Math 111 - 10/31/11

5.2) Exponential Functions

Learn Math Equation: f(( x)) = 2(^ x)^ , g(( x)) = 5(^ x)^ , h(( x)) =(((/ 1,/ 2)/))(^ x)^ , j(( x)) = 1 . 3 7(^ x)^ ,s.

any Learn Math Equation: a(^ x)^ where Learn Math Equation: a > 0 usually Learn Math Equation: as= 1

Remember:

Learn Math Equation: a(^ 0)^ = 1

Learn Math Equation: a(^ - 1)^ =(/ 1,/ a)/

Learn Math Equation: a(^(/ 1,/ 2)/)^ =(r,r a)r

Learn Math Graph:400 400 -6.544999998062849 1.0 7.455000001937151 -4.865000020712614 1.0 9.134999979287386 g true Function 1.8549999 3.1850002 false \2(^\x)^ Point 0.0 1.0 Point 1.0 2.0 Point 2.0 4.0 Point 3.0 8.0 Point -1.0 0.5 Function -0.9799999 5.6000004 false \5(^\x)^ Color red Function 5.04 2.975 false \1\.\1(^\x)^ Color blue

Cont, one-to-one, Domain is all reals, Range Learn Math Equation:(( 0 ,s8)), Asymptote at Learn Math Equation: y = 0, Learn Math Equation:(( 0 , 1)) y-intercept

Learn Math Graph:400 400 -7.0 1.0 7.0 -7.0 1.0 7.0 g true Function 1.5749999 3.045 false (((/\1,/\2)/))(^\x)^

Applications of exponential functions - Compound interest.

Suppose you invest $1000 at 10% interest (Compounded yearly.)

Learn Math Equation: 1 0 0 0 + 0 . 1(( 1 0 0 0)) = 1 0 0 0(( 1 . 1 0)) = 1 1 0 0

Learn Math Equation: 1 1 0 0 + 0 . 1(( 1 1 0 0)) = 1 1 0 0(( 1 . 1 0)) = 1 0 0 0(( 1 . 1))(( 1 . 1)) = 1 2 1 0

Learn Math Equation: 1 2 1 0 + 0 . 1(( 1 2 1 0)) = 1 0 0 0(( 1 . 1))(( 1 . 1))(( 1 . 1)) = 1 0 0 0(( 1 . 1))(^ 3)^ = 1 3 3 1

5 years: Learn Math Equation: 1 0 0 0(( 1 . 1))(^ 5)^

In general (for yearly compounding)

Learn Math Equation: A(( t)) = P(( 1 + r))(^ t)^

A: Amount you have now

t: What year it is

P: Original amount

r: interest rate

What about $1000 at 10% compounded quarterly.

You actually get Learn Math Equation:(/ 1 0,/ 4)/ = 2 . 5 % interest four times a year.

In one year

Learn Math Equation: 1 0 0 0(( 1 . 0 2 5))(( 1 . 0 2 5))(( 1 . 0 2 5))(( 1 . 0 2 5)) = 1 0 0 0(( 1 +(/ 0 . 1 0,/ 4)/))(^ 4)^

so in five years

Learn Math Equation: 1 0 0 0(( 1 +(/ 0 . 1 0,/ 4)/))(^ 4s* 5)^

In general (for compounding n times per year)

Learn Math Equation: A(( t)) = P(( 1 +(/ r,/ n)/))(^ ts* n)^

A: Amount you have now

t: What year it is

P: Original amount

r: interest rate

So $5000 at 4% compounded monthly for 10 years would be

Learn Math Equation: A = 5 0 0 0(( 1 +(/ 0 . 0 4,/ 1 2)/))(^ 1 0s* 1 2)^ =s.

What about $1 at 100% for one year?

Yearly: Learn Math Equation: 1(( 1 + 1))(^ 1)^ = 2 . 0 0

Quarterly: Learn Math Equation: 1(( 1 +(/ 1,/ 4)/))(^ 1s* 4)^s~ 2 . 4 4

Monthly: Learn Math Equation: 1(( 1 +(/ 1,/ 1 2)/))(^ 1s* 1 2)^s~ 2 . 6 1

Daily: Learn Math Equation: 1(( 1 +(/ 1,/ 3 6 5)/))(^ 1s* 3 6 5)^s~ 2 . 7 1

Hourly: Learn Math Equation: 1(( 1 +(/ 1,/ 3 6 5s* 2 4)/))(^ 1s* 3 6 5s* 2 4)^s~ 2 . 7 2

These numbers are approaching a "special" number

Learn Math Equation: es~ 2 . 7 1 8 2 8 2s.

Cont: Learn Math Equation: 1s* e(^ 1)^

So when we have continuously compounding interest...

Learn Math Equation: A = P e(^ r t)^

Learn Math Graph:400 400 -7.0 1.0 7.0 -7.0 1.0 7.0 g true Function 2.03 6.79 false \e(^\x)^ Function 1.9250001 3.6399999 false \2(^\x)^ Color red Function 0.105 3.885 false \3(^\x)^ Color blue

We can also transform any of these graphs just like before

Learn Math Graph:400 400 -7.0 1.0 7.0 -7.0 1.0 7.0 g true Function 2.8 3.1850002 false \2(^\x)^ Function -3.8500001 4.515 false \2(^\x)^\+\3 Color red Function 4.515 1.75 false \2(^\x\-\3)^ Color blue Point 0.0 1.0 Point 3.0 1.0 Color blue Point 0.0 4.0 Color red Function 1.5749999 -1.995 false \-((\2(^\x)^)) Color green Point 0.0 -1.0 Color green