So far we have worked with rational bases for exponential functions. For most real-world phenomena, however,
e is used as the base for exponential functions. Exponential models that use
$\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ as the base are called
continuous growth or decay models . We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and fluid dynamics.
The continuous growth/decay formula
For all real numbers
$\text{\hspace{0.17em}}t,$ and all positive numbers
$\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}r,$ continuous growth or decay is represented by the formula
$$A(t)=a{e}^{rt}$$
where
$a\text{\hspace{0.17em}}$ is the initial value,
$r\text{\hspace{0.17em}}$ is the continuous growth rate per unit time,
and
$\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is the elapsed time.
If
$\text{\hspace{0.17em}}r>0\text{\hspace{0.17em}}$ , then the formula represents continuous growth. If
$\text{\hspace{0.17em}}r<0\text{\hspace{0.17em}}$ , then the formula represents continuous decay.
For business applications, the continuous growth formula is called the continuous compounding formula and takes the form
$$A(t)=P{e}^{rt}$$
where
$P\text{\hspace{0.17em}}$ is the principal or the initial invested,
$r\text{\hspace{0.17em}}$ is the growth or interest rate per unit time,
and
$t\text{\hspace{0.17em}}$ is the period or term of the investment.
Given the initial value, rate of growth or decay, and time
$\text{\hspace{0.17em}}t,$ solve a continuous growth or decay function.
Use the information in the problem to determine
$\text{\hspace{0.17em}}a$ , the initial value of the function.
Use the information in the problem to determine the growth rate
$\text{\hspace{0.17em}}r.$
If the problem refers to continuous growth, then
$\text{\hspace{0.17em}}r>0.$
If the problem refers to continuous decay, then
$\text{\hspace{0.17em}}r<0.$
Use the information in the problem to determine the time
$\text{\hspace{0.17em}}t.$
Substitute the given information into the continuous growth formula and solve for
$\text{\hspace{0.17em}}A(t).$
Calculating continuous growth
A person invested $1,000 in an account earning a nominal 10% per year compounded continuously. How much was in the account at the end of one year?
Since the account is growing in value, this is a continuous compounding problem with growth rate
$\text{\hspace{0.17em}}r=\mathrm{0.10.}\text{\hspace{0.17em}}$ The initial investment was $1,000, so
$\text{\hspace{0.17em}}P=1000.\text{\hspace{0.17em}}$ We use the continuous compounding formula to find the value after
$\text{\hspace{0.17em}}t=1\text{\hspace{0.17em}}$ year:
Radon-222 decays at a continuous rate of 17.3% per day. How much will 100 mg of Radon-222 decay to in 3 days?
Since the substance is decaying, the rate,
$\text{\hspace{0.17em}}17.3\%$ , is negative. So,
$\text{\hspace{0.17em}}r\text{}=\text{}-\mathrm{0.173.}\text{\hspace{0.17em}}$ The initial amount of radon-222 was
$\text{\hspace{0.17em}}100\text{\hspace{0.17em}}$ mg, so
$\text{\hspace{0.17em}}a=100.\text{\hspace{0.17em}}$ We use the continuous decay formula to find the value after
$\text{\hspace{0.17em}}t=3\text{\hspace{0.17em}}$ days:
Using the data in
[link] , how much radon-222 will remain after one year?
3.77E-26 (This is calculator notation for the number written as
$\text{\hspace{0.17em}}3.77\times {10}^{-26}\text{\hspace{0.17em}}$ in scientific notation. While the output of an exponential function is never zero, this number is so close to zero that for all practical purposes we can accept zero as the answer.)
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic.
Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation
of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15)
it's standard equation is x^2 + y^2/16 =1
tell my why is it only x^2? why is there no a^2?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations