Math 127 Formula and Examples Sheet Exam #1
Average Rate
Δ
Y
f(b)
–
f(a)
of Change
Δ
X
b
–
a
Slope Intercept Form
yy
0
= m(xx
0
)
Half Life (t
1/2
)
=
 ln(2)
Doubling Time
=
ln(2)
k
k
Periodic Function
= P
0
a
t
a = e
k
k = ln(a)
Continuous Function
= P
0
e
kt
r = a1
a = r + 1
Decay when rate is negative, Growth rate is positive.
EXAMPLES:
1.
Prozac has a halflife of 3 days. What percent
remains after a week?
P(t) = P
0
e
kt
t
1/2
= ln(2)
k = ln(2)
k = .23
k
3
P(7) = P
0
e
k(7)
= P
0
(.198) ≈
P
0
(.2), only 20% of original
substance remains after a week.
2.
A substance loses 10% of its mass in 5 hours. Find
the halflife of this substance. (Exponential Decay)
P(5) = P
0
e
5k
r = 1  .1 = .9 (of original substance)
.9P
0
= P
0
e
5k
.9 = e
5k
ln(.9) = ln(e
5k
)
5kln(e) = ln(.9)
*
NOTE:
ln(e) = 1
k = ln(.9) = .021
t
1/2
=
ln(2) = 32.9 hours
5
.021
3.
A painting is supposedly by Vermeer (16321675)
and it contains 99.5% of its carbon14. Its halflife is
5,730 years. Is the painting fake?
t
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This note was uploaded on 03/03/2010 for the course MATH 127 taught by Professor Rudvalis during the Fall '07 term at UMass (Amherst).
 Fall '07
 rudvalis
 Rate Of Change, Slope

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