Tyler Aniol
Business Statistics 3355-902
Professor Gary Krause
Project 1
Due February 17, 2016
Aniol, Tyler
Page 1 of
Graph 1.1 Histogram Grocery Store Sample Data
20
18
16
14
12
10
Frequency within the Spending Range 8
6
4
2
0
Money Spent from the Sample
Now lets begin seeing the derivative as a function related to our original function that
calculates instantaneous rates of change for the original function.
It works like this:
The original function takes x and computes the second coordinate for the graph
Slopes, Difference Quotients, and derivatives are connected.
So lets start with a given functions graph and put in ( x, f ( x) and ( x h, f ( x h) .
Lets get the set-up from average rate of change through Difference Quotient through
derivative using x and
More shortcuts:
Product Rule!
If f(x) is a string of factors, you may use the Product Rule. For the simplest case,
suppose f(x) = h(x)(p(x).
f ' ( x ) h ' ( x ) p( x ) h ( x ) p' ( x )
f (x) x (x 4)
Example 13, cont.:
here:
h (x )
p( x ) x 4
x
h' (x)
We
IROC 2002
f '(5) 300(5) 770 730
While the derivative is a large negative number. Note that REVENUE is still positive but
it is decreasing with each additional thousand metric tons. The revenue for the next week
will be 730 LESS than the preceeding week.
A
One way is to multiply the functions and use the Sum rule.
The other way is to use the Product Rule.
Lets do it both ways.
Multiply and use the Sum Rule
1
3
1
f (x) x (x 4) x 2 (x1 4) x 2 4x 2
The derivative is, then,
1
1
3 2
x 2x 2
2
Heres some room for
In fact, given this point (1, 1) and m = 3, I can come up with the FORMULA for this
tangent line using the old point slope formula:
y y0 m(x x 0 )
y 1 = 3 (x 1)
This is nice to know and a standard question.
i.e. y = 3x 2.
Heres a picture of that line on t
Now for the Difference Quotient:
f (x h) f (x) 10 x h 10 x 10 x (10 h 1)
. Note that 10 x isnt affected by the limit
h
h
h
so lets look at
10h 1
h
and lets note that ln10 2.30258509.
h = .01
h = .001
h = .0001
2.32929
2.305238
2.302850
h = .01
h = .001
h
*
The annual US production of talc, a soft mineral, in thousands of metric tons, is given by
f (t ) 150t 2 770t 10, 400
where t is time in years and t = 0 corresponds to 1997.
A.
Graph this function from 1997 to 2007.
B.
Find the annual production in 2000
Example 7:
Find the equation of the tangent line at x = 3 for f ( x ) x 3 x 2 5 .
f ' ( x ) 3x 2 2 x
f ' (3) 33 = m
f(3) = 27 + 9 5 = 31. This is the point to use: (3, 31) in the point slope equation:
y y 0 m( x x 0 ) gives y 31 = 33(x 3)
The tangent line
Shortcuts to getting the derivative:
First lets do it the LONG way. Then well look at pictures and a shortcut.
Given f ( x) x 3 Find its derivative.
We will follow these steps.
Find the DQ
Take the limit of the DQ as h 0
Finding the Difference Quotient:
f
Now lets talk about the typical polynomial question:
Given this polynomial, find:
Domain, range, intercepts, derivative, sketch, equation of tangent line at a given point.
Example 8:
Given: f ( x ) x 2 6 x 5
Domain:
Vertex:
x-intercepts:
y-intercept:
all
Incentive compensation plans are used to attract and retain top managerial talent as well as to
align the interests of management with shareholders T
Industrial groups are organizations comprised of companies in different industries with common
ownership
Jesus Alvarez
1. Calculate the present value of $1,000 zero-coupon bond with 5 years to maturity if the required
annual interest rate is 6%.
1000/ (1+.06) ^5=747.258
2. A lottery claims its grand prize is $10 million, payable over 20 years at $500,000 per
Jesus Alvarez
2/25/14
2.
3.
A. More because you have more money
b. More because it becomes more liquid
c. Less because its expected return relative to other investments has gone
down
d. Less because its expected return has is going to go down
True, becaus
Answers
1. Fresh out of college, you are negotiating with your prospective new employer. They offer
you a signing bonus of $1,000,000 today or a lump sum payment of $1,250,000 three
years from now. If you can earn 7% on your invested funds, which bonus sh
An example of quotient rule:
f (x)
1
x
Domain:
Range:
all Real numbers except zero
all Real numbers except zero
no intercepts
VA
HA
x = 0, discontinuous here
y=0
The top polynomial is 1; its derivative is zero.
