MATH 151 Engineering Math I, Spring 2014
JD Kim
Week10 Section 4.4, 4.5
Section 4.4 Derivatives of Logarithm Functions
We know that it is differentiable because it is the inverse of the differentiable
function y = ex .
1
d
ln x =
dx
x
Ex1) Find the derivative
1-1) f (x) = cos(ln x)
1-2) y = ln(1 + ln x)
1-3) g(x) = ln
r
3x + 2
3x − 2
1
x+1
1-4) f (x) = ln √
x−2
d
1
loga x =
dx
x ln a
f ′ (x)
d
loga f (x) =
dx
f (x) ln a
Ex2)
d
log(2 + sin x)
dx
2
d x
a = ax ln a
dx
d f (x)
a
= af (x) ln a · f ′ (x)
dx
Ex3)
d x2
10
dx
Ex4) Find the derivative of f (x) = tan5 x + 5tan x
Ex5) Find the derivative of f (x) = log3 (5 − x4 ).
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Logarithmic Differentiation
The calculation of derivatives of complicated functions involving products, quotients, or powers can often be simplified by taking logarithms.
Steps in Logarithmic Differentiation
1. Take logarithms of both sides of an equation y = f (x).
2. Differentiate implicitly with respect to x.
3. Solve the resulting equation for y ′.
√
x3/4 x2 + 1
Ex6) Differentiate y =
(3x + 2)5
Ex7) Differentiate these;
7-1) f (x) = (sin x)cos x
4
7-2) y =
ex (x2 + 2)3
(x + 1)4 (x2 + 3)2
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Section 4.5 Exponential Growth and Decay
In many natural phenomena, quantities grow or decay at a rate proportional to
their size.
Definition If y(t) is the value of quantitiy at time t and if the rate of change of
y with respect to t is proportional to its size y(t) at any time, then
dy
= ky
dt
where k is a constant.
Then the quantity y(t) at time t is given by
y(t) = y0 ekt
where y0 is the initial quantity and k is a constant. The GOAL is to find k.
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Ex8) A bacteria culture starts with 1000 bacteria, and after half an hour the
population is tripled.
a) Find an expression for the number of bacteria after t hours.
b) Find the number of bacteria after 20 minutes.
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Half-Life
The half-life of a substance is the amount of time it takes for half of the substance
to disintegrate.
Ex9) The half-life of radium-226 (226
88 Ra) is 1590 years.
a) A sample of radium-226 has a mass of 100 mg. Find a formula for the mass of
226
88 Ra that remains after t years.
b) When will the mass be reduced to 30 mg?
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Ex10) After 3 days a sample of radon-222 decayed to 58% of its original amount.
What is the half-life of radon-222? How long will it take the sample to decay to 10%
of its original amount?
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Ex11) A curve passes through the point (0,7) and has the property that the slope
of the curve at every point p is half the y-coordinate of p. Find the equation of the
curve.
10
Ex12) The rate of change of atmospheric pressure P with respect to altitude
h is proportional to P , provided that the temperature is constant. At a specific
temperature the pressure is 101 kP a at sea level and 86.9 kP a at h = 1000 m. what
is the pressure at an altitude of 3500 m
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Compound Interest
If A0 dollars is invested at r % compounded n times a year, then the amount in
r
the account after t years is given by A = A0 (1 + )nt .
n
Ex13) If $4000 is invested at 8 % compounded monthly, how much money is in
the account at the end of 6 years?
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Continuous Compound Interest
If P dollars is invested at r % compounded continuously, then the amount in the
account after t years is given by A = P ert .
Ex14) How much money should be invested now at 6 % compounded continuously
in order to have $30, 000 after 18 years from now?
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Definition
The rate of cooling of an object is proportional to the temperature difference
between the object and the temperature of the object’s surroundings. If y(t) is the
dy
= k(y − T ), where y is the temperature
temperature of the object at time t, then
dt
of the object at time t and T is the room temperature (the temperature of the
room in which the object is cooling). The solution of this equation, which gives the
temperature of the object at time t, is y(t) = (yo − T )ekt + T , where y0 is the initial
temperature of the object.
Ex15) A thermometer is taken from a room where the temperature is 20◦ C to the
outdoors, where the temperature is 5◦ C. After one minute, the temperature reads
12◦ C. Use Newton’s Law of cooling to answer the following questions.
a) What will the reading of the thermometer be after 2 minutes?
b) When will the thermometer read 6◦ C?
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