Tables of Values and Plots

The following limit is of special interest. We examine the behavior of this function near 0.

$\blacktriangleright$ Evaluate

$\lim\limits_{{x\rightarrow 0}}^{}$${\dfrac{{\sin x}}{{x}}}$ = 1


\begin{example}
To see numerical evidence that $\lim_{x\rightarrow 0}\frac{\sin...
...333$. Note that the function values seem to
approach 1.\medskip
\end{example}

You can generate a table of values by applying a function to a vector of domain values and then concatenating matrices, or you can do it in one step by defining appropriate auxiliary functions. Following are examples of each.

$\blacktriangleright$ To create a table of values for the function y = f (x) by defining auxiliary functions

  1. Define the function f (x)

  2. Define a function g(n) that will provide a sample of values of the independent variable.

  3. Define the function h(i, j) = (2 - j)g(i) + (j - 1)f (g(i))

  4. From the Matrices submenu, choose Fill Matrix.dtbpF3.5051in2.06in0ptfillmat2.wmf

  5. Set Columns to 2, and and set Rows to match the size of your sample.

  6. Under Fill With, choose Defined By Function.

  7. In the input box for function name, enter h.

  8. Choose OK.


\begin{example}
Make a table of values for the sine function.
\end{example}

$\blacktriangleright$To create a table of values for the function y = sin x

  1. With the insertion point in the equation f (x) = sin x, choose Define + New Definition.

  2. With the insertion point in the equation g$\left(\vphantom{ i}\right.$i$\left.\vphantom{ i}\right)$ = i * 10-2, choose Define + New Definition.

  3. With the insertion point in the equation h(i, j) = (2 - j)g(i) + (j - 1)f (g(i)), choose Define + New Definition.

  4. Choose expanding brackets and leave the insertion point in the input box.

  5. From the Matrices submenu, choose Fill Matrix.

  6. Set Columns to 2, and and set Rows to 10.

  7. Under Fill With, choose Defined By Function.

  8. In the input box for function name, enter h.

  9. Choose OK.

The result is the matrix below on the left. To get the matrix on the right, choose Evaluate Numerically.

$\left[\vphantom{
\begin{array}{cc}
\frac{1}{100} & \sin \frac{1}{100} \\
...
...{100} \\
\frac{1}{10} & \sin \frac{1}{10}
\end{array}
\allowbreak }\right.$$\begin{array}{cc}
\frac{1}{100} & \sin \frac{1}{100} \\
\frac{1}{50} & \sin...
...}{100} & \sin \frac{9}{100} \\
\frac{1}{10} & \sin \frac{1}{10}
\end{array}$$\left.\vphantom{
\begin{array}{cc}
\frac{1}{100} & \sin \frac{1}{100} \\
...
...{100} \\
\frac{1}{10} & \sin \frac{1}{10}
\end{array}
\allowbreak }\right]$ = $\left[\vphantom{
\begin{array}{cc}
.0\,\allowbreak 1 & 9.\,\allowbreak 999\,...
...
.\,\allowbreak 1 & 9.\,\allowbreak 983\,3\times 10^{-2}
\end{array}
}\right.$$\begin{array}{cc}
.0\,\allowbreak 1 & 9.\,\allowbreak 999\,8\times 10^{-3} \\ ...
...{-2} \\
.\,\allowbreak 1 & 9.\,\allowbreak 983\,3\times 10^{-2}
\end{array}$$\left.\vphantom{
\begin{array}{cc}
.0\,\allowbreak 1 & 9.\,\allowbreak 999\,...
...
.\,\allowbreak 1 & 9.\,\allowbreak 983\,3\times 10^{-2}
\end{array}
}\right]$

$\blacktriangleright$ To generate a table of values by concatenating matrices

1.
Click itbpF0.3009in0.3009in0.0701inmatrix.wmf (or choose Insert + Matrix)

2.
Choose one column and any number of rows.

3.
Fill in domain values in the rows.

4.
Put parentheses around the matrix.

5.
Evaluate f (your matrix).


It is also useful to examine the graph of y = ${\frac{{\sin x}}{{x}}}$ on an interval containing 0. The graph gives additional strong evidence that $\lim_{{x\rightarrow 0}}^{}$${\frac{{\sin x}}{{x}}}$ = 1.

$\blacktriangleright$ Plot 2D + Rectangular

${\dfrac{{\sin x}}{{x}}}$

dtbpFU3in2.0003in0pt y = ${\frac{{\sin x}}{{x}}}$