The bottom polynomial is x; its derivative
Think of replacing the actual curve with a short section of a linedoes the line have
positive or negative slope if the graph is increasing? decreasing?
Lets look at some slopes of the lines tangent to our parabola:
(x, f(x)
f '(x) 2x 4
(1, 5)
6
negative
(
What does it mean when the derivative is zero?
It means that the instantaneous rate of change is going from a negative number to a
positive number (or vice versa) and that youve found a turn-around point.
What does it mean when the derivative is positive?
Lets talk slope of the tangent lines.
The Difference Quotient:
f (x h) f (x) e x h e x e x (e h 1)
h
h
h
Since e x isnt affected by the limit, Ill work with the factor that has an h in it:
Now lets take the limit as h approaches zero:
eh 1
h
Notice that t
Popper 11, Question 5
*
If $100 is invest in an account that earns 6% compounded annually, then the amount in
the account after t years is given by
C ( x) 3 x A(t ) 100(1.06)t
Find the amount after 5 years and the instaneous rate of change at that time.
A
Take the derivative of f(x) and set it equal to zero:
f '(x) 12x 3 24x 2 60x 72 0
Note that theres a common factor of twelve, divide it out:
f '(x)
x 3 2x 2 5x 6 0
12
Use the Rational Root theorem and synthetic division to find that the factors of the
de
Application problems!
Always important!
*
The revenue in dollars from the sale of x car seats for infants is given by
R ( x ) 60 x 0.025 x 2
0 x 2, 400
A.
Graph the revenue curve over its domain.
B.
Find the revenue for a production level of 1,000 car sea
*
Suppose that in a given gourmet food shop people are willing to buy x pounds of
chocolate candy per day at $p per quarter pound, as given by the following equation:
x 10
180
p
2 p 10
What is the instantaneous rate of change of demand with respect to pr
AROC from 500 to 1000:
f (500) 23750
f (1000) 35000
AROC
35000 23750
22.5
500
The slope of the secant line joining these two graph points in 22.5.
This means that you are getting, on average, $22.5 per baby seat when production
changes from 500 seats to
Where are the TAPs? Set the derivative equal to zero and solve for x!
f '(x) 3x 2 26x 9 (3x 1)(x 9) = 0
[at the TAPs the slope of the tangent line = 0]
x = 1/3 and x = 9
What are the graph points?
f (x) x 3 13x 2 9x 117
f(1/3) =
1 13 9
392
117 =
118.52
*
The number of female newborn deaths per 100,000 births in France is given by
M (t ) 0.005t 2 0.65t 22.8
Where t is in years and t = 0 corresponds to 1980.
A.
Graph this function from 1980 to 2005.
B.
Find the number of newborn girls who died in 2001.
C.
Heres a table of values:
inputs (x)
-3
outputs (y)
9
-2
4
-1
1
0
0
1
1
2
3
4
9
4
16
5
25
1 9
8
4
1 3 2
The average rate of change from 3 (initial) to 1 is
The average rate of change from 1 (initial) to 4 is
16 1 15
3
4 1 5
Now, Im using the function x 2
How do you find turn around points?
Set the derivative = 0 and solve for x.
Then use that x in the original formula to get the y value of the graph point.
Example 9:
f (x) x 3 13x 2 9x 117 (x 3)(x 3)(x 13)
Domain:
All Real numbers
Range:
All Real numbers
Example 14:
f (x) e x ( x 2 4x)
Domain:
all Real numbers
Leftward asymptotic to x = 0.
x-intercepts:
0 and 4
Where are the turn-around points?
f(x) = E(Q)
the derivative is E(Q) + E(Q)
recall that the derivative of e x is e x .
f '(x) e x ( x 2 4x) e x (
Retirement Project
Fall 2016
Name:
Abraham Rubio
Student ID: K00344510
Where a cell is highlighted in yellow, you must set up the relationships-that is, set up the equations or use the spreads
functions-to compute the required values. For instance, you ca
Abraham Rubio
10/12/16
Chapter 5
Executive Summary
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important factor is to understand what the costumers want and identify how we can affect how
they think and react. Mar
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Chapter 6
Chapter 6 takes a closer look at market strategies driven by customer value, they vary
depending on what customers need and want and categorize them based on different marketing.
Businesses know t
Retirement Project
Spring 2017
Name: Abraham Rubio
Student IK00344510
complete the computation. Do not enter values (numbers) in these cells. The computations should be based on
information provided in other cells-that is, you should refer to the location
Abraham Rubio
Executive summary ch. 2
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chapter 2 explains the company wide stratigic planing and its four steps, it also
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goes into depth of how marketing wo