Recitation 15

Exercise: Use the guidelines to sketch the curve.

  1. y=\frac{x}{x-1}
  2. y=\frac{x-1}{x^2}
  3. y=1/(1+e^{-x})
  4. y=x\ln x

The following steps are helpful when sketching curves. These are general guidelines for all curves, so each step may not always apply to all functions.

  1. Domain: Find the domain of the function. This will be useful when finding vertical asymptotes and determining critical numbers.
  2. Intercepts: Find the x- and y-intercepts of the function, if possible. To find the x-intercept, we set y = 0 and solve the equation for x. Similarly, we set x = 0 to find the y-intercept.
  3. Symmetry: Determine whether the function is an odd function, an even function or neither odd nor even. If f(-x) = f(x) for all x in the domain, then f is even and symmetric about the y-axis. If f(-x) = -f(x) for all x in the domain, then f is odd and symmetric about the origin.
  4. Asymptotes: Find the asymptotes of the function using the methods described above. First attempt to find the vertical and horizontal asymptotes of the function. IF necessary, find the slant asymptote.
  5. Intervals of Increase and Decrease: Using the methods described above, determine where f'(x) is positive and negative to find the intervals where the function is increasing and decreasing.
  6. Local Maximum/Minimum : Find the critical numbers of the function. Remember that the number c in the domain is a critical number if f'(c) = 0 or f'(c) does not exist. Use the first derivative test to find the local maximums and minimums of the function.
  7. Concavity and Points of Inflection : We must determine when f”(x) is positive and negative to find the intervals where the function is concave upward and concave downward. Inflection points occur whenever the curve changes in concavity.
  8. Sketch : Using the information obtained from steps A to G, we can sketch the curve. First, we draw dashed lines for the asymptotes of the function. Then plot the x- and y-intercepts, maximum and minimum points and points of inflection on the graph. Sketch the curve between the points, using the intervals of increase and decrease and intervals of concavity. Be sure that the graph behaves correctly when approaching asymptotes.

The sketches for the curves can be found in the following links.

  1. y=x/(1-x).
  2. y=(x-1)/x^2.
  3. y=1/(1+e^(-x)).
  4. y=x ln x.

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