Mrs. B's Average Rate of Change & Concavity

Front

What is the average rate of change of a function over an interval?

Back

The average rate of change is the slope of the secant line passing through two given points on the graph of the function.

Front

What does it mean for a function f(x) to be increasing at an increasing rate?

Back

It means that the slope of the tangent line is increasing, indicating that the function's rate of increase is itself increasing.

Front

What does it mean for a function f(x) to be decreasing at a decreasing rate?

Back

It means that the slope of the tangent line is becoming less negative, indicating that the function's rate of decrease is slowing down.

Front

How can you determine if a function is concave up?

Back

A function is concave up if its second derivative is positive, indicating that the slope of the tangent line is increasing.

Front

How can you determine if a function is concave down?

Back

A function is concave down if its second derivative is negative, indicating that the slope of the tangent line is decreasing.

Front

What is the significance of a point where a function transitions from concave down to concave up?

Back

This point is called an inflection point, where the function changes its concavity.

Front

What does it mean for a function to be decreasing?

Back

A function is decreasing if, as x increases, the value of f(x) decreases.

Front

What does it mean for a function to be concave down?

Back

A function is concave down if it curves downward, and the slope of the tangent line decreases as x increases.

Front

What is the graphical representation of the average rate of change?

Back

It is represented by the slope of the secant line connecting two points on the graph of the function.

Front

What is the difference between increasing at an increasing rate and increasing at a decreasing rate?

Back

Increasing at an increasing rate means the function's growth accelerates, while increasing at a decreasing rate means the function grows but at a slowing pace.