Multiplication property of 0)         C. 3 (xy)  =  __________ (Associative)                               D. x (y - 3)  =  _________(Distributive)                             

Relations, Functions, and Function Notation

Definition of a Relation, Domain, and Range

Examples

A. Consider the relation that sends a student to that student's age.

B. Consider the relation that sends a student to the courses that student is taking.C. Consider the relation that sends a parent to the parent's child.D. Consider the relation that sends a key word either to its matches from the Yahoo

search engine or to the statement "No matches found."

  Each of these are examples of relations.  The definition below formalizes this idea.

Definition: A relation is a correspondence between two sets (called the domain and the range) such that to each element of the domain, there is assigned one or more elements of the range.

Remark:  To define a relation three things must be designated:  the range set, the domain set and the rule of assignment.Non-Example

Let the domain be the set of all LTCC students and the range be the set of all math course offerings at LTCC.  Then the map that takes a student and sends the student to the math ourse he or she is taking is not a relation since there are students who are not taking math courses.

Example 

        (2,3), (2,4), (3,7), and (5,2) 

defines a relation with        Domain:  {2,3,5} and        Range:  {2,3,4,7}

Example

A circle represents the graph of a relation with domain consisting of x-values from the left side of the circle to the right side.  The range consists of y-values from the bottom to the top.

Functions

A function is a special type of relation where every input has a unique output.

Definition:   A function is a correspondence between two sets (called the domain and the range) such that to each element of the domain, there is assigned exactly one element of the range.  

Example :        (3,3), (4,3), (2,1), (6,5) is a function with  

     Domain: {2,3,4,6}  and        Range: {1,3,5}

Non-Example:

        (2,1), (5,6), (2,3), (6,7) 

is not a function since 2 gets sent to more than one value.

Function Notation

  DefinitionWe write f (x) to mean the function whose input is x.

Example:

If        f(x)  =  2x – 3 then        f(4)  =  2(4) - 3  =  5

We can think of f as the function that takes the input multiplies it by 2 and subtracts 3. Sometimes it is convenient to write f(x) without the x. Thus:

        f( ) = 2( ) - 3

whatever is in the parentheses, we put inside.  For example:

        f(x - 1)  =  2(x - 1) - 3

and

           f(x + 4) - f(x)               [2(x + 4) - 3] - [2(x) - 3]                                          =                                                                       4                                    4

              2x + 8 - 3 - 2x + 3          =                                            Distributing the 2 and the -                            4

                8        =             = 2                                Combining Like Terms               4

Example

If 

        f(x)  =  7

then 

        f(12)  =  7

Here f is called the constant function.  Whatever comes in to f, the number 7 comes out.

Function Arithmetic

We define the sum, difference, product and quotient of functions in the obvious way.

Example

If 

                         x + 1        f(x)  =                                                  x - 1and 

        g(x) = x2 + 4

then

                               x + 1        (f + g)(x) =                   + (x2 + 4)                               x - 1

                               x + 1        (f - g)(x) =                   - (x2 + 4)                               x - 1

                            x + 1        (f g)(x) =                 (x2 + 4)                            x - 1 and

       

The Quadratic Formula 

The Quadratic Formula

The solution to 

          ax

2

+ bx + c  =  0

is

                   

Example: Solve  3x2 - 2x + 5

  Solution:Here    a = 3         b = -2          c = 5

We have 

       

 

The Discriminant

We define the discriminant as

        D = b2  - 4ac

is a convenient measure of determining how many real roots (solutions) there are.  Notice that D is the expression inside of the square root sign in the quadratic formula.  Since the square root of a negative number produces only complex numbers, we see that if D is negative, then there will be no real roots.  If D is a positive number, then the quadratic formula will produce two roots (one for the plus and one for the minus).  If D is 0 then plus 0 and minus 0 are the same number, so we get only one root.  The table below summarizes.

  D Number of Real Roots

Positive 2

Negative 0

Zero 1

  Example:

How many roots are there for the equation:

        3x2 - 5x + 1  =  0

We have 

        D  =  25 - 12  >  0

hence there are two real roots.

The Sum and Product of the Roots

Since the two roots of a quadratic are

       

and

       

then if we add the two roots, we get:

       

and if we multiply the two roots we get

       

  Note that if a is 1 then the sum of the roots is -b and the product is c.  This relates to factoring when we find two numbers that add to b and multiply to c.

Example

What are the sum and product of the roots of

        4x2 -3x + 2

  Solution 

The sum is 

            b          3        -        =                       a          4

and the product of the roots is 

            c           2           1                  =          =                                                  a           4           2

 Determining the Quadratic Equation From the Roots

If we know the roots of a quadratic then it is easy to find the original quadratic by using the zero product formula in reverse.

  Example

Find an equation of a quadratic that has roots 2 and -4/3.

  Solution

We can write:

        (x - 2)(x - (-4/3))  =  (x - 2)(x + 4/3) 

                      4                  8                2           8        =  x2 +         x - 2x -         =  x2 -       x -                              3                  3                3           3

Arithmetic Sequences and Series

Arithmetic Sequences

Exercise:  Find the next term and the general formula for the following:

A. {2, 5, 8, 11, 14, ...}

B. {0, 4, 8, 12, 16, ...}

C. {2, -1, -4, -7, -10, ...}

For each of these three sequences there is a common difference.  In the first sequence the common difference is d = 3, in the second sequence the common difference is d = 4, and on the third sequence the common difference is d = -3.  We will call a sequence an arithmetic sequence if there is a common difference.  

The general formula for an arithmetic sequence is

an = a1  + (n - 1)d

Example

What is the difference between the fourth and the tenth terms of {2,6,10,14,...)

