Aim: Other Bases Course: Calculus Do Now: Aim: How do we differentiate and integrate the exponential...
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Transcript of Aim: Other Bases Course: Calculus Do Now: Aim: How do we differentiate and integrate the exponential...
Aim: Other Bases Course: Calculus
Do Now:
Aim: How do we differentiate and integrate the exponential function with other bases?
Find the extrema for
w/o calculator
( )2
x xe ef x
Aim: Other Bases Course: Calculus
Definition of Expo Function to Base a
If a is a positive real number (a 1) and x is any real number, then the exponential function to the base a is denoted by ax and is defined by
if a = 1, then y = 1x = 1 is a constant function.
(ln )x a xa e
Familiar properties
0 1x y x y
xx y
y
yx xy
a
a a a
aa
a
a a
Aim: Other Bases Course: Calculus
Definition of Log Function to Base a
If a is a positive real number (a 1) and x is any real number, then the logarithmic function to the base a is denoted by logax and is defined by 1
log lnlna x x
a
Familiar properties
log 1 0
log log log
log log
log log log
a
a a a
na a
a a a
xy x y
x n x
xx y
y
logb M log c M
log c b
Aim: Other Bases Course: Calculus
Properties of Inverses
Properties of Inverses
log
if and only if log
for 0
log for all
a
xa
x
xa
y a x y
a x x
a x x
Solve for x:
2
13 log 4
81x x
3 3
43
1log 3 log
81
log 3
4
x
x
x
2log 4
4
2 2
1
21
16
x
x
x
Aim: Other Bases Course: Calculus
More Rules
Derivatives for Bases Other than e
Let be a positive real number ( 1)
and let be a differentiable function of .
1. ln
2. ln
13. log
ln
14. log
ln
x x
u u
a
a
a a
u x
da a a
dxd du
a a adx dxd
xdx a x
d duu
dx a u dx
(ln )x a xa e
1log ln
lna x xa
Aim: Other Bases Course: Calculus
Model Problems
Find the derivative of each:
1. y = 2x
2. y = 23x
3. y = log10cosx
' 2 ln2 2x xdy
dx
3 3 3' 2 ln2 2 3 3ln2 2x x xdy
dx
10
sin' log cos
ln10 cos
1tan
ln10
d xy x
dx x
x
lnx xda a a
dx
lnu ud dua a a
dx dx
1
loglna
d duu
dx a u dx
Aim: Other Bases Course: Calculus
Opt 1
Integrating non-e Expo Functions
1. Convert to base e using
2. Integrate directly using
(ln )x a xa e
1
lnx xa dx a C
a
Evaluate:
2x dx
12
ln2x C
ln 2 xe dx 1
ln2ue du
(ln 2)1 12
ln2 ln2x xe C C
Two Options
u = (ln2)x du = (ln2)dx
ln2
dudx
Opt 2
Aim: Other Bases Course: Calculus
Exponential Growth: Interest Formulas
Exponential growthCompound Interest
y = a • bxExponential function
Exponential growthin general terms
y = A(1 + r)t
A P(1 r
n)nt
e 11
n
f nn
n
Exponential growthContinuous compoundingContinuous growth/decay
k is a constant (±)
A Pe rt
N Noekt
P = amount of deposit, t = number of years, A = balance after t years, r = annual interest rate, n = number of compoundings per year.
1 1lim 1 lim
x x
x x
xe
x x
Aim: Other Bases Course: Calculus
Model Problem
A deposit of $2500 is made in an account that pays an annual interest rate of 5%. Find the balance in the account at the end of 5 years if the interest is compounding a) quarterly, b) monthly, and c) continuously
) (1 )ntra A P
n 4 50.05
2500(1 ) $3205.094
) (1 )ntrb A P
n 12 50.05
2500(1 ) $3208.4012
) rtc A Pe 0.05 52500 $3210.06e
Aim: Other Bases Course: Calculus
Model Problem
A bacterial culture is growing according to the logistics growth function
where y is the weight of the culture in grams and t is the time in hours. Find the weight of the culture after a) 0 hours, b) 1 hour, and c) 10 hours. d) What is the limit as t approaches infinity?
0.4
1.25, 0
1 0.25 ty te
a) t = 0, 0.4 0
1.25 1.251gram
1.251 0.25y
e
b) t = 1, 0.4 1
1.251.071 .
1 0.25y g
e
Aim: Other Bases Course: Calculus
Model Problem
A bacterial culture is growing according to the logistics growth function
where y is the weight of the culture in grams and t is the time in hours. Find the weight of the culture after a) 0 hours, b) 1 hour, and c) 10 hours. d) What is the limit as t approaches infinity?
0.4
1.25, 0
1 0.25 ty te
c) t = 10, 0.4 10
1.251.244 .
1 0.25y g
e
d) t , 0.4
1.25 1.25lim 1.25 .
1 01 0.25 ttg
e
Aim: Other Bases Course: Calculus
The Product Rule