Exponential and Logarithmic Models:
Exponential Growth:
Function:
y = C ekt, k > 0
Features:
a) Asymptotic to y = 0 to the left.b) Passes via (0,C).c) C is the initial value.d) Increases devoid of bound to right.
Note:
A few things that exponential growth is used to model comprise the population growth, bacterial growth and the compound interest.
When you are lucky adequate to be given the initial value, that is the value if x = 0, then you already know the value of constant C. The only thing essential to complete the model is to have one extra point on the graph. Plug the values for x, y and C, and solve it for k.
Alternatively, almost similar to cheating, you can put the x-values to List 1, the y-values to List 2, and select the ExpReg option on TI-82 calculator.
Exponential Decay (decreasing form):
y = C e-kt, k > 0
a) Asymptotic to y = 0 to the right.b) Passes via (0,C).c) C is the initial value.d) Reducing, but bounded beneath by y = 0.
Exponential decay can be used to model the radioactive decay and depreciation.
Exponential decay models reduce very quickly, and then the level off to become asymptotic towards x-axis.
Similar to the exponential growth model, if you know the initial value then rest of the model is fairly simple to complete.Exponential Decay (increasing form):
y = C (1 - e-kt), k > 0
a) Asymptotic to y = C to the right.b) Passes via (0,0).c) C is the upper limit.d) Rising, however bounded above by y = C.
The Exponential decay models of this form can model the sales or learning curves where there is an upper limit. This is completed by subtracting the exponential expression from one and multiplying by upper limit.
Exponential decay models of this form will rise very quickly at first, and then level off to become asymptotic to upper limit.
Similar to the other exponential models, when you know upper limit, then the rest of model is fairly simple to complete.
Calculator will not fit the increasing model including exponential decay directly.
Gaussian Model:
y = a exp (-(x-c)/b)2
exp(x) is the other way of writing ex
a) Asymptotic to the x axis to left and right.b) Passes via (0,a)c) a controls the height of curved) Centered around x = ce) b controls the spread.f) Bell shaped.
The Gaussian model is utilized quite a bit in Statistics to model distribution of scores.
The Gaussian model is introduced by German mathematician Carl Friedrich Gauss and named after them. This is the same Gauss who introduced the Fundamental Theorem of Algebra. This is symmetric about its mean, x = c. To the right of origin, it reduces slowly at first, then more quickly, and then levels off and become asymptotic to x-axis.
Similar to the Exponential Decay model, the Gaussian model can be turned to an increasing function by subtracting the exponential expression from one and then multiplying the upper limit.
Logistics Growth Model:
y = a/(1 + b e-kx), k > 0
a) Asymptotic to y = a to the right,b) Asymptotic to y = 0 to the left,c) Passes via (0, a/(1+b))d) Slow growth, followed by the moderate growth and followed by the slow growth.
The logistics model starts with a slow growth, followed by the period of moderate growth, and then back to the period of slow growth. It consists of an upper limit which can’t be exceeded.
The Logistics model can be employed to approximate the sales and advertising (a little advertising produces a little growth in sales, more advertising produces moderate growth in sales and finally there reaches a point of saturation in which additional advertising advantages little in terms of sales) or the population growth where there is no capacity for unlimited growth (employ the exponential growth model if you require that).
Logarithmic Model:
y = a + b ln x
a) Rises without bound to the right.b) Passes via (1, a).c) Very fast growth, followed by the slower growth.d) Common log will grow slower than the natural log.e) b controls the rate of growth.
The logarithmic model has a period of fast increase, followed by the period where the growth slows, however the growth continues to raise without bound. This makes the model unsuitable where there requires being an upper bound. The major difference between this model and the exponential growth model is that, the exponential growth model starts slowly and then rises very fast as time increases.
Some of the physical applications encompass logarithmic models: The magnitude of earthquakes, intensity of sound and the acidity of a solution.
Earthquakes:
R = log I
The Richter scale is utilized to measure the intensity of earthquake. The real model is a little more complicated, however it simplifies to the equation shown. R is the magnitude on Richter scale of earthquake. I is the intensity of earthquake measured relative to the reference value. That reference value is the minimum seismic activity which can be measured, and consists of the value I0 = 1.
Each and every very rise of 1 in Richter scale signifies the magnitude of earthquake is 10 times greater.
Sound Intensity:
β = 10 log (I/I0)Sound level, measured in decibels, is given by the formula shown above. The reference value, I0 is the smallest sound intensity that can be heard by the human ear and is roughly equivalent to 1x10-16 watts per square centimeter.