We have

        a10 - a4  =  (10 - 4)d  =  6(4)  =  24

Arithmetic Series

First we see that

        1+ 2 + 3 + ... + 100  =  101 + 101 + ... + 101 (50 times)  =  101(50)

In general

                                             n(n + 1)        1 + 2 + 3 + ... + n  =                                                                      2

Example: What is         S  =  1 + 4 + 7 + 10 + 13 +... +  46

Solution

        S  =  1 + (1 + 1(3)) + (1 + 2(3))  + (1 + 3(3)) + ... + (1 + 15(3))

        =  (1 + 1 + ... + 1) + 3(1 + 2 + 3 + ... + 15)

        =  16 + 3(15)(16)/2

In General

                                 d(n - 1)(n)        Sn = n (a1)+                                                                    2

        = 1/2 [2n(a1) + d(n - 1)(n)] 

        = 1/2[2n(a1)+ dn2 - dn]

        = (n/2)[2(a1)+ dn - d]= (n/2)[2(a1) + d(n - 1)]

Or Alternatively

Sn = n/2(a1 + an)

Example

How much will I receive over my 35 year career if my starting salary is $40,000, and I receive a 1,000 salary raise for each year I work here?

Solution

We have the series:

        40,000 + 41,000 + 42,000 + ... + 74,000

        =  35/2 (40,000 + 74,000)  =  $1,995,500

 

Geometric Sequences and Series 

Geometric Sequences

Find the general term of the following:

A. 1, 2, 4, 8, 16, ...

B. 27, 9, 3, 1, 1/3, ...

C. 3, 6, 12, 24, 48, ...

D. 1/2, -1, 2, -4, 8, ...

Definition of a Geometric Sequence

A Geometric Sequence is a sequence which the ratio of the common terms is equal.

The general term is

                    an  =  a1rn-1  , where r is the common ratio.

Example

Find the general term of the geometric series such that

        a5  =  48 

and 

        a7  =  192

Solution

We have that 

        an  =  a1rn  

which gives the equations

        48  =  a1r4 ,         192  =  a1r6

Dividing the two equations, we get:    

        4  =  r2

Hence 

        r  =  2         or         r  =  -2

Substituting back into the first equation, we get

        48  =  16a1

So that 

        a1 = 3

Hence the general term of the sequence is

        an  =  (3)(2)n-1          or         an  =  (3)(-2)n-1  

Geometric Series

  Theorem 

         

Example

Find the sum

        5 + 10 + 20 + 40 + ... + 2560

Solution:     

        a1  =  5     and     r  =  2     and     n  =  10 

so that

                     a1(1 - rn)        Sn  =                                             (1 - r)

                   5(1 - 210)                            =                                                   1 - 2

        =  5115

For an infinite geometric series if  |r| <1 then  

       

Example

How much is going to taxes?  Suppose that we track a tax refund of $100.  Each time money is spent 8% goes towards taxes and the rest gets spent again.  How much of the original $100 will go back to taxes?

Solution

        a1 = 8       r = 0.92    (The next amount to be taxed is 92% less than the current amount)

                               a1                  S   =                                          1 - r 

                          8               8                =                 =             =  $100                    1 - 0.92         0.08

Hence all of the refund will eventually find its way back to the government coffers.

Business Economics Applications

Review of Revenue, Cost and Profit

We define the revenue R to be the total amount of money coming into the company, the cost C the total amount of money coming out of the business, and the profit P is the revenue minus the cost.  When we say marginal, we mean the derivative with respect to x the number of items sold.  

For example the marginal cost is

        Marginal Cost = dC/dx

If we let p be the price per unit, then we have

Revenue = Price*Units SoldR = px

Example:  

Suppose you own a snow board rental shop and have determined that the demand equation for your snow boards is

        p = $20 - x/10  

(At $20 per rental you wont sell any)  and the cost equation is

        C = 50 + 3x

($50 fixed costs and $3 per snow board rental)  What is the marginal profit in terms of x?  What price should you charge to maximize profits?

 Solution:

The revenue is

        R = px = (20 - x/10)(x) = 20x - x2/10

So that profit is

        P = R - C =  (20x - x2/10) - (50 + 3x) = 17x - x2/10 - 50

The marginal profit is

        dP/dx  = 17 - x/5

To find the maximum profit we set the marginal profit equal to zero and solve:

        17 - x/5 = 0

        x = 105

Thus the price we should set is

        p = 20 - 105/10 = $9.95 per rental.

Exercise:

Suppose that the cost for a truck driver is $7.50 per hour and that the cost to operate the truck is v2/50000 per mile where v is the average speed of the truck.  How fast should you recommend your driver to drive in order to minimize the total costs? 

Average

Recall that the average is the total divided by the number of items.  Hence, the average cost is the total cost C divided by the average cost x.  

Example:  

Find the minimum average cost if

        C = 2x2 + 5x + 18

Solution:  

The average cost is

        A = C/x = 2x + 5 + 18/x

        A' = 2 - 18/x2

We set         2 - 18/x2  = 0 

to get          x = 3

Since          A'' = 36/x3 

plugging in 3 gives a positive value.  By the second derivative test, we see that 3 is a minimum.  The minimum average cost is 

        Cmin  2(9) + 5(3) + 18 = 51

 Exercise:  

Find the maximum average revenue if the demand equation is

        p = 500 + 10x - x2

Elasticity

We define the price elasticity of demand by 

Definition

        elasticity = (rate of change of demand)/(rate of change of price)or        = (p/x)/(dp/dx)

We say that a product is elastic if || > 1, inelastic if || < 1

 

The idea is that a product is elastic if a drop in price results in a significant rise in demand.  A product is inelastic if a drop in price does not result in a significant rise in demand.  

Example:  

The demand function for a product is 

        p = 50 + x - x2   

determine the elasticity when x = 4.

