Section 4.2 Families of Curves

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Transcript Section 4.2 Families of Curves

Section 4.2
Families of Curves
• We have seen before that knowledge of one
function can provide knowledge of many
others
• For instance, what we know about y = x3 can
give us information about y = x3, y = 5x3,
y = (x - 10)3, or y = 3(x + 2)3 - 7
• We call these families of functions
• This helps us in their use in mathematical
modeling
• The position of an object moving vertical
under the influence of gravity is given by
h(t )  4.9t 2  v0t  h0
• Where t is time in seconds and h is the height
above the ground
• What do you notice about this function?
• Curves of the form y = Asin(Bx)
– This family is used to model a wave
• Curves of the form y  e
 ( x a ) 2 / b
– This family is related to the normal density
function used in probability and statistics
• Curves of the form y  a(1  e
bx
)
– This family of functions represents a quantity that
is increasing but leveling off
• The temperature of an object placed in a refrigerator
will cool off quickly at first, but slow down as it
approaches the temperature of the refrigerator
Examples
 ( x a ) 2 / b
• Find a function of the form y  e
for
b > 0 with a local maximum at x = 2 and points
of inflection at x = 1 and x = 3
– This is number 11 from 4.2 in the book
• Number 29 from 4.2 in the book