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 it mean if f'(x) > 0 for an interval?

Back

If f'(x) > 0 for an interval, it means that the function f(x) is increasing on that interval.

Front

What does it mean if f'(x) < 0 for an interval?

Back

If f'(x) < 0 for an interval, it means that the function f(x) is decreasing on that interval.

Front

What is a critical point?

Back

A critical point of a function is a point where the derivative is either zero or undefined. It is a potential location for relative extrema.

Front

What is the Second Derivative Test?

Back

The Second Derivative Test is a method used to determine the concavity of a function at a critical point. If f''(x) > 0, the function is concave up (local minimum); if f''(x) < 0, the function is concave down (local maximum). If f''(x) = 0, the test is inconclusive.

Front

What is a local maximum?

Back

A local maximum is a point on the graph of a function where the function value is higher than all nearby points.

Front

What is a local minimum?

Back

A local minimum is a point on the graph of a function where the function value is lower than all nearby 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 determined by the sign of the second derivative.

Front

What does it mean if f''(a) < 0 at a critical point a?

Back

If f''(a) < 0 at a critical point a, it indicates that the function has a local maximum at that point.

Front

What does it mean if f''(a) > 0 at a critical point a?

Back

If f''(a) > 0 at a critical point a, it indicates that the function has a local minimum at that point.