Solution: 

We see that 

        p = 38,     dp/dx = 1 - 2x = -7 

hence

        = (38/4)/(-7) = -38/28 

so that the product is elastic since its absolute value is larger than 1.

 Exercise:  Determine the elasticity for          p = x2 /(100x - 1) at x = 10.

Matrices

I. Definition of a Matrix

An m by n matrix is an array of numbers with m rows and n columns.

Example 1:

4 5

0 15

-9 3

is a 3 by 2 matrix.

Example 2:

Consider the system of equations

        2x -  y  + 3z = 5         x   +       4z = 3        5x - 7y + 3z = 7

Then the matrix

2 -1 3 5

1 0 4 3

5 -7 3 7

is called the augmented matrix associated to the system of equations. 

II. Solving Linear Systems Using Matrices

We can solve a linear system by writing down its augmented matrix and performing the row operations that we did last time.

Example:  

Solve

        2x -  y + z = 3          x + y + z = 2                 y - z = -1

Solution:

We write the associated augmented matrix:

2 -1 1 3

1 1 1 2

0 1 -1 -1

Now begin solving by performing row operations:

R1 <-> R2 R2 - 2R1 -> R2 R2 <-> R3

1 1 1 2

2 -1 1 3

0 1 -1 -1

1 1 1 2

0 3 -1 -1

0 1 -1 -1

1 1 1 2

0 1 -1 -1

0 3 -1 -1

R1  - R2 -> R1

R3 + 3R2 -> R3 R3 -> -1/4 R3

R1 - 2R3 -> R1R2 + R3 -> R2

1 0 2 3

0 1 -1 -1

0 0 -4 -4

1 0 2 3

0 1 -1 -1

0 0 1 1

1 0 0 1

0 1 1 0

0 0 1 1

R1 - 2R3 -> R1R2 + R3 -> R2

1 0 0 1

0 1 0 0

0 0 1 1

We can now put the matrix back in equation form:

        x = 1,     y = 0     and     z = 1

Note:  If we had seen a bottom row that was of the form 0 0 0 a where a is a nonzero constant, then there would be no solution.  If a had been 0 there would be infinitely many solutions.

III. Addition and Scalar Multiplication of Matrices

We can only add matrices that are of the same dimensions, that is if

A = 1 2 ,  B = 2 3 ,  C =  1 3

3 44 1

5 97 2

then only A + C makes sense.  We write

A + C =1 + 1 2 + 3

3 + 7 4 + 2 = 

2 5

10 6

For any matrix, we can multiply a matrix by a real number as in the following example (Same B as above):

5B =

10 15

20 5

25 45

We define the zero matrix to be the matrix with only zeros for entries.  For example, the 2 by 2 zero matrix is

0 0

0 0

IV. Multiplication of Matrices

To multiply matrices, unfortunately the definition is not the obvious one.  We can only multiply matrices where the number of columns of the first matrix is the same as the number of rows of the second matrix.  The best way to learn how to multiply matrices is by example:

     Let A =3 4 2

0 1 -2and B =

7 -3

-2 1

0 5

Then   AB=3(7) + 4(-2) + 2(0) 3(-3) + 4(1) + 2(5)

0(7) + 1(-2) + -2(5) 0(-3) + 1(1) + -2(5) =

13 5

-12 -9

Exercises:  

Let  A =1 2

3 4, B =

4 2 1

-2 0 0

1 6 -1

, C = 

1 0

2 1

4 5

, D = 3 4 0

5 0 0, E = 

3 4 2

1 5 0

1 -1 2

VI. Evaluate each one that makes sense:

1)  A + B    2)  4C     3)  AB     4)  CD      5)  DC     6)  B + E     7)  A3

Applications of Matrices

Application 1

A)  Tables and chairs are made in the Mexico plant, the Brazil plant, and the US plant. The matrix below represents the quantity made per day.

 

A =

  Quantity

Mexico Brazil US

Tables 15 10 50

Chairs 30 12 75

Labor and material cost for 1997 are represented in the following matrix.

B =

Labor Material

Mexico 15 20

Brazil 12 10

US 30 5

In 1997, the costs have increases to

    C =

Labor Material

Mexico 17 25

Brazil 15 15

US 45 10

Find the following and describe what they mean:

1)  AB     2)  C - B     3)  AC     4)  A(C - B)     5) 365AC

Application 2

Suppose that you have two jobs, each contribute to two different mutual funds for retirement.  The first fund pays 5% interest and the second pays 8% interest.  Initially $5,000 is put into the funds and after one year there will be $5,300.  If the first fund

got half of the money from the first job and one third of the money from the second job, how much did each job contribute?

Determinants and Inverses

I. Determinants:

Consider row reducing the standard 2x2 matrix.  Suppose that a is nonzero.

a b

c d

1/a R1 ->  R1 R2 - cR1  -> R2

1 b/a

c d

1 b/a

0 d - cb/a

Now notice that we cannot make the lower right corner a 1 if

        d - cb/a = 0 

or

        ad - bc = 0

          Definition of the Determinant We call ad - bc the determinant of the 2 by 2 matrix

a b

c d

it tells us when it is possible to row reduce the matrix and find a solution to the linear system.

Example:  

The determinant of the matrix

3 1

5 2

is 

        3(2) - 1(5) = 6 - 5 = 1

II. Determinants of Three by Three Matrices

We define the determinant of a triangular matrix

 

a d e

0 b f

0 0 c

by 

        det = abc

Notice that if we multiply a row by a constant k then the new determinant is k times the old one.  We list the effect of all three row operations below.

                     Theorem

The effect of the the three basic row operations on the determinant are as follows 1. Multiplication of a row by a constant multiplies the determinant by that

constant.2. Switching two rows changes the sign of the determinant.

