Chapter 1
Section 1.1
f (x) means the value of the function f
also called the "INPUT.”
at x . Be careful f (x) doesn’t mean
Let’s start with a function f ( x)  2 x  4
f
times
x.
x is
f (1) 
f (5) 
f( )
f (h) 
f ( x  h) 
y  2 x  5
f (2)  22  5  1
Write
as an ordered pair:
Write
as a function:
Group work:
1) Write
f ( x)  2 x  5
x, y   2,1
as a function.
2) Given g ( x)  2 x  4 , Find:
a) g(2)
b) x when g(x) = 5
c) Fill in the blank (4,___)
d) If g represents the number of gadgets in Jim's car after x days, then what is g (5) and what does it
mean?
1
3) Compute the indicated values of the given function.
a)
; H(-4)
b) G(t)=
Difference Quotient
Find
for the following:
i)
ii)
iii)
2
Composite Functions
 f  g x  f gx
Given:
f ( x)  x 2  4 , g ( x )  3 x  4 , p  x   4  x 2
1) Find:
g  f x
 f  px
g  p2
2) Find
 f  g 2
 f  g 4
g  f 3
g  f 1
 f  g 3
3) Find
3
4) Find a functions f(x) and g(x) such that the
5) Find the value of x where
6) Apply the formulas for determining a child's body surface area and a child's dose of medication. S is
a child's body surface area. k is the weight of the child in kilograms. C is the child's dose of medication
in milligrams.
a) Find the body surface area, to the nearest hundredth, of a child with a weight of 26 kg.
b) What is the child's dose, to the nearest unit, for a child that weighs 26 kg.
c) Write a composite function.
4
Domain and Range
Domain:
Range:
Determine the domain and range for the following:
2,3, 4,1, 5,3
y
1
2x  2
y  x4
 1,0, 2,1, 2,3
y
3x  5
3x  1
y   2x  3
1,0, 2,1, 3,3
y  x2
y
2 x
x2
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Business Formulas
x = The number of units.
p = The price of a unit.
Demand – D(x)=p, The price charged when x units are demanded.
Supply – S(x)=p, The price charged when x units are supplied.
Revenue – R(x)=px=xD=(price)(quantity sold), The money obtained from selling x units.
Cost– C(x), The cost to produce x units.
Profit– P(x), The profit obtained from selling x units.
Average Cost– AC(x)= , cost per unit
Average Revenue– AR(x)=
Average Profit– AP(x)=
, Revenue per unit
, Profit per unit
Profitability
a) Find the revenue and profit functions
b) Graph the revenue, cost, and profit functions in your calculator.
c) Find all values of x for which production of the commodity is profitable.
6
Applications of functions.
1) Arthur, the manager of a furniture factory, finds that the cost of production q bookcases during the
morning production run is
dollars. On a typical workday,
bookcases
are produced during the first t hours of a production run for
a) Express C in terms of t.
b) How much will have been spent on production by the end of the 3 rd hour?
c) What is the average cost of production in terms of q.
d) Express AC in terms of t.
e) What is the average cost of production during the 3rd hour?
f) Arthur has a budget of $11,000, when will this limit be reached?
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It is estimated that t years from now, the population of a certain suburban community will be
thousand people.
a) What is the domain of the function?
b) What will the population of the community be 9 years from now?
c) By how much will the population increase during the 9th year? This is the time between the 8th and
the 9th year.
d) What happens to the population as t gets larger and larger? Hint: plug it in your calculator?
e) What does d mean in real life? What reasons could you give for this situation?
8
Section 1.2
The Distance Formula:
Find the distance between the points (3 , 1) and (6 , 5) using a right triangle.
Hint: 1. plot the points, 2. use the line between the two points as the hypotenuse.
Now we are going to find the formula
a2  b2  c2
Distance formula: D =
1. Find the distance between (1, 0) and (3, 2). Leave your answer in simple radical form
2. Find the distance between (-1, 2) and (-3, 4). Leave your answer in simple radical form.
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PARENT GRAPHS
Sketch the shape of each in the box below. There is no grid. I just want the shape.
f x   2 x  1
f x   x 2
f x 
f x   x 3
x
f x   3 x
f x   x
Sketch the graph of the given function, include all x and y intercepts, and label the maximum or
minimum values if possible.
1)
2)
x-intercept
x-intercept
y-intercept
y-intercept
10
3)
x-intercept
y-intercept
Points of Intersection
Substitution
Elimination
Non-linear substitution
11
Piece-Wise Graphs
x  1
3x  1 if
 2
Graph the function f  x    x
if  1  x  4 , and evaluate f  3 , f  1 , f  0  , f  4  , f  7 
 x  5 if
x4

x
y
x
y
x
y
Optimal Selling Price
x = The number of units.
p = The price of a unit.
