Glossary
Concave down
A function is concave down if its graph opens downward, meaning the slopes of its tangent lines are decreasing. This occurs when the second derivative of the function is negative.
Example:
The function is always concave down, resembling an inverted U-shape, as its second derivative is .
Concave up
A function is concave up if its graph opens upward, meaning the slopes of its tangent lines are increasing. This occurs when the second derivative of the function is positive.
Example:
The function is always concave up because its second derivative, , is always positive.
Concavity
Concavity describes the direction a function's graph opens, either upward or downward. It is determined by the rate at which the slopes of the tangent lines are changing.
Example:
The graph of exhibits concavity that is always upward, forming a wide U-shape.
Inflection point
An inflection point is a point on the graph of a function where the concavity changes (from concave up to concave down or vice versa). At this point, the second derivative is typically zero or undefined, and its sign changes.
Example:
For , the origin is an inflection point because the graph changes from concave down to concave up at that specific location.
Second derivative
The second derivative of a function, denoted as $f''(x)$ or $\frac{d^2y}{dx^2}$, is the derivative of its first derivative. It provides information about the concavity of the original function.
Example:
If a car's position is given by , its velocity is , and its acceleration is the second derivative, .