3. Replacing one row by that row + a multiply of another row has no effect on the determinant.

To find the determinant of a matrix we use the operations to make the matrix triangular and then work backwards.

Example:

Find the determinant of

2 6 10

2 4 -3

0 4 2

We use row operations until the matrix is triangular.

        1/2 R1 <-> R1     (Multiplies the determinant by 1/2)

1 3 5

2 4 -3

0 4 2

          R2 - 2R1 -> R2     (No effect on the determinant)

1 3 5

0 -2 -13

0 4 2

Note that we do not need to zero out the upper middle number.  We only need to zero out the bottom left numbers.

R3 + 2R2 ->  R3     (No effect on the determinant)  

1 3 5

0 -2 -13

0 0 -24

Note that we do not need to make the  middle number a 1.

The determinant of this matrix is 48. Since this matrix has 1/2 the determinant of the original matrix, the determinant of the original matrix has 

        determinant = 48(2) = 96.

III. Inverses

We call the square matrix I with all 1's down the diagonal and zeros everywhere else the identity matrix.  It has the unique property that if A is a square matrix with the same dimensions then

        AI = IA = A

Definition

If A is a square matrix then the inverse A-1 of A is the unique matrix such that

AA-1 = A-1A = I  

Example:  

Let 

          A =2 5

1 3

then

          A-1  =3 -5

-1 2

Verify this!

                    Theorem

The inverse of a matrix exists if and only if the determinant is nonzero.

To find the inverse of a matrix, we write a new extended matrix with the identity on the right.  Then we completely row reduce, the resulting matrix on the right will be the inverse matrix.

Example:

2 -1

1 -1

First note that the determinant of this matrix is 

        -2 + 1 = -1

hence the inverse exists.  Now we set the augmented matrix as

2 -1 1 0

1 -1 0 1

R1 <-> R2   R2 - 2R1 -> R2 R1 + R2  -> R1

1 -1 0 1

2 -1 1 0

1 -1 0 1

0 1 1 -2

1 0 1 -1

0 1 1 -2

Notice that the left hand part is now the identity.  The right hand side is the inverse. Hence

          A-1 =1 -1

1 -2

IV. Solving Equations Using Matrices

Example:

Suppose we have the system

        2x - y = 3        x - y = 4

Then we can write this in matrix form

        Ax = b

where

          A =2 -1

1 -1x =

x

yand      b =

3

4

We can multiply both sides by A-1:

        A-1A x = A-1b

or 

        x = A-1b

From before,

          A-1 =  1 -1

1 -2

Hence our solution is

  -1

-5

or 

        x = -1     and     y = 5

Linear Systems

I.  Geometry of Systems of Equations

We know that for two by two linear systems of equation, the geometry is that of two lines that either intersect, are parallel, or are the same line.  If they intersect then there is exactly one solution, if they are parallel then there are no solutions, and if they are the same line, then there are infinitely many solutions.  For three by three systems, the situation is different.  The solution set is either  the empty set, a point, a line, or a whole plane.  For four by four systems, the geometry becomes four dimensional and is rough to comprehend, but is still useful.

II. The Algebra of Linear Systems.

In this class we will perform algebra on linear systems in a new way.  For example, for a three by three system, we line the equations up to form three rows.  We will manipulate the rows to simplify the equations.  There are three operations, called row operations that we can perform:

Row Operations

A. We can multiply an entire row by a nonzero constant  

          cRi ->  Ri  B. We can interchange two rows.  

          Ri <-> Rj  

C. We can replace one row with that row + a multiple of another row

          cRj  + Ri  -> Ri  

Exercises:

1.        2x + y     =  4        x - 2y + z  =  0        3x - 4y + 2z  = 1

2. 4x - 3y - z = 0x - 3y + 2z = 73x + 9y - z = -2

3. 3x - 2y + z = 34x + y = 111y - 4z = -9

I. Application

Your breakfast consists of orange juice, cereal, and eggs with the following nutritional information:

OJ Cereal Eggs

Protein 0% 10% 20%

Vitamin C 20% 15% 0%

Calories 100 120 100

If you must have 30% protein, 30% Vitamin C and 300 calories for your breakfast, How many servings of OJ, Cereal, and Eggs should you have?

Solution

Let 

        x = the number of servings of OJ

        y = the number of servings of Cerial

        z = the number of servings of eggs

Then

                    10y   + 20z   = 30        20x   + 15y               = 30        100x + 120y + 100z = 300

Solving,       x                     = 3/2, y   = 0 ,   z     = 3/2

We conclude that the breakfast should consist of 1.5 of a serving of OJ, no cereal,  and 1.5 servings of eggs.

Rates of Change

Velocity - Recall that the difference between speed and velocity is that velocity has direction and speed does not.  In other words, the speed is the absolute value of velocity.  We have seen that the secant line can be used to approximate the velocity.  The formula for average velocity is

If we want the instantaneous velocity we take the limit as tf  approaches ti.  This is just the alternate form of the derivative.  This leads to the definition below

Definition

Let s(t) be the position function, then the instantaneous velocity at v(t) is the derivative of the position function

          v(t)  =  s'(t)

Example  

The height of a pine cone t seconds after falling from a tree is given by

        s = -16t2 + 100

A.   Find the average velocity in the interval [0,2]

B.   Find the instantaneous velocity for t = 0 and t = 1

C.   How long will it take to hit the ground

D.   Find the velocity when it hits the ground

Solution:

A.   We compute 

                      s(b) - s(a)              100 - 36        vave =                          =                          =  32 ft/sec                          b - a                         2

B.            s'(t) = -32t          so       s'(0) = 0         and         s'(1) = -32

C.    We solve          -16t2 + 100 = 0     or      t = 2.5

D.       s'(2.5) = -80

  Marginals

In economics the key terms are revenue, cost,  and profit.  We use the word marginal to indicate the additional cost of producing one more.  In calculus terms marginal means the derivative.

