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


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


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


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


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


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


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


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


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


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


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


The Product Rule

aim: other bases course: calculus do now: aim: how do we differentiate and integrate the exponential...

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