Download - Calc 5.5a
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Define exponential functions that have bases other than eDifferentiate and integrate functions that have other basesUse Exponential functions to model compound interest and exponential growth
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The base of the natural exponential function is e. This base can be used to assign meaning to a general base a
The laws of exponents apply here as well:1.a0 = 12.axay = ax+y
3.ax/ay = ax – y
4.(ax)y= axy
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Ex 1 p. 360 Radioactive Half-life ModelThe half-life of carbon-14 is about 5715 years. A sample contains 1 gram of carbon-14. How much will be present in 10,000 years?
Solution: Let t = 0 represent the present time and y represent the amount (in grams) of carbon-14. Using a base of ½, you can model y by the equation
57151
2
t
y
If t = 0, y = 1 gram. If t = 5715, then y = ½, which would be correct.
1000057151
.29734682
y
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Remember, this is just the change of base rule you’ve seen before, just in a new setting!
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Logarithmic Properties still apply
a
a a a
a a a
a
1. log 1 0
2. log log log
3. log log log
4. log logna
xy x y
xx y
y
x n x
Exponential functions and logarithmic functions are inverse functions
( ) xf x a( ) logag x x
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So as review, we’ll work with these properties with bases other than base eEx. 2a, p. 361
13
243x Take the log base 3 to each side
3 3
1log 3 log
243x
53log 3x
5x
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Ex. 2b, p. 361
27 log x Exponentiate each side using base 2
2log72 2 x72 x
128 x
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When thinking of these derivatives, it is often helpful to think of them as natural exponential things or as natural log things.
lna xxd da e
dx dx
ln lna x de a x
dx
ln ln lna x xe a a a
logad
xdx
1ln
ln
dx
dx a
1 1
ln a x
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Ex 3 p. 362 Differentiating functions to other bases.Find the derivative of each function.
. 2xa y ' (ln 2)2xy 5. 2 xb y 5' (ln 2)2 (5)xy 55ln 2 2 x
. logsinc y x log with no base shown is a common log, base 10cos
'(ln10)sin
xy
x cot
ln10
x
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Sometimes an integrand will work with a exponential function involving another base than e. When this occurs, we can do one of two things – convert to base e using the formula and integrate, or integrate directly, using the following formula:
(ln )x a xa e1
lnx xa dx a C
a
Ex 4 p. 363 Integrating an exponential function with base 33xdx
13
ln 3x C
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When the power rule was introduced in Ch. 2, we limited it to rational exponents. Now the rule is extended to cover all real exponents.
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This next example compares the derivatives of four different functions involving exponents. Be CAREFUL!
2d.
dxea e 0 Constant rule
2d.
dxxb e 22 Exponential rulexe
d.
dxec x
1 Power ruleeex
. xdd xdx
xy x Logarithmic differentiation
required!ln ln xy x lnx x' 1
ln (1)y
x xy x
1 ln x ' 1 lny y x 1 lnxx x
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5.5a p.366/ 3-60 mult 3