Example

Suppose that the cost of producing x burgers per hour is 

        C(x)  =  1000/x + x         for         x > 35

The burgers cost 2.50 each.  Find the marginal cost, marginal revenue, and marginal profit of producing 40 burgers

Solution:  

The marginal cost is 

        C'(x) = (1000x -1 + x)' = -1000x -2 + 1

so that 

        C'(40) = -1000/1600 + 1 = .375

so that the additional cost of producing 40 burgers is about 38 cents

  Revenue is price times number sold

            R  =  2.5x 

so 

            R' =  2.5 

so that the marginal revenue is $2.50.

The profit is 

            Profit  =  Revenue - Cost 

so the marginal profit is 

        marginal revenue - marginal cost 

        =  2.5 - .38  =  2.12

Example

You are the owner of the Tahoe View Inn and have experimented with different pricing strategies.  When you charged $80 per night, you were able to average 50 occupied rooms and when you charged $90 per night, you were able to average 45 occupied rooms.  Find the marginal revenue at an average of 50 occupied rooms assuming the demand function is linear.  Then interpret its meaning.

  Solution

We need to first find the demand function.  Since it is linear, we need to find its slope.  We calculate

                      90 - 80        m  =                          =  -2                              45 - 50

Using the point slope formula for the equation of a line gives

        p - 80  =  -2(x - 50)  =  -2x + 100

        p  =  -2x + 180

Since

        Revenue  =  (Price)(Quantity) 

we have

        R(x)  =  (-2x + 180)(x)  =  -2x2 + 180x

To find the marginal revenue we calculate

        R'(x)  =  -4x + 180

Now plug in x  =  50 to get 

        R'(50)  =  -4(50) + 180  =  -20

Since the marginal revenue is negative, this says that increasing the occupancy rate will decrease revenue at a rate of $20 per unit.  We should strive to decrease the occupancy rate by raising the price.  Later on, we will learn how to find the price at which the maximal revenue will be achieved.

 Money, Mixture, Motion, and Inequalities

Money Problems

Example

You have 40 coins in nickels and dimes.  How many dimes do you have if you have a total of $2.85?

Solution:

Let 

        d = the number of dimes you havethen

        40 - d  =  the number of nickels that you have.

The total money that you have is

        10d + 5(40 - d)  =  285            Value of dimes + Value of nickels = 285

        10d + 200 - 5d  =  285             Distributing the 5

        5d + 200  =  285                       10d - 5d = 5d

        5d  =  85                                    Subtracting 200 from both sides

        d  =  17                                       Dividing by 5

We have 17 dimes.

Example

You are the manager of the new Tahoe Stadium.  You sell your VIP seats for $200 each and your general admission seats for $75.  Your stadium holds 10,000 people, and you need to earn at least $1,000,000.  If you sell out, how many of your seats should you designate as VIP seats?

Solution:  

Let 

        x  =  number of VIP seats

then

        10,000 - x  =  number of general admission seats.  

The money from the VIP seats is  

        200x

and the money from the general admission seats is

        75 (10,000 - x)  

Hence

200x + 75(10,000 - x)   =  1,000,000       VIP money + general Ad money = 1,000,000

200x + 750,000 - 75x   =  1,000,000       Distributing the 75 

125x + 750,000  =  1,000,000                  200x - 75x = 125x

125x  =  250,000                                         Subtracting 750,000

x  =  2,000                                                    Dividing by 125

We designate 2,000 seats as VIP seats. 

Mixture Problems

Example

Vodka contains 40% alcohol and wine contains 10% alcohol.  You want to make a new drink that is 20% alcohol using vodka and wine.  How much of each should you use to make 15 ounces of this drink?

Solution:

Our answer should be

We let 

        x  =  number of ounces of  vodka

Then 

        15 - x  is the number of ounces of wine

Note that the amount of alcohol in the final mixture is 

        15 (0.2)  =  3                    15 ounces times 20% alcohol = 3

Hence we can write

        0.4x + 0.1(15 - x)  =  3       vodka alcohol + wine alcohol = total alcohol

Multiplying by 10 to get rid of the decimal, we get:

        4x + (15 - x) = 30

        4x + 15 - x = 30 

        3x = 15                         4x - x = 3x  and 30 - 15 = 15 

        x = 5                             dividing by 3

Hence we pour 5 ounces of vodka and 10 ounces of wine.  (It is not recommended to try this at home).

Motion Problems

Example

Suppose that I am walking from school at 3 miles per hour and start at 12:00.  At 12:30, you start riding your bike at 18 miles per hour to find me.   At what time do you find me?

Solution:   Let 

        t = the time after 12:00

We use the formula 

        distance = rate times time

Then my distance from school is        3t

To find your distance from school, multiply the rate, 18 by the time since you left, t - 1/2.

        18 (t - 1/2)

We set the two equal to each other:

        3t  =  18(t - 1/2) 

        3t  =  18t - 9                                              distributing through

        -15t  =  -9                                                  subtracting 18t from both sides

        t  =  9/15  =  3/5  =  36/60 or 36 minutes.      dividing both sides by -15

Linear Inequalities

Definition

A linear inequality is one that can be reduced to

          ax + b < 0 or           ax + b > 0 or            ax + b < 0 or           ax + b > 0

Step by step method for solving linear inequalities:

1. Simplify both sides (distribute and combine like terms).

2. Bring the x's to the left and the constants to the right.3. Divide by the coefficient (changing the inequality if the sign of the inequality is

negative).4. Plot on a number line (remember holes and dots).

Example

Solve

        2(x - 9)  <  3(2x - 10)

Solution

        2x - 18  <  6x - 30            Distributing the 2 and the 3

        -4x  <  -12                          Subtracting 6x and adding 18

        x  >  3                               Dividing by 4

Plot on a number line with a hole at 3 and an arrow to the right of 3.

