Derivative Graphs Flashcard

Front

What does it mean for a function to be differentiable at a point?

Back

A function is differentiable at a point if it has a defined derivative at that point, which means the slope of the tangent line can be calculated. This requires the function to be continuous and not have any sharp turns or vertical tangents at that point.

Front

What is the relationship between continuity and differentiability?

Back

A function must be continuous at a point to be differentiable there, but continuity alone does not guarantee differentiability. A function can be continuous but not differentiable if it has sharp turns or cusps.

Front

What does a vertical tangent indicate about the derivative of a function?

Back

A vertical tangent indicates that the derivative does not exist at that point because the slope of the tangent line approaches infinity.

Front

What does g''(3) = -8 imply about the function g(x) at x=3?

Back

It implies that the function g(x) is concave down at x=3.

Front

How is the concavity of a function determined?

Back

The concavity of a function is determined by its second derivative. If g''(x) > 0, the function is concave up; if g''(x) < 0, the function is concave down.

Front

What is the significance of the second derivative in relation to a function's graph?

Back

The second derivative provides information about the concavity of the function's graph and can indicate points of inflection where the concavity changes.

Front

What are the intervals of concavity for a function?

Back

Intervals of concavity are ranges of x-values where the function is either concave up or concave down, determined by the sign of the second derivative.

Front

What does it mean if a function is concave up on an interval?

Back

If a function is concave up on an interval, it means that the graph of the function is curving upwards, and the slope of the tangent line is increasing.

Front

What does it mean if a function is concave down on an interval?

Back

If a function is concave down on an interval, it means that the graph of the function is curving downwards, and the slope of the tangent line is decreasing.

Front

How can you find the intervals where a function is concave down?

Back

To find the intervals where a function is concave down, determine where the second derivative is negative.