# 6.1 Exponential functions  (Page 7/16)

 Page 7 / 16

An initial investment of $100,000 at 12% interest is compounded weekly (use 52 weeks in a year). What will the investment be worth in 30 years? about$3,644,675.88

## Using the compound interest formula to solve for the principal

A 529 Plan is a college-savings plan that allows relatives to invest money to pay for a child’s future college tuition; the account grows tax-free. Lily wants to set up a 529 account for her new granddaughter and wants the account to grow to $40,000 over 18 years. She believes the account will earn 6% compounded semi-annually (twice a year). To the nearest dollar, how much will Lily need to invest in the account now? The nominal interest rate is 6%, so $\text{\hspace{0.17em}}r=0.06.\text{\hspace{0.17em}}$ Interest is compounded twice a year, so $\text{\hspace{0.17em}}k=2.$ We want to find the initial investment, $\text{\hspace{0.17em}}P,$ needed so that the value of the account will be worth$40,000 in $\text{\hspace{0.17em}}18\text{\hspace{0.17em}}$ years. Substitute the given values into the compound interest formula, and solve for $\text{\hspace{0.17em}}P.$

Lily will need to invest $13,801 to have$40,000 in 18 years.

Refer to [link] . To the nearest dollar, how much would Lily need to invest if the account is compounded quarterly?

$13,693 ## Evaluating functions with base e As we saw earlier, the amount earned on an account increases as the compounding frequency increases. [link] shows that the increase from annual to semi-annual compounding is larger than the increase from monthly to daily compounding. This might lead us to ask whether this pattern will continue. Examine the value of$1 invested at 100% interest for 1 year, compounded at various frequencies, listed in [link] .

Frequency $A\left(t\right)={\left(1+\frac{1}{n}\right)}^{n}$ Value
Annually ${\left(1+\frac{1}{1}\right)}^{1}$ $2 Semiannually ${\left(1+\frac{1}{2}\right)}^{2}$$2.25
Quarterly ${\left(1+\frac{1}{4}\right)}^{4}$ $2.441406 Monthly ${\left(1+\frac{1}{12}\right)}^{12}$$2.613035
Daily ${\left(1+\frac{1}{365}\right)}^{365}$ $2.714567 Hourly ${\left(1+\frac{1}{\text{8766}}\right)}^{\text{8766}}$$2.718127
Once per minute ${\left(1+\frac{1}{\text{525960}}\right)}^{\text{525960}}$ $2.718279 Once per second ${\left(1+\frac{1}{31557600}\right)}^{31557600}$$2.718282

These values appear to be approaching a limit as $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ increases without bound. In fact, as $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ gets larger and larger, the expression $\text{\hspace{0.17em}}{\left(1+\frac{1}{n}\right)}^{n}\text{\hspace{0.17em}}$ approaches a number used so frequently in mathematics that it has its own name: the letter $\text{\hspace{0.17em}}e.\text{\hspace{0.17em}}$ This value is an irrational number, which means that its decimal expansion goes on forever without repeating. Its approximation to six decimal places is shown below.

## The number e

The letter e represents the irrational number

The letter e is used as a base for many real-world exponential models. To work with base e , we use the approximation, $\text{\hspace{0.17em}}e\approx 2.718282.\text{\hspace{0.17em}}$ The constant was named by the Swiss mathematician Leonhard Euler (1707–1783) who first investigated and discovered many of its properties.

## Using a calculator to find powers of e

Calculate $\text{\hspace{0.17em}}{e}^{3.14}.\text{\hspace{0.17em}}$ Round to five decimal places.

On a calculator, press the button labeled $\text{\hspace{0.17em}}\left[{e}^{x}\right].\text{\hspace{0.17em}}$ The window shows Type $\text{\hspace{0.17em}}3.14\text{\hspace{0.17em}}$ and then close parenthesis, $\text{\hspace{0.17em}}\left[\right)\right].\text{\hspace{0.17em}}$ Press [ENTER]. Rounding to $\text{\hspace{0.17em}}5\text{\hspace{0.17em}}$ decimal places, $\text{\hspace{0.17em}}{e}^{3.14}\approx 23.10387.\text{\hspace{0.17em}}$ Caution: Many scientific calculators have an “Exp” button, which is used to enter numbers in scientific notation. It is not used to find powers of $\text{\hspace{0.17em}}e.$

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
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