Skip to main content

Workbook for Stewart’s "Calculus"

Cory Wilson

Section 3.5 Summary of Curve Sketching

Subsection 3.5.1 Before Class

https://mymedia.ou.edu/media/3.5-1/1_lwewtbkr
Figure 26. Pre-Class Video 1

Subsubsection 3.5.1.1 Summary of Graphing

Guidelines for Curve Sketching.
  1. Find the domain of the function.
  2. Identify \(x-\) and \(y-\)intercepts.
  3. Locate symmetry in the graph, i.e. determine if the function is even, odd, periodic, etc.
  4. Locate any asymptotes: vertical or horizontal.
  5. Determine the intervals where the function is increasing or decreasing.
  6. Find local maxima/minima.
  7. Determine the intervals where the function is concave up/concave down.
With this information, you can confidently sketch the graph.
Example 3.5.1.
Sketch the graph of \(f(x) = x^3 + 3x^2\)
Solution.
The graph of \(f(x) = x^3 + 3x^2\) on the interval \([-4,2]\text{.}\)
Example 3.5.2.
Sketch the graph of \(g(x) = \dfrac{x^2}{\sqrt{x+1}}\)
Solution.
The graph of \(g(x) = \dfrac{x^2}{\sqrt{x+1}}\) on the interval \([-1,2]\text{.}\)

Subsection 3.5.2 Pre-Class Activities

Example 3.5.3.

Sketch the graph of \(y=2+3x^2-x^3\)
Solution.
The graph of \(y=2+3x^2-x^3\) on the interval \([-2,4]\text{.}\)

Example 3.5.4.

Sketch the graph of the function \(y = \dfrac{x^2 + 5x}{25-x^2}\)
Solution.
The graph of \(y = \dfrac{x^2 + 5x}{25-x^2}\) on the interval \([-8,10]\text{.}\)

Subsection 3.5.3 In Class

Example 3.5.5.

Sketch the graph of \(y = \dfrac{x}{x-1}\)
Solution.
The graph of \(y = \dfrac{x}{x-1}\) on the interval \([-4,4]\text{.}\)

Example 3.5.6.

Sketch the graph of \(f(x) = \dfrac{x^2}{x^2-4}\)
Solution.
The graph of \(y = \dfrac{x^2}{x^2-4}\) on the interval \([-4,4]\text{.}\)

Example 3.5.7.

Sketch the graph of \(y = \dfrac{x^3}{x^3+1}\)
Solution.
The graph of \(y = \dfrac{x^3}{x^3+1}\) on the interval \([-3,3]\text{.}\)

Example 3.5.8.

Sketch the graph of \(g(x) = \dfrac{x}{\sqrt{x^2-1}}\)
Solution.
The graph of \(g(x) = \dfrac{x}{\sqrt{x^2-1}}\) on the interval \([-4,4]\text{.}\)

Example 3.5.9.

Sketch the graph of \(y=\dfrac{x}{x^2-4}\)
Solution.
The graph of \(y=\dfrac{x}{x^2-4}\) on the interval \([-5,5]\text{.}\)

Example 3.5.10.

Sketch the graph of \(y=\dfrac{x^3}{x-2}\)
Solution.
The graph of \(y=\dfrac{x^3}{x-2}\) on the interval \([-4,4]\text{.}\)

Example 3.5.11.

Sketch the graph of \(y = \dfrac{\sin x}{1+\cos x}\)
Solution.
The graph of \(y = \dfrac{\sin x}{1+\cos x}\) on the interval \([-3\pi,3\pi]\text{.}\)

Subsection 3.5.4 After Class Activities

Example 3.5.12.

Sketch the graph of \(y = \dfrac{1+5x-2x^2}{x^2-1}\)
Solution.
The graph of \(y = \dfrac{1+5x-2x^2}{x^2-1}\) on the interval \([-3,3]\text{.}\)
<Prev^TopNext>