What is a good example of exponential function in every day life?
552015
The only thing that comes to mind is population growth.  The more people (or whatever organisms) there are, the faster their population grows.
962855
should be e^u*du/dx\n\nd[e^(x^2)] = e(x^2)*2*x
634042
Theres a poster in my maths class that says "a trillion steps towards an infinitae pi" So i assume the best so far is a trillion number after the decimal point. WOW!
371301
1. Solving AC circuits that have resistors, inductors, and capacitors in them.\n2. Signal processing\n3. Solving differential equations.\n\nA smart guy by the name of Euler showed that e^(i*x) = cos(x) + i*sin(x), thus allowing the conversion of a complex number in rectangular form (a + b*i) into polar form (r*e^(i*x)). It also allows us to write sinusoids in terms of complex exponentials, as in:\ncos(x) = (1/2)(e^(i*x)-e^(-i*x))\nIt turns out, perhaps unexpectedly, that complex exponentials are oftentimes easier to deal with then sinusoids. It also turns out, according to a guy by the name of Fourier, that most "well-behaved" functions can be written in terms of sinusoids (and thus can be written in terms of complex exponentials as well). This paved the way for a lot of neat applications. Google "complex exponential" (or any of the terms or names I dropped here) if you want more details.
139157
Try to fit your data with Excel’s exponential fitting curves,\nIt will give you Coefficients which make the exponential looks like a line.\n\nOr try the experiment more rapidly.
863364
Not sure if I understood your question. maybe you meant a series of functions. Suppose that, for each n you have a function f_n and, for every x in the domain of the functions f_n, you define S(x) = f_1(x) + ....f_n(x) +...provided this series converges. Then, you have a series of functions, since for every x S(x) is given by the limi of a series.\n\nFor example, consider s(x) = 1 + x  x^2/(2!) ....+ x^n(n!)....|This is a power series, in this case it converges for every real x. This is the definitioj of the exponencial function e^x
392380
All kinds of simple, easy calculus examples in real life (especially engineering!).  These depend on the level of math knowledge, but for University Calc. II:\n\nOptimization: (i.e. give a box with a volume, find minimum surface area) or (here's a certain amount of fencing, make the largest enclosure you can).  (Lagrangian Multipliers? did I even spell that right?  We looked at those when there are a lot of constraints on the problem (i.e. relationship between box dimensions must follow .....).\n\nForces: for example, average fluid pressure on a dam, given a function of distributed force, or fluid pressure on a submerged object. (Can be done in single or multiple integrals).\n\nRelated Rates:  i.e. two cars approach at right angles, different speeds, when will they meet.  Or, dropping a ball near a light source, find speed of shadow as function of angle made with light source, distance from ground, etc.  Differetial calculus example.\n\nArea and Volume: Double and triple integrals to find the volumes and areas of shapes. (Or single integrals, volumes of revolution, areas from curve to axis, if multi-variable calculus not covered).\n\nCurve Fitting: really good real-life use, take a set of data points from an experiment, select a curve type (i.e. quadratic, cubic, straight line, logarithmic, etc) and use minimization to get the best curve fit (set up an equation for the residuals, sum of (function) - (data point), and take derivatives with respect to each unknown coefficient in the function, set all derivatives equal to zero, solve simulatneously for curve coefficients. Now we can predict the outcome of the experiment at intermediate values, (as long as the curve fit is good).\n\nCentre of Mass, Centroids, etc: Find the balancing point on certain objects or groups of objects using integration. Can be single or multiple integrals, depending on complexity of shape.\n\nFields:  i.e. graviational fields, electrical fields, etc.  Or make up some fields.  Integration to find work required by a particle to traverse a path through some field.  Conservative and non-conservative fields.  Can use as an example of energy needed to move satellites along paths (equations) in space near earth's gravity, an electron through an electrical field, etc.  I guess some of these examples the field is conservative and it becomes kinda trivial if you work with end points, but don't have to.\n\nThese are some of the applications we used in our Calculus II-type course.  The hardest part is always setting the problem up, not solving, (which makes for good application problems I guess).  If multi-variable calculus isn't covered yet, then some of these examples would be kinda rough (i.e. fields).  Anyway, hope these are useful. \n\nCheers,
356955
