Concave Down
Definition
Concave down refers to the shape of a graph where it curves downward, resembling a frowning face. It indicates that the second derivative of a function is negative.
Analogy
Picture yourself skiing down a mountain slope with steep descents. The slope forms downward curves, similar to how concave down graphs look.
Related terms
Inflection Point: An inflection point is a point on the graph where the concavity changes from concave up to concave down or vice versa.
Decreasing Function: A decreasing function is one in which the y-values decrease as x-values increase.
Second Derivative Test: The second derivative test is used to determine whether a critical point on a graph corresponds to a local maximum or minimum by examining the sign of the second derivative.
"Concave Down" appears in:
Subjects (1)
Study guides (4)
AP Calculus AB/BC - 4.6 Approximating Values of a Function Using Local Linearity and Linearization
AP Calculus AB/BC - 5.3 Determining Intervals on Which a Function is Increasing or Decreasing
AP Calculus AB/BC - 5.6 Determining Concavity
AP Calculus AB/BC - 5.9 Connecting a Function, Its First Derivative, and its Second Derivative
Practice Questions (5)
How does the tangent line approximation change when the function is concave down?
Which of the following pairs of parametric equations are concave down at t = 1?
A particle is moving on the xy-plane starting at t = 0. Its motion can be described by the parametric functions y(t) = 3t^3 - 2t and x(t) = ln(t). At what time, t, is the path of the particle neither concave up nor concave down?
A particle moves on the xy-plane. Its motion can be described by the parametric functions y(t) = sin(t) and x(t) = cos(t). For which t in the interval 0 ≤ t < 2π is the path of the particle concave down?
A video game character's attack and defense increase over time (t > 0) is given by the parametric equations A(t) = 3t^2+5t and D(t) = 2t^2+8t. James the gamer plots these values on a plot with attack as the x-axis and defense as the y-axis. Is the resulting curve concave up or concave down?
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