- Topic
- Curve sketching
- Difficulty
- Standard
- Suggested time
- 3 min
- Marks
- 2
Curve sketching practice question 21
The graph of the derivative of a function is shown. Which of the graphs below could be the graph of ?
rises where and falls where . Read the sign of on each side of its zeros.
The zeros of are where has horizontal tangents. Which candidates turn at and ?
Theory UBC Math 100 §7CLP-1 §3.6Key results
Worked solution
Strategy
Turn the graph of f′ into a sign chart, then pick the candidate that rises and falls accordingly and turns exactly where f′ = 0.
Working
The parabola is above the x-axis outside its zeros and below it between them.
First derivative test at each zero of f′.
Two candidates turn near x = ±1.2, so their horizontal tangents are in the wrong places; of the two that turn at ±2, only one rises, falls, then rises.
Answer
The answer is The graph that rises to a high point at x = −2, falls to a low point at x = 2, then rises..
Check
f′ is lowest, −4, at x = 0, so f should fall most steeply there; Graph A does, halfway between its turning points at ±2.
Where this goes wrong
Choosing a graph that crosses the x-axis where f′ is zero. The zeros of f′ give the turning points of f; where f itself crosses the axis cannot be read from f′ at all, since raising or lowering a graph keeps every slope.
Leans on
The derivative as the slope of a tangent line, from The definition of the derivative
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