Questions: Curve Sketching

Question 1

A function f satisfies f'(x) > 0 for x < 2, f'(2) = 0, and f'(x) < 0 for x > 2. Which conclusion does the first derivative test support?

• A. f has a local minimum at x = 2
• B. f has a local maximum at x = 2
• C. f has an inflection point at x = 2
• D. f is constant near x = 2

Answer: (B)

When f' changes from positive to negative at a critical point, the function changes from increasing to decreasing — the hallmark of a local maximum. A local minimum would require f' to change from negative to positive. An inflection point is a sign change in f'' (concavity), not f'. The first derivative test reads the sign change of f', not its value.

Question 2

A student sketches a rational function and draws the curve crossing its horizontal asymptote at x = 5. The curve then levels off toward the asymptote as x → ∞. Is this sketch valid?

• A. No — a function can never cross its horizontal asymptote
• B. Yes — crossing a horizontal asymptote is permitted; asymptotes describe end behavior, not local behavior
• C. No — horizontal asymptotes only exist if the function approaches them monotonically
• D. Yes — but only if the function is a polynomial

Answer: (B)

A horizontal asymptote describes the limit as x → ±∞, not what happens at finite x values. A function can cross its horizontal asymptote any number of times and still approach that value at infinity. The common misconception is treating asymptotes as uncrossable barriers. Only vertical asymptotes (where the function is undefined) cannot be crossed.

Question 3

If f'(c) = 0, then f is expected to have a local extremum at x = c.

• T. True
• F. False

Answer: False

A critical point where f'(c) = 0 is only a *candidate* for a local extremum. The first derivative test requires f' to actually *change sign* at c. If f'(x) > 0 on both sides of c (or < 0 on both sides), c is neither a local max nor a local min — it may be an inflection point with a horizontal tangent, like x = 0 for f(x) = x³.

Question 4

A curve sketch that correctly identifies all local extrema, inflection points, asymptotes, and end behavior is a faithful representation of the function, even without precise coordinates for most points.

• T. True
• F. False

Answer: True

The purpose of curve sketching is to capture the qualitative shape using derivative information, not to plot exact values. If you correctly show where the function is increasing vs. decreasing, where it is concave up vs. down, where it has extrema and inflection points, and how it behaves at the boundaries of its domain, the sketch accurately represents the function's essential structure. Exact coordinates are secondary.

Question 5

Explain why the systematic curve-sketching checklist allows you to draw an accurate graph without computing f(x) at arbitrary points.

Model answer: Each step of the checklist extracts derivative information that constrains the graph's shape: the sign of f' determines where the function rises and falls, the sign of f'' determines concavity, critical points locate extrema, and limits at infinity determine end behavior. Together, these constraints leave only one possible qualitative shape for the function.

Plotting f(1.5) or f(3.2) tells you only the height at a single point — it could be part of an increasing stretch, a local peak, or an inflection. By contrast, knowing f' > 0 on (1, 4) tells you the function is strictly increasing throughout that entire interval, which is far more informative. The checklist accumulates qualitative constraints that collectively determine the graph's global shape, making point-by-point plotting redundant for sketching purposes.