3.3 Even and odd
The function f is an even function if
f(­x) = f(x) for all x in the domain of f
The function f is an odd function if
f(­x) = ­f(x) for all x in the domain of f
3.4 Variation
Direct Variation
•
Inverse variation
•
Joint Variation •
Combined Variation
•
Examples:
v varies directly as the square of s
v = ks2
`
t varies directly as s
t = ks
y is inversely proportional to x
y = k
x
y varies directly as x and inversely as the square of p.
y = kx
p2
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Direct Variation
• The words “y varies directly with x” or “y is directly proportional to x” mean that y = kx for some real constant k.
• The number k is called the constant of proportionality.
• Y varies directly with x is equivalent to y=mx + b with b = 0.
Examples: • Your Pay varies directly with the number of Hours worked. p = k h
• The Distance traveled on an interstate with cruise control on varies directly with the Time you travel.
d = k t
• v varies directly as the square of s
v = ks2
The Weight (V) of an object on Venus varies directly as its weight (E) on Earth. A person weighing 120 lb on Earth would weigh 106 lb on Venus. How much would a person weighing 150 lb on Earth weigh on Venus?
Solve for V when E is 150
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Pressure on a Diver
The pressure on a diver in the ocean is directly proportional to the depth of the diver below the oceans surface. If the pressure at a depth of 20ft is 8.9 pounds per square inch, find the pressure at a depth of 50ft.
P = kd
Direct Variation as the nth Power
If y varies as the nth power of x, then
y = kxn
where k is a constant
Example: The distance s that an object falls from rest varies directly as the square of the time t that it has been falling. If an object falls 144 feet in 3 seconds, how far will it fall in 7 seconds?
s = k t2
144 = k 32
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Inverse Variation:
• The words “y varies inversely with x” or “y is inversely proportional to x”, mean that for some real­number constant k.
Two variables are said to vary inversely or be inversely proportional if their product is constant k. Is equivalent to Hyperbola ­the graph of inverse variation
Definition of Inverse Variation nth power
The variable y varies inversely as the nth power of x, or y is inversely proportional to the nth power of x, iff
y = k
xn
where k is the variation constant
Example: Loudness of a Sound
The impact of sound on the human ear , measured in decibels, from a stereo speaker is inversely proportional to the square of the distance of the listener from the speaker. If the loudness is 35 decibels at a distance of 10ft from the speaker, what is the loudness when the listener is 6ft from the speaker?
L = k
d2
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Joint Variation
The variable z varies jointly as the variables x and y iff
z = kxy
Example: The volume of a cone varies jointly with the square of the radius and the height.
V = k r2 h
Example:
The cost of insulating the ceiling of a house varies jointly as the thickness, in inches, of the insulation and the area, in square feet, of the ceiling. It costs $175 to insulate a 2100­square­foot ceiling with insulation that is 4 inches thick. C = k t A
175 = k* 4*2100
175 = 8400k
k = .0208
Find the cost of insulating a 2400­square­foot ceiling with insulation that is 6 inches thick.
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Combined Variation:
Example
#39. Automotive Technology
The force needed to keep a car from skidding on a curve varies jointly as the weight of the car and the square of its speed and inversely as the radius of the curve. It takes 2800 pounds of force to keep an 1800­pound car from skidding on a curve with a radius of 425 ft and traveling 45 mph. What force is needed to keep the same car from skidding when it takes a similar curve with radius 450 ft at 55 mph?
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The time it takes to build a highway varies directly with the length of the road, but inversely with the number of workers. If it takes 100 workers 4 weeks to build 2 miles of highway, how long will it take 100 workers to build 20 miles of highway?
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