Revenue – R(x)=px=xD=(price)(quantity sold), The money obtained from selling x units.
Cost– C(x), The cost to produce x units.
Profit– P(x), The profit obtained from selling x units.
Warm-up
i) Fifty tires are manufactured for $20 each. What is the total manufacturing costs?
ii) It is estimated that if the tires sold for p dollars apiece, consumers will buy 1,560-12p of them each
month.
The number of units= _____________________
The price of a unit=______________
If the tires sold for $40 each, how many of the tires were sold?
iii) If 1080 tires were produced and purchased, what is the profit made on the 1080 tires?
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1) A manufacturer can produce tires at a cost of $20 each. It is estimated that if the tires sold for p
dollars apiece, consumers will buy 1,560-12p of them each month. Express the manufacturer's monthly
profit as a function of price, graph this function, and use the graph to determine the optimal selling price.
How many tires will be sold each month at the optimal price?
The number of units= _____________________
The price of a unit=______________
Cost= $20(number of units)=
Revenue= (price of a unit)(number of units)=
Profit= Revenue- Cost=
Optimal selling price=
Number of units sold at the optimal price=
2) A bookstore can obtain an atlas from the publisher at a cost of $10 per copy and estimates that if it
sells for p dollars per copy, approximately 20(22-x) copies will be sold each month. Express the
bookstore's monthly profit from the sale of the atlas as a function of price, graph this function, and
estimate the optimal selling price. How many copies will be sold each month at the optimal selling
price?
The number of units= _____________________
The price of a unit=______________
Cost=
Revenue= (price of a unit)(number of units)=
Profit= Revenue- Cost=
Optimal selling price=
Number of units sold at the optimal price=
3) Suppose that when the price of a certain commodity is p dollars per unit, then x hundred units will be
purchased by consumers, where p = -0.05x+38. The cost of producing x hundred units is
hundred dollars. Express the profit P as a function of x.
The number of units= _____________________
The price of a unit=______________
Cost=
Revenue= (price of a unit)(number of units)=
Profit= Revenue- Cost=
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Motion of a projectile.
4) A projectile is fired at an inclination of 45° to the horizontal. The height of the projectile is given by
x2
 x , where x is the horizontal distance from the firing point.
the function h  x   
500
a) Find the maximum height of the projectile.
b) How far from the firing point will the projectile strike the ground?
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Section 1.3: linear equations
m=Slope = ______________=_______________=______________
m=
m=
m=
m=
Find the slope of the line going through the points (4,2) and (-3,0).
Consider the following graph of a function.
On the chart below specify the intervals over which …
1.
The function is positive, negative, or zero.
2.
The function is increasing, decreasing, or constant.
3.
The rate of change between any two points in the interval is positive,
AB
BC
CD
DE
EF
FG
negative, or zero.
GH
HI
1. function
pos, neg, 0
2. function
inc, dec, constant
3. rate of change pos,
neg, 0
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Find the slope and y-intercept for the following:
Finding the equation of the line.
y  mx  b
1st find the slope
2nd find the y-intercept
1. Given a point (3,4) on the line and a slope of -3. Find the equation of the line.
2. Given two points on the line (3,2) and (4,3). Find the equation of the line.
16
Linear Applications
1. The average price of a movie ticket is P(t )  0.16t  4.3 , where t is the number of years since 1990.
P(t ) 
0.16
t  4.3
1 __________
a) Find P(5) . Verbally interpret.
c) Slope = _____. Verbally interpret.
b) When was the movie ticket price $6.70?
2. After four months of use, Biffs Spiffy Spam-Free computer had dropped to $1100 in value. After ten
months, the value had declined to $620.
a) Find a linear equation that models the data.
b) Slope = _____. Verbally interpret.
c) y-intercept. = (
,
) Verbally interpret.
d) How long will it take the computer’s value to drop to $100?
17
Find the linear equation that models the data.
3. World grain production was 1241 million tons in 1975 and 2048 million tons in 2005, and has been
increasing at an approximately constant rate. Find a linear function for world grain production, P, in
million tons, as a function of t, the number of years since 1975.
4. A manufacturer buys $20,000 worth of machinery that depreciates linearly so that its trade-in value
after 10 years will be $1,000. Express the value of the machinery as a function of its age.
Form an equation, then answer the question.
5. Jimmy “the hands” charges customer $14 monthly plus 50 cents per minute for backrubs.
6. a) Tony's tires has operating costs of $500 per day with a startup cost of $35000. Write a cost
equation to describe Tony's costs.
b) Tony's tires has a daily average revenue of $1025. Write a revenue equation to describe Tony's daily
revenue.
c) Create a profit function.
d) When will Tony's tires become profitable?
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Section 1.4
Optimization problems
1) A dairy farmer plans to enclose a rectangular pasture adjacent to a river. To provide enough grass
for the herd, the pasture must contain 180,000 square meters. No fencing is required along the river.
