In mathematics, the exponential function is the function that is noted exp and is equal to its own derivative and takes the value 1 in 0. Moreover, it is used to model phenomena in which a constant difference on the variable leads to a constant ratio. The student will need to develop new skills with this chapter. In addition, he will learn calculation methods that will allow him to progress throughout the school year.
The exponential function is the only continuous function in ℝ that turns a sum into a product. Moreover, this function takes the value e in 1. It is a special case of functions of this type called exponentials of base a. Also, it can be determined as a limit of a sequence or using an integer series. The elementary applications of real or complex exponential functions concern the solution of differential equations. In addition, the student must read the course often to fully understand and complete the class exercises at home.
Any function with an expression of the form f(x) = Aeλx is also sometimes called an exponential function.
I. Definition and variations of the exponential function.
Let be a strictly positive real.
A function f defined for any real by is an exponential function.
An exponential function f defined on by with >0 is :
- strictly increasing on if, and only if, >1;
- strictly decreasing on if, and only if, 0 < < 1;
- constant on if, and only if, = 1.
II. Algebraic properties of the exponential function.
For all positive real numbers and and for all strictly positive real numbers and , we have :
Let and be two strictly positive real numbers and a non-zero integer.
The equation has as a unique positive solution the real = called the n-th root of .
If a quantity undergoes an evolution of rate , then it reaches the same value by undergoing successive evolutions of the same rate where a non-zero natural number.
The number is called the average rate of the successive changes in the overall rate .
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