       

 Mean, Mode, Median, and Standard Deviation

The Mean and Mode

The sample mean is the average and is computed as the sum of all the observed outcomes from the sample divided by the total number of events.  We use x as the symbol for the sample mean.  In math terms, 

       

where n is the sample size and the x correspond to the observed valued.

  Example

Suppose you randomly sampled six acres in the Desolation Wilderness for a non-indigenous weed and came up with the following counts of this weed in this region:

        34, 43, 81, 106, 106 and 115 

We compute the sample mean by adding and dividing by the number of samples, 6.

             34 + 43 + 81 + 106 + 106 + 115                                                                     =  80.83                                        6

We can say that the sample mean of non-indigenous weed is 80.83.

The mode of a set of data is the number with the highest frequency.  In the above example 106 is the mode, since it occurs twice and the rest of the outcomes occur only once.

The population mean is the average of the entire population and is usually impossible to compute. We use the Greek letter for the population mean. 

Median, and Trimmed Mean

One problem with using the mean, is that it often does not depict the typical outcome.  If there is one outcome that is very far from the rest of the data, then the mean will be strongly affected by this outcome.  Such an outcome is called and outlier.  An alternative measure is the median.  The median is the middle score.  If we have an even number of events we take the average of the two middles.  The median is better for describing the typical value.  It is often used for income and home prices.

Example

Suppose you randomly selected 10 house prices in the South Lake Tahoe area.  Your are interested in the typical house price.  In $100,000 the prices were

        2.7,   2.9,   3.1,   3.4,   3.7,  4.1,   4.3,   4.7,  4.7,  40.8

If we computed the mean, we would say that the average house price is 710,000.  Although this number is true, it does not reflect the price for available housing in South Lake Tahoe.  A closer look at the data shows that the house valued at 40.8 x $100,000  =  $4.08 million skews the data.  Instead, we use the median.  Since there is an even number of outcomes, we take the average of the middle two

      3.7 + 4.1                        =  3.9            2

The median house price is $390,000.  This better reflects what house shoppers should expect to spend.

        

There is an alternative value that also is resistant to outliers.  This is called the trimmed mean which is the mean after getting rid of the outliers or 5% on the top and 5% on the bottom.  We can also use the trimmed mean if we are concerned with outliers skewing the data, however the median is used more often since more people understand it.

Example:

At a ski rental shop data was collected on the number of rentals on each of ten consecutive Saturdays: 

        44, 50, 38, 96, 42, 47, 40, 39, 46, 50.

  To find the sample mean, add them and divide by 10:

         44 + 50 + 38 + 96 + 42 + 47 + 40 + 39 + 46 + 50                                                                                        = 49.2                                        10

Notice that the mean value is not a value of the sample.

To find the median, first sort the data:

        38, 39, 40, 42, 44, 46, 47, 50, 50, 96

Notice that there are two middle numbers 44 and 46.  To find the median we take the average of the two.

                             44 + 46        Median  =                      =  45                                  2

Notice also that the mean is larger than all but three of the data points.  The mean is influenced by outliers while the median is robust.

Variance,  Standard Deviation and Coefficient of Variation

The mean, mode, median, and trimmed mean do a nice job in telling where the center of the data set is, but often we are interested in more.  For example, a pharmaceutical engineer develops a new drug that regulates iron in the blood.  Suppose she finds out that the average sugar content after taking the medication is the optimal level.  This does not mean that the drug is effective.  There is a possibility that half of the patients have dangerously low sugar content while the other half have dangerously high content.  Instead of the drug being an effective regulator, it is a deadly poison.  What the pharmacist needs is a measure of how far the data is spread apart.  This is what the variance and standard deviation do.  First we show the formulas for these measurements.  Then we will go through the steps on how to use the formulas.

We define the variance to be 

       

and the standard deviation to be

       

Variance and Standard Deviation: Step by Step

1. Calculate the mean, x.

2. Write a table that subtracts the mean from each observed value.3. Square each of the differences.4. Add this column.5. Divide by n -1 where n is the number of items in the sample  This is the variance.6. To get the standard deviation we take the square root of the variance.  

Example

The owner of the Ches Tahoe restaurant is interested in how much people spend at the restaurant.  He examines 10 randomly selected receipts for parties of four and writes down the following data.

        44,   50,   38,   96,   42,   47,   40,   39,   46,   50

He calculated the mean by adding and dividing by 10 to get

        x  =  49.2

Below is the table for getting the standard deviation:

  x x - 49.2 (x - 49.2 )2  

44 -5.2 27.04

50 0.8 0.64

38 11.2 125.44

96 46.8 2190.24

42 -7.2 51.84

47 -2.2 4.84

40 -9.2 84.64

39 -10.2 104.04

46 -3.2 10.24

50 0.8 0.64

Total   2600.4

Now 

        2600.4                         =  288.7        10 - 1

Hence the variance is 289 and the standard deviation is the square root of  289 = 17.

What this means is that most of the patrons probably spend between $32.20 and $66.20.

The sample standard deviation will be denoted by s and the population standard deviation will be denoted by the Greek letter .

The sample variance will be denoted by s2 and the population variance will be denoted by 2.

The variance and standard deviation describe how spread out the data is.  If the data all lies close to the mean, then the standard deviation will be small, while if the data is spread out over a large range of values, s will be large.  Having outliers will increase the standard deviation.

One of the flaws involved with the standard deviation, is that it depends on the units that are used.  One way of handling this difficulty, is called the coefficient of variation which is the standard deviation divided by the mean times 100%

                                  CV  =           100%                               

In the above example, it is 

         17                   100%   =  34.6%        49.2

This tells us that the standard deviation of the restaurant bills is 34.6% of the mean.

