Glossary
Concave Down
A function is concave down on an interval if its graph opens downwards, resembling an inverted cup, which occurs when its second derivative is negative.
Example:
The initial upward trajectory of a rocket, before gravity significantly slows its acceleration, might be concave down.
Concave Up
A function is concave up on an interval if its graph opens upwards, resembling a cup, which occurs when its second derivative is positive.
Example:
The path of a ball thrown upwards, after reaching its peak, is concave up as it accelerates downwards.
Decreasing Function
A function is decreasing on an interval if its y-values fall as its x-values increase, which means its first derivative is negative.
Example:
The amount of medication in a patient's bloodstream is a decreasing function over time as the body processes it.
First Derivative (f'(x))
Represents the instantaneous rate of change of a function, indicating its slope and whether the original function is increasing or decreasing.
Example:
If f'(x) is positive, the original function f(x) is currently climbing uphill.
First Derivative Test
A method used to determine relative extrema of a function by analyzing the sign changes of its first derivative around critical points.
Example:
If f'(x) changes from positive to negative at x=c, the First Derivative Test confirms a relative maximum at x=c.
Function (f(x))
A mathematical relation where each input has exactly one output. In calculus, it often represents a quantity's value based on an independent variable.
Example:
If a car's position is given by f(t) = t^2, then at t=3 seconds, its position is 9 meters.
Increasing Function
A function is increasing on an interval if its y-values rise as its x-values increase, which means its first derivative is positive.
Example:
A population growth model shows an increasing function when the birth rate consistently exceeds the death rate.
Point of Inflection
A point on a function's graph where its concavity changes (from concave up to down or vice versa), and the second derivative is zero or undefined.
Example:
The point where a growth curve transitions from accelerating growth to decelerating growth is often a point of inflection.
Relative Maximum
A point on a function's graph where the function changes from increasing to decreasing, resulting in a local high point.
Example:
The peak of a mountain range on a topographical map represents a relative maximum elevation.
Relative Minimum
A point on a function's graph where the function changes from decreasing to increasing, resulting in a local low point.
Example:
The lowest point a roller coaster reaches in a dip is a relative minimum before it climbs again.
Second Derivative (f''(x))
Represents the rate of change of the first derivative, indicating the concavity of the original function.
Example:
When f''(x) is negative, the function f(x) is bending downwards, like a frown.
Second Derivative Test
A method used to determine relative extrema of a function by evaluating the sign of the second derivative at a critical point where the first derivative is zero.
Example:
If f'(c) = 0 and f''(c) > 0, the Second Derivative Test indicates a relative minimum at x=c.
x-intercept (in context of derivatives)
The point(s) where a graph crosses the x-axis, meaning the function's value is zero at that point. For f'(x), these indicate critical points of f(x); for f''(x), they indicate potential inflection points of f(x).
Example:
If the graph of f'(x) has an x-intercept at x=2, it means f(x) has a critical point (a local max or min) at x=2.