Quiz: Curve Sketching
Test your understanding of curve sketching with these review questions.
1. A complete curve analysis should include all of the following EXCEPT:
- Domain and range
- Intercepts and asymptotes
- The exact area under the curve
- Intervals of increase/decrease and concavity
Show Answer
The correct answer is C. Curve sketching involves domain, intercepts, symmetry, asymptotes, first derivative analysis (increase/decrease, extrema), and second derivative analysis (concavity, inflection points). Area requires integration, not sketching.
Concept Tested: Complete Curve Analysis
2. If f'(x) > 0 and f''(x) < 0, the graph is:
- Increasing and concave up
- Increasing and concave down
- Decreasing and concave up
- Decreasing and concave down
Show Answer
The correct answer is B. f' > 0 means increasing, f'' < 0 means concave down. The graph rises while curving downward—it's getting less steep as it rises.
Concept Tested: f f' f''
3. From a graph of f'(x), you can determine:
- The exact values of f(x)
- Where f is increasing or decreasing
- The y-intercept of f
- The area under f
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The correct answer is B. From f', you can tell where f is increasing (f' > 0) or decreasing (f' < 0), and locate critical points (f' = 0). You cannot determine exact f values without more information.
Concept Tested: Graph from Derivative
4. If the graph of f' crosses the x-axis from positive to negative, f has:
- A local minimum
- A local maximum
- An inflection point
- A vertical asymptote
Show Answer
The correct answer is B. When f' goes from positive to negative, f goes from increasing to decreasing, indicating a local maximum. This is the First Derivative Test in graphical form.
Concept Tested: Derivative from Graph
5. Which features should be plotted first when sketching a curve?
- Random points on the curve
- Intercepts, asymptotes, and critical points
- The area under the curve
- Only the maximum and minimum
Show Answer
The correct answer is B. Start with key features: intercepts (where graph crosses axes), asymptotes (boundaries), and critical points (extrema). These provide the framework for the sketch.
Concept Tested: Domain Analysis
6. If f is an even function, its graph has symmetry about:
- The x-axis
- The y-axis
- The origin
- The line y = x
Show Answer
The correct answer is B. Even functions satisfy f(−x) = f(x), meaning the graph is symmetric about the y-axis. You only need to analyze x ≥ 0 and reflect.
Concept Tested: Symmetry Analysis
7. An inflection point on f corresponds to what on f'?
- A zero of f'
- A maximum or minimum of f'
- Where f' is undefined
- Where f' crosses the x-axis
Show Answer
The correct answer is B. An inflection point of f is where f'' = 0 or undefined and concavity changes. This corresponds to a local extremum of f' (where f'' = 0).
Concept Tested: Connecting Three Graphs
8. For the function f(x) = x/(x² + 1), what is the horizontal asymptote?
- y = 1
- y = 0
- y = x
- No horizontal asymptote
Show Answer
The correct answer is B. As x→±∞, f(x) = x/(x² + 1) ≈ x/x² = 1/x → 0. The degree of denominator exceeds the numerator, so y = 0 is the horizontal asymptote.
Concept Tested: Asymptote Analysis
9. The graph of f''(x) tells you about:
- Where f is increasing or decreasing
- The concavity of f
- The y-intercept of f
- The maximum value of f
Show Answer
The correct answer is B. The sign of f'' determines concavity: f'' > 0 means f is concave up, f'' < 0 means concave down. Where f'' = 0 and changes sign indicates inflection points.
Concept Tested: Concavity Analysis
10. When analyzing local extrema, which analysis is most direct?
- Find where f(x) = 0
- Find where f'(x) = 0 or undefined, then test
- Find where f''(x) = 0
- Plot many points
Show Answer
The correct answer is B. Local extrema occur at critical points where f'(x) = 0 or undefined. Then use First or Second Derivative Test to classify each as max, min, or neither.
Concept Tested: Local Extrema Analysis