What dimensions will use the least amount of fencing? (minimize the perimeter function)
Primary equation: Perimeter
Secondary equation:
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2) A storage box with a square base must have a volume of 80 cubic centimeters to hold all of the slime
I collected as a child. The top and bottom cost $0.10 per square centimeters and the sides cost $0.05 per
square centimeter. Find the dimensions that will minimize cost.
Primary equation: Cost
Secondary equation:
3) A company has received an order from the city recreation department to manufacture 8,000
Styrofoam kickboards for its summer swimming program. The company owns several machines, each
of which can produce 30 kickboards an hour. The cost of setting up the machines to produce these
particular kickboards is $20 per machine. Once the machines have been set up, the operation is fully
automated and can be overseen by a single production supervisor earring $19.20 per hour. Express the
cost of producing the 8,000 kickboards as a function of the number of machines used, and estimate the
number of machines the company should use to minimize cost.
Primary equation: Cost
Secondary equation: time
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Supply and Demand
a) Find the value of x, for which equilibrium occurs and the corresponding equilibrium price p.
b) Sketch the graphs of the supply and demand curves on the same graph, and shade the market shortage
and the market surplus.
c) For what values of x is there a market shortage?
d) For what values of x is there a market surplus?
e) Where does the supply line cross the y-axis. Discuss the significance of this.
by hand
1)
and
Calculator
2)
and
21
Break-Even Analysis- this time, you create the linear equations.
1) A greeting card company can sell cards for $2.75 each. The company's total cost consists of a fixed
overhead of $12,000 plus variable cost of 35 cents per card.
The number of units= _____________________
The price of a unit=______________
Cost=
Revenue=
Profit= Revenue- Cost=
Will the company have a profit or loss if it sells 5000 cards?
When will the company break-even (become profitable)
How many cards will the company need to sell to make a profit of $9,000?
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Section 1.5/1.6
Left hand limit
Look at the graph below and find the limit as x approaches a from the left (-).
Right hand limit
Look at the graph below and find the limit as x approaches a from the right (+).
Limit
The previous examples are of functions that are not continuous.
List the values for each of the functions were they are not continuous.
23
Finite Limits that exist.
1) Graph:
Draw a picture.
, Use an x-interval of [-2,1], hit zoom fit. Find the value of x=-1.
Limit from the left
Limit from the right
Limit
Is the function continuous at
2)Graph:
?
, Use an x-interval of [2,4], hit zoom fit. Find the value of x=3.
Draw a picture.
Limit from the left
Limit from the right
Limit
Is the function continuous at
?
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3) Graph:
, Use an x-interval of [0,2], hit zoom fit. Find the value of x=1. Find the domain.
Simplify the function and substitute x=1.
Draw a picture.
Simplified:
Domain:
Simplified with value of x=1:
Limit from the left
Limit from the right
Limit
Is the function continuous at
?
Limits that don't exist.
1)Graph:
, Use an x-interval of [3,5], hit zoom fit. Find the value of x=4. Find the domain.
Draw a picture.
Domain:
Limit from the left
Limit from the right
Limit
Is the function continuous at
?
25
Limits involving infinity.
1) Graph:
Draw a picture.
, hit zoom 6.
Limit as x approaches neg. infin.
2) Graph:
Limit as x approaches infin.
, hit zoom 6.
Draw a picture.
Limit as x approaches neg. infin.
3) Graph:
Limit as x approaches infin.
, hit zoom 6.
Draw a picture.
Limit as x approaches neg. infin.
Limit as x approaches infin.
26
Finding limits analytically.
Plug the number in!
If
Find:
27
Simplify first!
Can't substitute-Use calculator or left and right values at 4.
Look at leading term. Arm movements....
Divide by the
Use this:
Limit is infin. or neg. inf....divide everything but leading terms and use the rule.
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Applications:
1) Alicia, the plant manager of a plant, determines that when x% of the plant's capacity is being used,
the total cost of operations C hundred dollars, where
The company is has an ideal capacity of 70%. What costs should the company expect to approach as
the move towards the ideal capacity?
2) A big box store determines the cost, in dollars, of creating a competing "star wars" tooth brush
called a "space wars" toothbrush is
, where x is the number of toothbrushes.
a) Find the average cost.
b) Find the limit of the average cost as x approaches infinity. What does this mean?
3) The population of a community is given by the function
, where t is in years.
What can you expect the population to be in the long run?
4) Suppose the cost to remove x% of an oil spill is given by the function
where C(x) is the cost in thousands of dollars. What happens as
?
29
Piece-Wise Functions
1)
,
Limit from the left
x
Limit from the right
x
y
y
Limit
Is the function continuous at
2)
?
,
Limit from the left
x
Limit from the right
x
y
y
Limit
Is the function continuous at
?
30