Concavity
Definition
Concavity describes whether a graph opens upward (concave up) or downward (concave down). It indicates whether the graph is curving upwards like an "U" shape or downwards like an "n" shape.
Analogy
Think about riding your bike along different types of hills. If you ride on an uphill slope, your path will be concave up because it curves upward. On a downhill slope, your path will be concave down because it curves downward.
Related terms
Inflection Point: An inflection point occurs where there is a change in concavity on a graph. At this point, the graph transitions from being concave up to concave down or vice versa.
Second Derivative Test: The second derivative test helps determine if critical points on a graph represent local maxima or minima by analyzing changes in concavity.
Point of Inflection: A point of inflection is a specific type of inflection point where the graph changes concavity and has no local maxima or minima.
"Concavity" appears in:
Study guides (3)
Additional resources (4)
Practice Questions (11)
If the second derivative of a function is negative, what can be said about the concavity of the graph?
If the second derivative of a function is positive, what can be said about the concavity of the graph?
How can the Second Derivative Test be used to determine concavity?
When determining the concavity of a function, which of the following is correct?
What is the concavity of a function if its second derivative is zero?
What does the concavity of a function indicate about its rate of change?
When determining the concavity of a function, which of the following is true?
The Second Derivative Test is used to analyze the concavity of a function and determine whether a critical point is a relative minimum, relative maximum, or neither. This test is based on the principle that:
The Second Derivative Test is a valuable tool for determining extrema because it allows us to analyze the concavity of a function at critical points. By examining the sign of the second derivative, we can identify whether a critical point corresponds to a relative minimum, relative maximum, or what else?
What information can be obtained from a slope field about the concavity of a function?
Henry draws a heart defined by the parametric equations x(t) = sin(t) and y(t) = 1 + cos(t) - cos^2(t). What is the concavity of the curve at the point (sqrt(3)/2, 5/4)?
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