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exponential growth curve definition
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From Wikipedia
A growth curve is an empirical model of the evolution of a quantity over time. Growth curves are widely used in biology for quantities such as population size, body height or biomass. Values for the measured property can be plotted on a graph as a function of time; see Figure 1 for an example.
Growth curves are employed in many disciplines besides biology, particularly in statistics, which has an extensive literature on growth curves. In mathematical statistics, growth curves are often modeled as being continuousstochastic processes, e.g. as being sample paths that almost surely solve stochastic differential equations.
Bacterial growth
In this example (Figure 1, see Lac operon for details) the number of bacteria present in a nutrientcontaining broth was measured during the course of an 8 hour cell growthexperiment. The observed pattern of bacterial growth is biphasic because two different sugars were present, glucose and lactose. The bacteria prefer to consume glucose (Phase I) and only use the lactose (Phase II) after the glucose has been depleted. Analysis of the molecular basis for this biphasic growth curve led to the discovery of the basic mechanisms that control gene expression.
Cancer cell growth
Cancer research is an area of biology where growth curve analysis [http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Search&db=books&doptcmdl=GenBookHL&term=%22growth+curve%22%5BAll+Fields%5D+AND+cmed6%5Bbook%5D+AND+351845%5Buid%5D&rid=cmed6.section.10278] plays an important role. In many types of cancer, the rate at which tumors shrink following chemotherapy is related to the rate of tumor growth before treatment. Tumors that grow rapidly are generally more sensitive to the toxic effects that conventional anticancer drugs have on the cancer cells. Many conventional anticancer drugs (for example, 5Fluoro Uracil) interfere with DNA replication and can cause the death of cells that attempt to replicate their DNA and divide. A rapidly growing tumor will have more actively dividing cells and more cell death upon exposure to such anticancer drugs.
In the example shown in Figure 2, a tumor is found after the cell growth rate has slowed. Most of the cancer cells are removed by surgery. The remaining cancer cells begin to proliferate rapidly and cancer chemotherapy is started. Many tumor cells are killed by the chemotherapy, but eventually some cancer cells that are resistant to the chemotherapy drug begin to grow rapidly. The chemotherapy is no longer useful and is discontinued.
The growth of children
Children who fall significantly below the normal range of growth curves for body height [http://www.cdc.gov/nchs/data/nhanes/growthcharts/set1clinical/cj41c021.pdf] can be tested for growth hormone deficiency and might be treatable with hormone injections [http://www.nlm.nih.gov/medlineplus/ency/article/001176.htm].
Exponential growth
Some growth curves for certain biological systems display periods of exponential growth. Typically, periods of exponential growth are of limited duration due to depletion of some ratelimiting resource.
Bacterial growth curve. In autecological studies, bacterial growth in batch culture can be modeled with four different phases: lag phase (A), exponential or ...
In mathematics, two quantities are proportional if they vary in such a way that one of them is a constantmultiple of the other.
Symbol
The mathematical symbol 'âˆ�' is used to indicate that two values are proportional. For example, A âˆ� B.
In Unicode this is symbol U+221D.
Direct proportionality
Given two variables x and y, y is '(directly) proportional to x (x and y vary directly, or x and y are in direct variation) if there is a nonzero constant k such that
 y = kx.\,
The relation is often denoted
 y \propto x
and the constant ratio
 k = y/x\,
is called the proportionality constant or constant of proportionality.
Examples
 If an object travels at a constant speed, then the distance traveled is proportional to the time spent traveling, with the speed being the constant of proportionality.
 The circumference of a circle is proportional to its diameter, with the constant of proportionality equal to Ï€.
 On a map drawn to scale, the distance between any two points on the map is proportional to the distance between the two locations that the points represent, with the constant of proportionality being the scale of the map.
 The force acting on a certain object due to gravity is proportional to the object's mass; the constant of proportionality between the mass and the force is known as gravitational acceleration.
