The techniques of sketching a curve consist of

Determining the domain of a curve
Finding first and second derivatives (to find relative max, relative minima and point of reflection)
Noting any symmetry
Finding any asymptotes
Noting some special values
Considering some limits as x and y go to infinity
Construct a table

These techniques are used to determine the curve’s trend. They may not be applicable in all cases, but, depending on the particular problem, some can always be used.

Example 1, Example 2, Return to Main Page

Example 3. Graph function. Then use the GraphFunc utility online to confirm the results.

Solution

1. The domain of f(x) is determined for all.

2. Derivatives:

.
. Solution is: x = 1.
At x = 1 =>.

3. x intercept(s): f(x) = 0 => x = 0.

y intercept(s): f(0) = 0.

4. Special limits:

(L’Hospital’s Rule)

and

5. Construct a table:

The Graph’s Trend shows that there is a max at x = 1 because the first derivative of f(x) changes sign from positive to negative as x passes through this point. The arrow lines show the prediction of the trend of the graph.

The information from the constructed table helps to draw the graph of f(x) as shown in Figure 1.

Figure 1 – The graph ofis the blue line.

· If the GraphFunc applet online is used to plot the graph and confirm the result, the valid syntax of the function be entered at the command line is x*exp(-x). The result is illustrated in Figure 2.

Figure 2.

Example 1, Example 2, Return to Main Page