There are 10 decibels to bel. While bel is the real unit, similar to meter, liter or gram, we can use decibel for all practical aims. The rise of 10 decibels is equal to the sound which is 10 times as strong in intensity. The increase of 20 decibels is equal to the sound intensity which is 100 times greater.
Acidity:
pH = - log [ H+]pH is the measure of acidity of a substance. It varies from 0 to 14, with acids ranging from 0 to 7, 7 being neutral, and the bases ranging from 7 to 14. The [H+] is the concentration of hydrogen ions and is measured in moles per liter. The more is the hydrogen ions, then smaller the pH (note that the negative sign is in front of log) and more acidity the solution is.
Upper Bounds:
Note that the Exponential Growth and Logarithmic models rise without bound to right. The Gaussian and Exponential Decay models both approach to the x-axis to the right. Merely the Logistics Growth model provides you an upper bound to the right. The Exponential Decay and Gaussian models can be made to contain an upper bound by subtracting the exponential expression (therefore making it decreasing rather than increasing) from one and multiplying by the upper bound.
Latest technology based Algebra Online Tutoring Assistance
Tutors, at the www.tutorsglobe.com, take pledge to provide full satisfaction and assurance in Algebra help via online tutoring. Students are getting 100% satisfaction by online tutors across the globe. Here you can get homework help for Algebra, project ideas and tutorials. We provide email based Algebra help. You can join us to ask queries 24x7 with live, experienced and qualified online tutors specialized in Algebra. Through Online Tutoring, you would be able to complete your homework or assignments at your home. Tutors at the TutorsGlobe are committed to provide the best quality online tutoring assistance for Algebra Homework help and assignment help services. They use their experience, as they have solved thousands of the Algebra assignments, which may help you to solve your complex issues of Algebra. TutorsGlobe assure for the best quality compliance to your homework. Compromise with quality is not in our dictionary. If we feel that we are not able to provide the homework help as per the deadline or given instruction by the student, we refund the money of the student without any delay.
tutorsglobe.com industrial production of penicillin assignment help-homework help by online penicillin production tutors
tutorsglobe.com geographic information systems assignment help-homework help by online humanities tutors
Determination of Boiling Point of Hydrocarbons tutorial all along with the key concepts of Factors influencing boiling point, Polarity, Branching, Micro Method Determination of Boiling Point of Hydrocarbons
Chemistry of Carbohydrates-Lipids-Nucleic Acid tutorial all along with the key concepts of Carbohydrates, Functions of Carbohydrates, Lipids, Nucleic Acids, Building Blocks of Nucleic Acids and Functions of Nucleic Acids
Oscillators tutorial all along with the key concepts of Kinds of Oscillator, Sine Wave Oscillators, Relaxation oscillators, Sweep oscillators, Significance of oscillators, Principle of Oscillators, Applications of the Multi wave oscillators
Motion of Charge Particles in Electric and Magnetic Field tutorial all along with the key concepts of Motion in an Electric Field, Cathode Ray Oscilloscope, Lorentz Force and its Applications and Cyclotron
Lagrange and Hamiltonian Mechanics tutorial all along with the key concepts of Frame of Reference and Constraints of Motion, Constraints of Motion, Generalized Coordinates and Degrees of Freedom
tutorsglobe.com interhalogen compounds assignment help-homework help by online p block elements tutors
solder capable polyurethane enamelled round copper wire. it comprises thermal capacity of 120°c, 130°c and 155°c. the diameter ranges from 0.08 mm to 1.00 mm and can be employed in transformers, electronic, meters, and communication devices.
tutorsglobe.com menstrual cycle assignment help-homework help by online functioning of female reproductive system tutors
www.tutorsglobe.com offer nuclear chemistry homework help, nuclear chemistry assignment help, nuclear chemistry solutions, online tutoring and instant answers for nuclear chemistry problems by online chemistry tutors.
organic spectroscopy tutorial all along with the key concepts of interaction of light and matter, properties of light, infrared absorption spectroscopy, infrared light absorption and molecular structure, determination of molecular structure, nuclear magnetic resonance spectroscopy
Theory and lecture notes of Costs of Inflation all along with the key concepts of Costs of Moderate Expected Inflation, Costs of Moderate Unexpected Inflation, Hyperinflation and Its Costs, Inflation Tax. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Costs of Inflation.
Theory and lecture notes of How to find Global Deadlocks all along with the key concepts of how to find global deadlocks, lock management pragmatics, local deadlock detector. Tutorsglobe offers homework help, assignment help and tutor’s assistance on How to find Global Deadlocks.
www.tutorsglobe.com offers Object Oriented Analysis homework help, assignment help, case study, writing homework help, online tutoring assistance by computer science tutors.
1956020
Questions Asked
3689
Tutors
1496112
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!