Properties
Since
 y = kx\,
is equivalent to
 x = \left(\frac{1}{k}\right)y,
it follows that if y is proportional to x, with (nonzero) proportionality constant k, then x is also proportional to y with proportionality constant 1/k.
If y is proportional to x, then the graph of y as a function of x will be a straight line passing through the origin with the slope of the line equal to the constant of proportionality: it corresponds to linear growth.
Inverse proportionality
As noted in the definition above, two proportional variables are sometimes said to be directly proportional. This is done so as to contrast direct proportionality with inverse proportionality.
Two variables are inversely proportional (or varying inversely, or in inverse variation, or in inverse proportion or reciprocal proportion) if one of the variables is directly proportional with the multiplicative inverse (reciprocal) of the other, or equivalently if their product is a constant. It follows that the variable y is inversely proportional to the variable x if there exists a nonzero constant k such that
 y = {k \over x}
The constant can be found by multiplying the original x variable and the original y variable.
Basically, the concept of inverse proportion means that as the absolute value or magnitude of one variable gets bigger, the absolute value or magnitude of another gets smaller, such that their product (the constant of proportionality) is always the same.
For example, the time taken for a journey is inversely proportional to the speed of travel; the time needed to dig a hole is (approximately) inversely proportional to the number of people digging.
The graph of two variables varying inversely on the Cartesian coordinate plane is a hyperbola. The product of the X and Y values of each point on the curve will equal the constant of proportionality (k). Since k can never equal zero, the graph will never cross either axis.
Hyperbolic coordinates
The concepts of direct and inverse proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates; the two coordinates correspond to the constant of direct proportionality that locates a point on a ray and the constant of inverse proportionality that locates a point on a hyperbola.
Exponential and logarithmic proportionality
A variable y is exponentially proportional to a variable x, if y is directly proportional to the exponential function of x, that is if there exist nonzero constants k and a
 y = k a^x.\,
Likewise, a variable y is logarithmically proportional to a variable x, if y is directly proportional to the logarithm of x, that is if there exist nonzero constants k and a
 y = k \log_a (x).\,
Experimental determination
To determine experimentally whether two physical quantities are directly proportional, one performs several measurements and plots the resulting data points in a Cartesian coordinate system. If the points lie on or close to a straight line that passes through the origin (0, 0), then the two variables are probably proportional, with the proportionality constant given by the line's slope.
Unrelated proportionality
Given two variables x and y, y
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Answers:I picked an easy one, #14: To reuse something is to use it again. An old, wornout tire might still be usable on a trailer, but the rubber from which it's made can be recycled to make something else. An used piece of paper can be reused if it's only been written on one side, but if it's shredded up and broken down, the materials can be recycled to make new paper.
Answers:See bacterial growth curve in Wiki link. The bacteria start dividing at 11.30am (ie after lag phase) and do so until 4.30pm, a total time of 5hours. The growth is exponential, which means that the number of organisms present at any time t (measured from time zero) is N(t) = N(0)*exp(Rt) where R is the growth rate constant. Rearrange and take logs to base e to find R : exp(Rt) = N(t)/N(0) = 1.8x10^10 / 2.75x10^5 = 6.55x10^4. Rt = loge (6.55x10^4) = 11.0898. R = 11.0898 / t = 11.0898 / 5 = 2.21796 per hour. Answer 2: mean growth rate constant = 2.22 per hour. (ie there are on average 2.22 cell divisions per hour.) The "mean generation time" is the time it takes for the colony to double its size. eg we start with N organisms and end with 2N. So using the above formula and value of R : 2N = N*exp(Rt) 2 = exp(Rt) loge(2) = 0.69314718 = Rt t = 0.69314718 / R = 0.69314718 / 2.21796 = 0.31251548 hour. Answer: mean generation time = 0.312hour = 18.8mins. During the 5hour growth phase there have been 2.22 divisions (=generations) per hour. So the number of generations is 5 x 2.22 = 11.1 generations. The additional information allows you to calculate the death rate constant in the same way, but the question does not ask for that.
