Flashcards on How to recognize the graph of the derivative from the graph of a function
How to Recognize the Graph of the Derivative from a Function
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Derivatives of Functions
33 cards
Card 1
Question: What does the derivative f'(x) represent graphically?
Answer: The slope of the tangent line to the function's curve at any given point.
Card 2
Question: How does an increasing interval of f(x) relate to its derivative f'(x)?
Answer: If f(x) is increasing, its tangent lines have positive slopes, so f'(x) must be positive (its graph lies above the x-axis).
Card 3
Question: What does it mean for f'(x) if f(x) is decreasing over an interval?
Answer: If f(x) is decreasing, its tangent lines have negative slopes, meaning f'(x) is negative (its graph lies below the x-axis).
Card 4
Question: What happens to the graph of f'(x) at a local maximum or minimum of a differentiable function f(x)?
Answer: At a local maximum or local minimum, the tangent line is horizontal (slope is zero), so f'(x) will cross the x-axis (f'(x) = 0).
Card 5
Question: How does concavity of f(x) relate to the behavior of f'(x)?
Answer: If f(x) is concave up, its slopes are increasing, meaning f'(x) is an increasing function. If f(x) is concave down, its slopes are decreasing, meaning
Card 6
Question: What feature will f'(x) have at an inflection point of f(x)?
Answer: At an inflection point, where concavity changes, f'(x) will have a local extremum (a local maximum or minimum).
Card 7
Question: What does a vertical tangent line in f(x) imply about f'(x)?
Answer: If f(x) has a vertical tangent line, f'(x) will have a vertical asymptote at that x-coordinate.
Card 8
Question: What happens to f'(x) if f(x) has a sharp corner or cusp?
Answer: The derivative f'(x) is undefined at that point, resulting in a discontinuity (often a jump or a hole) in the graph of f'(x).
Card 9
Question: If f(x) is a constant function, what is f'(x)?
Answer: If f(x) is a constant function, then f'(x) = 0 (the x-axis itself).
Card 10
Question: If f(x) is a linear function like f(x) = mx + b, what is its derivative f'(x)?
Answer: f'(x) = m (a horizontal line at height m).