Question

In your own words, state the properties of the natural exponential function.

Short answer

The key properties of the natural exponential function are: its asymptotic behaviour towards 0 as \( x \) approaches negative infinity, the derivative of the function is the function itself, the function increases rapidly for positive \( x \) and decreases rapidly for negative \( x \), and the function intercepts the y-axis at \( y=1 \), but has no x-intercepts.

Step-by-step solution

Identify the Asymptotic Behavior

The natural exponential function \( e^x \) has a horizontal asymptote at \( y=0 \). This means that as \( x \) approaches negative infinity, the function \( e^x \) approaches 0 but never reaches it.

Identify the Derivative

The derivative of the natural exponential function \( e^x \) is unique in that it is the function itself, i.e., the derivative of \( e^x \) is \( e^x \) again.

Identify the Rate of Change

The natural exponential function \( e^x \) increases rapidly for positive \( x \) (since the rate of change equals the function itself), and decreases rapidly (tending to zero) for negative \( x \).

Identify the Intercepts

The graph of \( y=e^x \) intersects the y-axis at \( y=1 \), i.e., when \( x=0, y=1 \). As far as x-intercepts are concerned, there are none, as the function never touches the x-axis.