# What is exponential growth in ecology?

In general, *exponential growth* occurs when the derivative of a function at a given point is proportional to the function's value. Within the biological sciences, one often sees exponential growth/decay (decay will occur when the derivative is negative) with regards to population; human population, for example, increases exponentially, as does the population of organisms in a bacterial culture.

However, within most ecosystems, population does not simply increase unbounded. There will be certain restrictions, such as carrying capacity (how large a population the region can support with the available resources) and predator-prey interactions.

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Exponential growth in ecology refers to a pattern of growth where a population increases at a constant relative growth rate over time. This means that the rate of increase of the population is proportional to its current size. In other words, the larger the population gets, the faster it grows. Exponential growth is often represented by the equation (N_t = N_0 \times e^{rt}), where (N_t) is the population size at time (t), (N_0) is the initial population size, (r) is the intrinsic rate of increase (the growth rate per individual), and (e) is the base of the natural logarithm. Exponential growth is a theoretical concept and is rarely observed in natural populations over long periods because it assumes unlimited resources, which is not typically the case in ecological systems. However, populations may exhibit exponential growth over short periods when resources are abundant and conditions are favorable.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- Let R be the region bounded by #y = 1/x#, #y = x^2#, #x = 0#, and #y = 2# and revolved about the x axis. How do you find the volume of rotation using: a) the method of cylindrical shells; b) the method of circular disks?
- What is the particular solution of the differential equation # 2yy' = e^(x-y^2) # with #y=-2# when #x=4#?
- How do you find the surface area of the solid obtained by rotating about the #x#-axis the region bounded by #x=1+2y^2# on the interval #1<=y<=2# ?
- How do you find the distance travelled from t=0 to #t=pi# by an object whose motion is #x=3cos2t, y=3sin2t#?
- What is a solution to the differential equation #dy/dx=(x+y)/(x-y)#?

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