atomthick said:Because it's the Taylor expansion of exp(x) near 0
exp(x) = 1 + x + (x^2)/2 + (x^3)/6 + ...
You can cut the series at any term you would like, however you can't equal it to 1 because there will be no parameter left to give values to...
HallsofIvy said:You can also get this approximation by replacing the curving graph by a tangent line.
An exponential function is a mathematical function in the form of f(x) = ax, where a is a constant and x is the variable. It is characterized by a rapid increase or decrease in value as x increases or decreases, respectively.
To approximate an exponential function, you can use a technique called linearization. This involves finding the tangent line to the curve at a certain point and using that line to estimate the value of the function at that point.
Approximating an exponential function can be useful in situations where the function is complex or difficult to work with, or when precise values are not needed. It can also be used to make predictions or estimate values for a given input.
Exponential functions are used to model many real-life phenomena, such as population growth, compound interest, radioactive decay, and bacterial growth. They are also commonly used in economics, physics, and engineering.
The accuracy of an approximation of an exponential function depends on the method used and the number of data points used to create the approximation. Generally, the more data points used and the closer they are to the point being approximated, the more accurate the approximation will be.