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Exponential functions

Growth and decay

Let's first look at a few examples of growth and decay processes to get an idea of ​​where exponential functions are applied.

Growth (processes)

  • If you invest money, the bank will give you interest. The money invested increases exponentially as long as you don't withdraw money. You use the to calculate how much money will come out of this Interest bill.
  • The multiplication of bacterial cultures is often described as exponential.

Disintegration (processes)

  • The Milk foam in a latte macchiato or the Head of foam on a beer disintegrate exponentially.
  • The decay of radioactive nuclides is called exponential.

To describe such processes, one uses Exponential function.

What is an exponential function?

The special thing about Exponential functions is that the independent size that variable, is in the exponent. An exponential function has the general form

$ f (x) = c \ cdot a ^ x $.

It is

  • $ x $ the variable,
  • $ a \ in \ mathbb {R} ^ + $ die Base the exponential function and
  • $ c \ in \ mathbb {R} $ is a constant. This stands for Initial value with exponential processes.

Usually one writes Exponential functions with the base $ e \ approx2.71828 $, the Euler's number. How this is done with the general basis $ a $ - how to switch between the general representation and the natural exponential function - you can see here:

$ f (x) = c \ cdot a ^ x = c \ cdot e ^ {\ ln (a ^ x)} $.

We used $ e ^ {\ ln (x)} = x $ and the calculation rule for logarithms: $ \ log (p ^ {q}) = q \ cdot \ log (p) $. This rule applies to any logarithm, regardless of the base. So that is

$ f (x) = c \ cdot e ^ {\ ln (a ^ x)} = c \ cdot e ^ {\ ln (a) \ cdot x} $.

The natural exponential function

A Exponential function with the base $ e $ is used as a natural exponential function denotes, for example $ f (x) = e ^ {x} $.

More generally, a natural exponential function can look like this:

$ f (x) = c \ cdot e ^ {kx} $.

It is

  • $ x $ the variable (the time is often used for $ x $, then the variable $ t $ is also used for "time"),
  • $ e $ die Euler's number and
  • $ c $ and $ k $ parameter. In connection with growth and decay processes taking into account the Decay law is $ c> 0 $ der Initial value.

Influence of the parameters on the function graph

In this context, let us only consider the parameters $ c $ and $ k $, which are particularly interesting.

The function graph of the function $ f (x) = 1 \ times e ^ {1 \ times x} = e ^ x $ with $ c = 1 $ and $ k = 1 $ can be seen here.

You can already do the following properties detect. The graph of $ f (x) = e ^ x $

  • is strictly monotonically growing,
  • intersects the y-axis at $ 1 $ and
  • lies completely above the x-axis ($ e ^ x> 0 $ for all $ x \ in \ mathbb {R} $) and in particular $ e ^ x \ neq 0 $ applies.

These properties apply to all $ k> 0 $, provided that $ c = 1 $. Let's look at a few other options.

  • At $ f (x) = e ^ {- x} $ with $ c = 1 $ and $ k = -1 $, the graph is mirrored on the y-axis: It is monotonically decreasing. The other properties are retained.
  • With $ f (x) = - e ^ {x} $ with $ c = -1 $ and $ k = 1 $, the graph is mirrored on the x-axis. It then declines monotonically, lies completely below the x-axis and intersects the y-axis at $ c $. It's $ f (x) <0 $.
  • For $ f (x) = - e ^ {- x} $ with $ c = -1 $ and $ k = -1 $, the graph for negative $ k $ is mirrored on the x-axis. The graph is monotonically increasing, lies completely below the x-axis and intersects the y-axis at $ c $.

Examples

Growth process

Paul invests $ c = 1500 ~ € $ at an interest rate of $ p = 1.65 \% $. How does capital develop over the years? First you determine that Growth factor $ a $:

$ a = 1 + \ frac {1.65} {100} = 1.0165 $.

So that is Growth function given by:

$ f (x) = 1500 \ cdot 1.0165 ^ x $.

Again, $ x $ is the time in years. $ f (x) $ indicates the capital after $ x $ years. You can also write this function with the base $ e $:

$ f (x) = 1500 \ cdot e ^ {0.016x} $.

The graph of this function is monotonically increasing and intersects the y-axis at $ 1500 $.

Decay process

The tree population of a forest in hectares decreases by $ 15 \% $ per year. At the start of the observation, the tree population is $ 250 acres. First you determine the base $ a $. This is the factor with which the decay can be described:

$ a = 1- \ frac {15} {100} = $ 0.85.

Now you can Decay function specify:

$ f (x) = 250 \ cdot 0.85 ^ x $.

The variable $ x $ stands for the time in years. $ f (x) $ indicates the tree population after $ x $ years. With the base $ e $ you can write this like this:

$ f (x) = 250 \ cdot e ^ {- 0.163x} $.

The graph of this function is monotonically decreasing and intersects the y-axis at $ 250 $.