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
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
and all positive numbers
and
continuous growth or decay is represented by the formula
where
is the initial value,
is the continuous growth rate per unit time,
and
is the elapsed time.
If
, then the formula represents continuous growth. If
, then the formula represents continuous decay.
For business applications, the continuous growth formula is called the continuous compounding formula and takes the form
where
is the principal or the initial invested,
is the growth or interest rate per unit time,
and
is the period or term of the investment.
Given the initial value, rate of growth or decay, and time
solve a continuous growth or decay function.
Use the information in the problem to determine
, the initial value of the function.
Use the information in the problem to determine the growth rate
If the problem refers to continuous growth, then
If the problem refers to continuous decay, then
Use the information in the problem to determine the time
Substitute the given information into the continuous growth formula and solve for
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
The initial investment was $1,000, so
We use the continuous compounding formula to find the value after
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,
, is negative. So,
The initial amount of radon-222 was
mg, so
We use the continuous decay formula to find the value after
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
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.)
Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ...
Step 2: Find each score's deviation from the mean. ...
Step 3: Square each deviation from the mean. ...
Step 4: Find the sum of squares. ...
Step 5: Divide the sum of squares by n – 1 or N.
The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.
Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400
a. what is the probability of getting more than 12,000 hits?
b. what is the probability of getting fewer than 9,000 hits?
Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400.
a. What is the probability of getting more than 12,000 hits