1.  The von Bertalanffy growth function (VBGF) introduced by von Bertalanffy in 1938 predicts the length of a shark as a function of its age, L = L(t): L(t) = Loo - (Loo - Lo) exp(-kt) \n2.  Examples of exponential growth\nBiology. \nMicroorganisms in a culture dish will grow exponentially, at first, after the first microorganism appears (but then logistically until the available food is exhausted, when growth stops). \nA virus (SARS, West Nile, smallpox) of sufficient infectivity (k > 0) will spread exponentially at first, if no artificial immunization is available. Each infected person can infect multiple new people. \nHuman population, if the number of births and deaths per person per year were to remain constant (but also see logistic growth). \nMany responses of living beings to stimuli, including human perception, are logarithmic responses, which are the inverse of exponential responses; the loudness and frequency of sound are perceived logarithmically, even with very faint stimulus, within the limits of perception. This is the reason that exponentially increasing the brightness of visual stimuli is perceived by humans as a smooth (linear) increase, rather than an exponential increase. This has survival value. Generally it is important for the organisms to respond to stimuli in a wide range of levels, from very low levels, to very high levels, while the accuracy of the estimation of differences at high levels of stimulus is much less important for survival. \nElectroengineering \nCharging and discharging of capacitors and changes in current in inductors are also exponential growth and decay phenomena. Engineers use a rule of five time constants to estimate when a steady state has been reached. \nComputer technology \nProcessing power of computers. See also Moore's law and technological singularity (under exponential growth, there are no such singularities). \nInternet traffic growth. \nWikipedia:Modelling Wikipedia's growth - an analysis of Wikipedia's growth rates based on article growth rates and other statistics \nInvestment. The effect of compound interest over many years has a substantial effect on savings and a person's ability to retire. See also rule of 72 \nPhysics \nAtmospheric pressure decreases exponentially with increasing height above sea level, at a rate of about 12% per 1000m. \nNuclear chain reaction (the concept behind nuclear weapons). Each uranium nucleus that undergoes fission produces multiple neutrons, each of which can be absorbed by adjacent uranium atoms, causing them to fission in turn. If the probability of neutron absorption exceeds the probability of neutron escape (a function of the shape and mass of the uranium), k > 0 and so the production rate of neutrons and induced uranium fissions increases exponentially, in an uncontrolled reaction. \nNewton's law of cooling  where T is temperature, t is time, and, A, D, and k > 0 are constants, is an example of exponential decay. \nMulti-level marketing \nExponential increases appear in each level of a starting member's downline as each subsequent member recruits more people.
95230
The compound interest formula is an exponential function:\n\nA=P(1+(r/n))^(n*t) where\nA=final amount\nP=principal or beginning amount\nr=interest rate (as a decimal)\nn=number of compounding per year\nt=time in years\n\nThe other exponential formula for compounding interest is used for continuously compounding interest or when (n*t) is equal to infinity. It is also the equation used when charting the growth or decay of bacteria.\n\nA=P*e^(r*t).\n\nPlease check the second source for info on how this equation relates to bacteria growth/decay.
999317
(1) Simple inverse is log X as long as log is defined for the specific value of X, except that it's also not defined at X=1 since log(1) = 0.\n(2) Inverse function: If Y = 1/logX, then X = exp(1/Y) assuming natural logarithms. This is defined except at Y=0, which can't hold for finite X.\n\nThe proofs of both these are trivial. They are obvious.
711884
Homework.\n\nShow it first for intervals. Then extend to simple functions. Then use the fact thats imple functions are dense in L1.
956580
In Excel you can graph the data and then add a trendline.  You can even have it display the equation for the trendline.  Some possible choices for fit are linear (line), exponential, polynomial of degree 2, 3, 4, 5, 6, etc.\n\nIf you give some sample data, I might be able to help further...
222928
There is no function that exists that has as its derivative e^(x^2).  As someone else pointed out, you can use the Maclaurin series representation and integrate term by term.  However, this series converges very slowly and would require a lot of terms for two place decimal accuracy, especially if x is large.
764897
use derivative rule for exponential as\n\nd/dx(A exp(bx))=(A*b)*exp(bx)\n\nyou got hint. No more Spoon Feeding
445844
FALSE DUDE... exponential no.'s rn't real nos themselves
888519
EXAMPLE #1 CHEMISTRY & BIOLOGY; \n\nGrowth: N(t) = n0e^(kt)         "n=number, k=rate, t=time"\n\nDecay: M(t) = m0(1/2)^(t/h)       "m=mass, t=time, h=half-life"\n\nEXAMPLE #2 BUSINESS & ECONOMICS;\n\nCompound interest \nwith e,"continuous":  P(t) = p0e^(rt)     "p=principle, r=rate, t=time"\n\nEXAMPLE #3 SOCIAL SCIENCES;\n\nPopulation Growth: P(t) = p0e^(kt)  "p=populaton, k=rate, t=time"\n\nThese are a VERY few examples of the uses of e(exponential) in everyday life. As we go further in the mathematics/science field we'll encounter more highly sophisticated usefullness of e.\n\nP.S. the letters used to symbolize subject may differ from individuals point of view but the function of growth & decay is mathematically the same.
895128
If by 'convert' to a linear function you mean approximate by a linear function, then a possible solution is given by the Taylor's expansion of the exponential function.\n\nConsider a point x0, then the linear approximation of y around x0 is given by:\n\ny(x) = e^(x0+a) + e^(x0+a) * (x+a-x0) ,\n\nwhich is linear in x.  Obviously this is only an approximation as the Taylor expansion of the exponential function will involve a very large number of higher order (i.e. quadratic, cubic etc) terms in the right hand side of the equation.