Problem of the Day - Calculator Allowed

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Problem of the Day - Calculator Allowed
If F is a continuous function and F'(x) f(x)
for all real numbers x, then

A) 2F(3) - 2F(1) B) ½F(3) - ½F(1) C) 2F(6) -
2F(2) D) F(6) - F(2) E) ½F(6) - ½F(2)
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Problem of the Day
If F is a continuous function and F'(x) f(x)
for all real numbers x, then

A) 2F(3) - 2F(1) B) ½F(3) - ½F(1) C) 2F(6) -
2F(2) D) F(6) - F(2) E) ½F(6) - ½F(2)
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Problem of the Day
If F is a continuous function and F'(x) f(x)
for all real numbers x, then

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What is the base of the natural log function?
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What is the base of the natural log function?
The base is e and this can be used to assign a
meaning to a general base a.
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Definition of Exponential Function to Base a
If a is a positive real number ?1and x is any
real number then the exponential function to base
a is
If a 1 then y 1x 1 is a constant function.
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These functions obey the usual laws of exponents.
a0 1 loga 1 0 (ax)y axy loga xy
logax logay axay axy loga xn n loga x
ax ax - y loga x loga x - loga y
ay y loga ax x
alogax x
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Definition of Logarithmic Function to Base a
If a is a positive real number (a ? 1) and x is
any positive real number, then the logarithmic
function to the base a is denoted by loga x and
is defined as
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ln x and ex are inverses of each other
and
loga x and ax are inverses of each other
y ax iff x loga y
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Solve for x
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Solve for x
log both sides
match
match
-3 x
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Solve for x
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Solve for x
exponentiate both sides
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Derivatives for Bases Other Than e
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y 2x
Examples
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Examples
y 2x
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y 23x
Examples
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Examples
y 23x
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y log10 cos x
Examples
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Examples
y log10 cos x
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Method 1
Method 2
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Integrals for bases other than e
Example
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What do you use when?
Constant Rule
Exponential Rule
Power Rule
Log Differentiation
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Applications of Exponential Functions -
Compound Interest
t number of years n number of times
compounded per yr P dollars invested r annual
interest rate A balance after t years
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Consider
(Proof is in appendix)
Thus continuous compounding yields
A Pert
(see example 6 on page 353)
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Attachments