Chebyshev's Theorem

A mathematician named Chebyshev came up with bounds on how much of the data must lie close to the mean.  In particular for any positive k, the proportion of the data that lies within k standard deviations of the mean is at least

                  1        1  -                         k2

For example, if k  =  2 this number is

                 1        1  -         =  .75                22 

This tell us that at least 75% of the data lies within 75% of the mean.  In the above example, we can say that at least 75% of the diners spent between 

        49.2 - 2(17)  =  15.2

and

        49.2 + 2(17)  =  83.2 

dollars.

and for Grouped Data

 Calculating the Mean from a Frequency Distribution

Since calculating the mean and standard deviation is tedious, we can save some of this work when we have a frequency distribution.  Suppose we were interested in how many siblings are in statistics students' families.  We come up with a frequency distribution table below.

Number of Children

1 2 3 4 5 6 7

Frequency 5 12 8 3 0 0 1

Notice that since there are 29 respondents, calculating the mean would be very tedious.  Instead, we see that there are five ones, 12 twos, 8 threes, 3 fours, and 1 seven.  Hence the total count of siblings is

        1(5) + 2(12) + 3(8) + 4(3) + 7(1)  =  72

Now divide by the number of respondents to get the mean.

                   72          =            =  2.5                  29

Extending the Frequency Distribution Table

Just as with the mean formula, there is an easier way to compute the standard deviation given a frequency distribution table.  We extend the table as follows:

Number of Children (x)

Frequency (f) xf x2f

1 5 5 5

2 12 24 48

3 8 24 72

4 3 12 48

5 0 0 0

6 0 0 0

7 1 7 49

Totals f   =  29 xf  =  72 x2f  =  222

Next we calculate

                                   (xf)2                  (72)2

        SSx  =  x2f  -                 =  222 -                                                        n                        29

        =  43.24

Now finally apply the formula

       

to get

   

      

Weighted Averages

Sometimes instead of the simple mean, we want to weight certain outcomes higher then others.  For example, for your statistics class, the following percentages are given

        Homework  =  150

        Midterm  =  450

        Project  =  100

        Final  =  300

Suppose that you received an 84% on your homework, a 96% on your midterms, a 98% on your project and an 78% on your final.  What is your average for you class?

To compute the weighted average, we use the formula

                                      xw        Weighted Average  =                                                               w

We have

        xw  =  .88(150) + .97(450) + .98(100) + .78(300)  =  900.5

and

        w  =  150 + 450 + 100 + 300  =  1000

Now divide to get your weighted average

            900.5                          =  .9005            1000

and for Grouped Data

Calculating the Mean from a Frequency Distribution

Since calculating the mean and standard deviation is tedious, we can save some of this work when we have a frequency distribution.  Suppose we were interested in how many siblings are in statistics students' families.  We come up with a frequency distribution table below.

Number of Children

1 2 3 4 5 6 7

Frequency 5 12 8 3 0 0 1

Notice that since there are 29 respondents, calculating the mean would be very tedious.  Instead, we see that there are five ones, 12 twos, 8 threes, 3 fours, and 1 seven.  Hence the total count of siblings is

        1(5) + 2(12) + 3(8) + 4(3) + 7(1)  =  72

Now divide by the number of respondents to get the mean.

                   72          =            =  2.5                  29

Extending the Frequency Distribution Table

Just as with the mean formula, there is an easier way to compute the standard deviation given a frequency distribution table.  We extend the table as follows:

Number of Children (x)

Frequency (f) xf x2f

1 5 5 5

2 12 24 48

3 8 24 72

4 3 12 48

5 0 0 0

6 0 0 0

7 1 7 49

Totals f   =  29 xf  =  72 x2f  =  222

Next we calculate

                                   (xf)2                  (72)2

        SSx  =  x2f  -                 =  222 -                                                        n                        29

        =  43.24

Now finally apply the formula

       

to get

   

      

Increasing and Decreasing Functions

Definition of Increasing and Decreasing

We all know that if something is increasing then it is going up and if it is decreasing it is going down.  Another way of saying that a graph is going up is that its slope is positive.  If the graph is going down, then the slope will be negative.  Since slope and derivative are synonymous, we can relate increasing and decreasing with the derivative of a function.  First a formal definition.

 Definition of Increasing and Decreasing

A function is increasing on an interval if for any x1 and x2 in the interval then

            x1 <  x2          implies          f(x1)  <  f(x2)

A function is decreasing on an interval if for any x1 and x2 in the interval then

            x1 <  x2          implies          f(x1)  >  f(x2)

 How does this relate to derivatives?  Recall that the derivative is the limit 

            f(x2) - f(x1)                                                   x2  -  x1             

If  x1 <  x2, then the denominator will be positive.  If also  f(x1) <  f(x2), then the numerator will be positive, hence the derivative will be positive.  On the other hand if  f(x1) >  f(x2), then the numerator will be negative and the derivative will be negative.  this leads us to the following theorem.

 Theorem on Derivatives and Increasing/Decreasing Functions

Let f be a differentiable function on the interval (a,b) then 

1. If f '(x) < 0 for x in (a,b), then f is decreasing there.2. If f '(x) > 0 for x in (a,b), then f is increasing there.3. If f '(x) = 0 for x in (a,b), then f is constant.

  

 Examples and Critical Numbers

Example

Determine the values of x where the function

        f(x)  =  2x3 + 3x2 - 12x + 7

 Solution

We first take the derivative

        f '(x)  =  6x2 + 6x - 12

To determine where the derivative is positive and where it is negative, find the roots.  Factor to get

        6(x2 + x - 2)  =  6(x - 1)(x + 2)

Hence the change in sign can occur when 

        x  =  1 and x  =  -2  

Now create some test values    

x f '(x)

-3 24

0 -12

2 24

 The derivative is positive outside of [-2,1] and is negative inside of [-2,1].  We can conclude that f is increasing outside of [-2,1] and decreasing inside of [-2,1].  The graph is shown below.

