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Section 4.5 Summary of Curve Sketching

Here we enhance the curve sketching strategy from Section 4.3 to add consideration of features like asymptotes, for which l'Hospital's Rule can be useful.

Not all ingredients will be needed for every sketch! For example, if the sketch is being done for a particular purpose [e.g. an optimization problem] only some of this information will be needed [e.g. local extrema but not inflection points and concavity].

Expanded Curve Sketching Strategy.
  1. Note any endpoints of the domain and points where the function is undefined, such as due to division by zero. At isolated points where \(f\) is undefined due to division by zero or such, compute any limits, including one-sided and infinite limits, like at vertical asymptotes.
  2. Note any symmetry, like being even [\(f(-x)=f(x)]\text{,}\) odd [\(f(-x)=-f(x)\)], or periodic. This allows drawing only part of the graph and then filling in the rest by “copying”.
  3. Compute the \(y\) intercept \(f(0)\) and if feasible, find the \(x\)-intercepts: points where \(y=0\text{.}\)
  4. Find any horizontal asymptotes, by checking for limits \(\displaystyle \lim_{x \to -\infty} f(x)\) and \(\displaystyle \lim_{x \to \infty} f(x)\text{.}\) If these limits exist (including infinite values), add these to the list of “interesting” \(x\) values.
  5. Compute the first derivative \(y'=f'(x)\text{,}\) and find the critical points, where \(y'\) is zero or DNE.
  6. Compute the second derivative \(y''=f''(x)\text{,}\) and find the points where \(y''\) is zero or DNE (possible inflections).
  7. Evaluate the function at all these “interesting” \(x\) values, and summarize on a table with a column for each of these \(x\) values and a column for each interval between them, as seen in Section 4.3. Include \(x\) values of \(\pm\infty\) if limits were found there.
  8. Determine the sign of \(y'\) and of \(y''\text{,}\) meaning positive, negative or zero, and add this information in the next rows of the table as \(+\text{,}\) \(-\text{.}\) or \(0\text{.}\) One way to do this is to evaluate \(y'\) and \(y''\) at one \(x\) value within each interval between “interesting” values. Increasing/decreasing behavior can also be checked by just comparing the values at the interesting \(x\) values to either side, and this can even work when one of those values is a limit as \(x \to \infty\) or \(x \to -\infty\text{.}\)
  9. In the final row of the table, draw a little fragment of curve with the correct increasing/decreasing behavior and correct concavity.
  10. Sketch the graph using the points in the top two lines, the shapes in the bottom line of this table, and any other information gathered, such as about horizontal and vertical asymptotes.

Study Examples 1–5 in the text.

We omit the final topic, Slant Asymptotes, and thus ignore Example 6.