Graphs & Derivatives

Front

What is the derivative of a function?

Back

The derivative of a function measures the rate at which the function's value changes as its input changes. It represents the slope of the tangent line to the graph of the function at a given point.

Front

What does f' represent in calculus?

Back

f' represents the first derivative of the function f, indicating the rate of change of f with respect to its variable.

Front

What is a critical point?

Back

A critical point of a function occurs where its derivative is zero or undefined, indicating potential locations for relative maxima, minima, or inflection points.

Front

What is an inflection point?

Back

An inflection point is a point on the graph of a function where the concavity changes, which can be identified where the second derivative is zero or undefined.

Front

When is a function increasing?

Back

A function is increasing on an interval if its derivative is positive throughout that interval.

Front

When is a function decreasing?

Back

A function is decreasing on an interval if its derivative is negative throughout that interval.

Front

What does f''(x) indicate?

Back

f''(x) is the second derivative of the function f, which provides information about the concavity of the function and can indicate points of inflection.

Front

How do you find relative maxima and minima using derivatives?

Back

To find relative maxima and minima, set the first derivative f' equal to zero to find critical points, then use the second derivative test to determine the nature of these points.

Front

What is the relationship between a function's increasing/decreasing behavior and its derivative?

Back

If a function is increasing, its derivative is positive; if it is decreasing, its derivative is negative.

Front

What does it mean if g'(-2) = 0?

Back

It means that x = -2 is a critical point of the function g(x), where the function may have a relative maximum, minimum, or an inflection point.