       

 We saw that the values of x such that the derivative is 0 was of special interest.  Other points where there could be a change from increasing to decreasing is where the derivative is undefined.  

We call c a critical number if either f '(c)  =  0 or f '(c) is undefined.  

 Example

Determine where the function below is increasing and where it is decreasing.

                              2            f(x)   =                 +   18x                            x - 1

                              2            f(x)   =                 +   18x                            x - 1

 Solution

Since f(x) is not continuous at x  =  1, it is also not differentiable there.  Hence x  =  1 is a critical point.  To find other critical points, we take a derivative.  It is helpful to use negative exponents instead of fractions here.

        f '(x)  =  [2(x - 1) -1 + 18x]'  =  -2(x - 1) -2 + 18  =  0

        18  =  2(x - 1) -2 Divide by 2 and multiplying by (x - 1)2

        9(x - 1)2  =  1    Take the square root of both sides

        3(x - 1)  =  1        or     3(x - 1) =  -1

        x  =  4/3    or     x  =  2/3

This gives us three critical points

        x  =  2/3        x  =  1        and        x  =  4/3

Now construct a table and determine positive intervals and negative intervals

     -2(x - 1) -2 + 18    

x f '(x)

0 Positive

.9 Negative

1.1 Negative

2 Positive

        We can conclude that f is increasing for values of x less than 2/3 and values of x greater then 4/3.  f is decreasing for values between 2/3 and 4/3 excluding x  =  1.  A graph of f is shown below.

       

Application

  The weight (in pounds of a newborn infant during its first three months of life can be modeled by 

        W  =  1/3 t3 + 5/2 t2 - 19/6 t + 8

where t is measured in months.  Determine when the infant was gaining weight and when it was losing weight.

 Solution

We are asked to find when the function is increasing and when it is decreasing.  We have

        W'  =  t2 + 5t - 19/6

Using the quadratic formula or a calculator we get

        0.56      or        -5.56

Since the domain is stated to be between 0 and 3, we use only 0.56.  Now construct a table

t W'(t)

0 Negative

1 Positive

Hence the function is increasing for t greater than 0.56 and decreasing for t smaller than 0.56

We can conclude that the infant was losing weight for the first 0.56 months of its life and then began gaining weight afterwards at least up to the third month.

Source: www.ltcconline.net/greenl/courses

 Modeling with First Order Differential Equations

Whenever there is a process to be investigated, a mathematical model becomes a possibility.  Since most processes involve something changing, derivatives come into play resulting in a differential equation.  We will investigate examples of how differential equations can model such processes.  

Example    A Polluted Pond

A pond initially contains 500,000 gallons of unpolluted water has an outlet that releases 10,000 gallons of water per day.  A stream flows into the pond at 12,000 gallons per day containing water with a concentration of 2 grams per gallon of a pollutant.  Find a differential equation that models this process and determine what the concentration of pollutant will be after 10 days.

Solution

    We let x(t) be amount of pollutant in grams in the pond after t days.

We use a fundament property of rates:

         Total Rate       =       Rate In   -   Rate Out

To find the rate in we use

             grams                gallons      grams              12,000      2                             =                                      =                                         day                     day         gallon                  1           1

 

        =  24,000 grams per day

 

To find the rate out we first notice that since there was initially 500,000 gallons of water in the lake and the water level is increasing at a rate of 2,000 gallons per day, the total number of gallons of water in the lake after t days is 

        gallons  =  500,000 + 2,000 t

The units for the rate out is grams per day.  We write

             grams                gallons      grams              10,000                 x                             =                                      =                                                                     day                     day         gallon                  1           500,000 + 2,000 t

 

                 10x        =                    grams per day               500 + 2t

Putting this all together, we get

             dx                                   10x                         =  24000   -                                  dt                                500 + 2t

This is a first order linear differential equation with 

                                10             p(t)  =                             g(t)  =  24,000                            500 + 2t

We have

           

Multiplying by the integrating factor and using the reverse product rule gives

            ((500 + 2t)5x)'  =  24,000(500 + 2t)5

Now integrate both sides to get

            (500 + 2t)5x  =  2,000(500 + 2t)6 + C

                                                          C            x   =  2000(500 + 2t)  +                                                                            (500 + 2t)5 

    Now use the initial condition to get

                                                C            x   =  2000(500)  +                                                            (500)5 

        C  =  -3.125 x 1019

Now plug in 10 for t and calculate x

                                                          -3.125 x 1019

            x   =  2000(500 + 2(10))  +                                                                                      (500 + 2(10)5 

            =  218,072 grams

A graph is given below

 

Example

You just won the lottery.  You put your $5,000,000 in winnings into a fund that has a rate of return of 4%.  Each year you use $300,000.  How much money will you have twenty years from now?

Solution

This is also a 

        Total Rate  =  Rate In - Rate Out 

problem.  Let 

        x  =  the balance after t years

The rate out is   300,000 and the rate in is .04x

We have the differential equation

        dx/dt  =  .04x  -  300,000

This is both linear first order and separable.  We separate and integrate to obtain

       

       

25ln(.04x - 300,000)  =  t + C1

        .04x  -  300,000  =  C2 et/25

        x   =  Cet/25 + 7,500,000

Now use the initial condition that when t  =  0, x  =  5,000,000

        5,000,000  =  C + 7,500,000

So that 

        x   =  -2,500,000 et/25 + 7,500,000

Plugging in 20 for t gives 

        x  =  1,936,148

You will have about 2 million dollars left.