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R = e r T 1 + r T

which is somewhat similar to [link] . Another view of the relation can be seen by approximating [link] by

x ( n + 1 ) - x ( n ) T = r x ( n ) ,

which gives

x ( n + 1 ) = x ( n ) + r T x ( n )
= ( 1 + r T ) x ( n )

having a solution

x ( n ) = x ( 0 ) ( 1 + r T ) n

This implies [link] also, and the method is known as Euler's method for numerically solving a differential equation.

These approximations are used often in modeling. For population models a differential evuation is often used, even though it is obvious thatbirths and deaths occur at random discrete times and populations can take on only integer values. The approximation makes sense only if we uselarge aggregates of individuals. We end up modeling a process that occurs at random discrete points in time by a continuous time mode, which is thenapproximated by a uniformly-spaced discrete time difference equation for solution on a digital computer!

The rapidity of increase of an exponential is usually surprising and it is this fact that makes understanding it important. There are several ways to describe the rate of growth.

If x = k e r t ,
then d x d t = k r e r t
or r = 1 x d x d t .

This states that r is the rate of growth per unit of x . For example, the growth rate for the U.S. is about 0.014 per year, or anincrease of 14 people per thousand people each year.

Another measure of the rate is the time for the variable to double in value. This doubling time, T d , is constant and can easily be shown to be given by

T d = 1 r log e 2 = 0 . 6931472 1 r

For example, doubling times for several rates are given by

r T d
.01 70
.02 35
.03 23
.04 17
.05 14
.06 12

The present world population is about three billion, and the growth rate is 2.1% per year. This gives

p ( t ) = 3 e 0 . 021 t

with p ( t ) measured in billions of people and t in years. This gives a doubling time of 33 years. While it is easy to talk of growthrate and doubling times, these have real predictive meaning only if the growth is exponential.

Two points of view

There are two rather different approaches that can be used when describing some physical phenomenon by exponential growth. It can be viewed as anempirical description of how some variables tend to evolve in time. This is a data-fitting view that is pragmatic and flexible, but does not givemuch insight or direction on how to conduct experiments or what other things might be implied.

The second approach primarily considers the underlying differential equation as a "law" of growth that results in exponential behavior. Thislaw has various assumptions and implications that can be examined for reasonableness or verified by independent experiment. While perhaps notso important for the first-order linear equation here, this approach becomes necessary for the more complicated models later.

These approaches must often be mixed. The data will imply a model or equation which will give direction as to what data should be taken, whichwill in turn imply modifications, etc. The process where structure is chosen and the parameters are chosen so that the model solution agreeswith observed data is a form of parameter identification. That was how [link] was determined.

The use of semi-log plots

When examining data that has been plotted in fashion, it is often hard to say much about its basic nature. For example if a time series is plotted on linear coordinates as follows

it would not be obvious if it were samples of an exponential, a parabola, or some other function. Straight lines, on the other hand, are easy to identify and so we will seek a method of displaying data that will use straight lines.

If x ( t ) is an exponential, then

x ( t ) = k e r t

Taking logarithms of base e for both sides of [link] gives

log x = log k + r t

If, rather than plotting x versus t , we plot the log of x versus t , then we have a straight line with a slope of r and an intercept of log k . It would look like

Actually using the logarithm of a variable is awkward so the variable itself can be plotted on logarithmic coordinates to give the same result. Graph paper with logarithmic spacing along one coordinate and uniform spacing along the other is called semi-log paper.

Consider the plot of the U.S. population displayed on semi-log paper in Figure 3. Note that there were two distinct periods of exponentialgrowth, one from 1600 to 1650, and another from 1650 to 1870. To calculate the growth rate over the 1650 – 1870 period, we can calculatethe slope.

r = log p ( t 1 ) - log p ( t 2 ) t 1 - t 2
= log 10 7 - log 10 5 1823 - 1670
= 16 . 118 - 11 . 513 153 = 0 . 03

During that period, there was a 3% per year growth rate or, in other terms, a 23-year doubling time.

An alternative is to measure the doubling time and calculate r from [link] . Still another approach is to measure the time necessary for the populationto increase by e = 2 . 72 ... The growth rate is the reciprocal of that time interval. Derive and check these for yourself.

The data displayed in Figures 1, 2 and 3 illustrates exponential growth and the use of semi-log paper.The books [link] [link] give interesting discussions of growth.

Analytical versus numerical solutions

In some cases, there is a choice between an analytical solution in the form of an equation and a numerical solution in the form of a sequence ofnumbers. A real advantage of an analytical form is the ability to easily see the effects of various parameters. For example, the exponentialsolution of [link] given in [link] directly shows the relation of the growth rate r in equation [link] to the exponent in solution [link] . If the equation were numerically solved, say on a digital computer or calculatorusing Euler's method given in [link] , it would take numerous experimental runs to establish the same relations.

On the other hand, for complicated equations there are no known analytical expressions for the solutions, and numerical solutions are the onlyalternatives. It is still worth studying the analytical solution of simple equations to gain insight into the nature of the numerical solutions of complex equations.

Assumptions

The linear first-order differential equation model that is implied by exponential growth has many assumptions that are worth noting here.First, the growth rate is constant, independent of crowding, food, availability, etc. It also assumes that age distribution within thepopulation is constant, and that an average birth and death rate makes sense. There are many factors one will want to include effects ofcrowding and resource availability, time delays in reproduction, different birth and death rates for different age groups, and many more. In thenext section we will add one complication the effects of a limit to growth.

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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Can you compute that for me. Ty
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emma Reply
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what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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Adjanou
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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Maurice
answer
Magreth
progressive wave
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Dynamics of social systems. OpenStax CNX. Aug 07, 2015 Download for free at https://legacy.cnx.org/content/col10587/1.9
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