Answers:The exponential function is a function in mathematics. The application of this function to a value x is written as exp(x). Equivalently, this can be written in the form ex, where e is the mathematical constant that is the base of the natural logarithm (approximately 2.71828182846) and that is also known as Euler's number. The exponential function is nearly flat (climbing slowly) for negative values of x, climbs quickly for positive values of x, and equals 1 when x is equal to 0. Its y value always equals the slope at that point.As a function of the real variable x, the graph of y = ex is always positive (above the x axis) and increasing (viewed lefttoright). It never touches the x axis, although it gets arbitrarily close to it (thus, the x axis is a horizontal asymptote to the graph). Its inverse function, the natural logarithm, ln(x), is defined for all positive x. In older sources it is often referred as an antilogarithm which is the inverse function of a logarithm. Sometimes, especially in the sciences, the term exponential function is more generally used for functions of the form cbx, where b, called the base, is any positive real number, not necessarily e. See exponential growth for this usage. In general, the variable x can be any real or complex number, or even an entirely different kind of mathematical object; see the formal definition below. Contents [hide] 1 Overview and motivation 2 Formal definition 3 Derivatives and differential equations 4 Continued fractions for ex 5 On the complex plane 6 Computation of ez for a complex z 7 Computation of ab where both a and b are complex 8 Matrices and Banach algebras 9 On Lie algebras 10 Double exponential function 11 Similar properties of e and the function ez 12 See also 13 References 14 External links [edit] Overview and motivation The exponential function is written as an exponentiation of the mathematical constant e because it is equal to e when applied to 1 and obeys the basic exponentiation identity, that is: It is the unique continuous function satisfying these identities for real number exponents. Because of this it can be used to define exponentiation to a non rational exponent. The exponential function has an analytic continuation which is an entire function, that is it has no singularity over the whole complex plane. The occurrence of the exponential function in Euler's formula gives it a central place when working with complex numbers. The definition has been usefully extended to some nonnumeric exponents, for instance as the matrix exponential or the exponential map. There are a number of other characterizations of the exponential function. The one which mainly leads to its pervasive use in mathematics is as the function for which the rate of change is equal to its value, and which is 1 at 0. In the general case where the rate of change is directly proportional (rather than equal) to the value the resulting function can be expressed using the exponential function as follows: gives If b = ek then this has the form cbx. Exponentiation with a general base b as in bx (called the exponential function with base b) is defined using the exponential function and its inverse the natural logarithm as follows: Its use in science is described in exponential growth and exponential decay. [edit] Formal definition Main article: Characterizations of the exponential function The exponential function (in blue), and the sum of the first n+1 terms of the power series on the left (in red).The exponential function ex can be defined, in a variety of equivalent ways, as an infinite series. In particular it may be defined by a power series: . Note that this definition has the form of a Taylor series. Using an alternate definition for the exponential function should lead to the same result when expanded as a Taylor series. Less commonly, ex is defined as the solution y to the equation It is also the following limit: The error term of this limitexpression is described by where, the polynomial's degree (in x) in the nominator with denominator nk is 2k. [edit] Derivatives and differential equations The importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives. In particular, That is, ex is its own derivative and hence is a simple example of a pfaffian function. Functions of the form Kex for constant K are the only functions with that property. (This follows from the PicardLindel f theorem, with y(t) = et, y(0)=K and f(t,y(t)) = y(t).) Other ways of saying the same thing include: The slope of the graph at any point is the height of the function at that point. The rate of increase of the function at x is equal to the value of the function at x. The function solves the differential equation y = y. exp is a fixed point of derivative as a functional. In fact, many d
Answers:I assume you want the curve to be only positive? Your idea was right, there s only one problem: the exponential function grows way too fast to see the oscillatory behaviour in it s entirety in normalsized images. Try to multiply the argument of the sine by some number. I used 10 and it gave me quite a wiggly line. I used gnuplot with the following instructions: set xrange [0:5] plot abs(sin(x*10))*2.718282**x I don t know if this is exactly what you wanted, but it sure looks interesting.
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