Chapter 12 – 4Curve-Sketching Techniques

1. Analyze f(x)

  • Determine the domain
  • Find intercepts
  • Find asymptotes

2. Analyze the first derivative

  • Find local minima and maxima
  • Find where the function is increasing and where it is decreasing

3. Analyze the second derivative

  • Find inflection points
  • Find where the function is concave up and where it is concave down

4. Sketch the graph

1. Analyze f(x)

  • Domain: all x except x = 3.
  • X-intercept at (-3,0). Y-intercept at (0, -1)
  • Vertical asymptote at x = 3,
  • Horizontal asymptote at y = 1,

2. Analyze the first derivative

  • is never zero and is undefined at x = 3

3. Analyze the second derivative

  • is never zero and is undefined at x = 3

4. Sketch the graph.

1. Analyze f(x)

  • Domain: all x
  • X-intercept at (3,0). Y-intercept at (0, 3)
  • Horizontal asymptote at y=0,

2. Analyze the first derivative

  • =0 at x = 2
  • local maximum at

3. Analyze the second derivative

4. Sketch the graph.

Black dots are the intercepts. The red dot is the local (and absolute) maximum. The blue dot is the inflection point.

Example 3. In Silicon Valley (in California), a number of computer-related manufacturing firms were found to be contaminating underground water supplies with toxic chemicals stored in leaking underground containers. A water quality control agency ordered the companies to take immediate corrective action and to contribute to a monetary pool for testing and cleanup of the underground contamination. Suppose that the required monetary pool (in millions of dollars) for the testing and cleanup is estimated to be given by

where x is the proportion of the total contaminant removed. Sketch the graph of this function.

1. Analyze f(x)

  • Domain:
  • X and Y-intercept at (0,0).
  • Vertical asymptote at x = 1,
  • Horizontal asymptote at y = -2,

2. Analyze the first derivative

  • is never zero and is undefined at x = 1

3. Analyze the second derivative

  • is never zero and is undefined at x = 1

4. Sketch the graph.

Figure 2. The Bigger Picture

Example 4.The total daily cost (in dollars) of producing x park benches is given by

(A) Sketch the graphs of the average cost function and the marginal cost function on the same set of coordinated axes. Include any oblique asymptotes.

(B) Find the minimum average cost.

The average cost function is given by

1. Analyze A(x)

  • Domain:
  • There is no X-intercept on the domain of this function.
  • Vertical asymptote at x = 0,
  • Oblique asymptote at ,

2. Analyze the first derivative

  • at

3. Analyze the second derivative

  • is never zero and is undefined at x = 0

4. Sketch the graph.

The marginal cost function is given by

1. Analyze M(x)

  • Domain:
  • There is no X-intercept on the domain of this function.
  • No asymptotes

2. Analyze the first derivative

  • is never zero

3. Analyze the second derivative

  • The marginal cost has no concavity (because it is linear).

4. Sketch the graph.