Section Subscript Notation
In calculus and differential equations, functions are usually written using parentheses. For example:
\begin{equation*}
f(x) = x^2 + 3x + 2, \qquad g(x) = x^5 \sin(x)
\end{equation*}
These definitions tell us how to evaluate the function for different inputs. We can plug in any real number:
\begin{align*}
f(7) \amp = 7^2 + 3 \cdot 7 + 2 = 72\\
f(-1.3) \amp = (-1.3)^2 + 3 \cdot (-1.3) + 2 = -0.21\\
f(w) \amp = w^2 + 3w + 2\\
f(x+5) \amp = (x+5)^2 + 3(x+5) + 2\\
\amp = x^2 + 13x + 42
\end{align*}
Subscript notation.
When the inputs to a function are restricted to whole numbers like \(n = 0, 1, 2, 3, \ldots\text{,}\) we often use subscript notation. This is especially common when working with sequences or numerical methods that step through time.
For example, suppose we define a function using subscript notation:
\begin{equation*}
a_n = n^2 + 3n + 2
\end{equation*}
This behaves similarly to \(f(x)\text{,}\) except that \(n\) must be a nonnegative integer. We would not evaluate \(a_n\) at a value like \(-1.3\text{.}\)
Here are some values of the sequence:
\begin{align*}
a_0 \amp = 0^2 + 3\cdot 0 + 2 = 2\\
a_1 \amp = 1^2 + 3\cdot 1 + 2 = 6\\
a_{13} \amp = 13^2 + 3\cdot 13 + 2 = 210\\
a_{200} \amp = 200^2 + 3\cdot 200 + 2 = 40602
\end{align*}
Youβll see subscript notation again when we use Eulerβs method later in this course, where we write approximations as \(y_0, y_1, y_2, \ldots\) to represent the values of a function at evenly spaced time steps.
Exercises Practice
Exercise Group.
Practice evaluating each subscripted expression using the given rule.
2.
If \(x_n = -12\text{,}\) evaluate \(x_{41}\text{.}\)3.
If \(y_n = (n + 1)!\text{,}\) evaluate \(y_3\text{.}\)You have attempted of activities on this page.
