Concavity and Rates of Change

Front

What is concavity in a function?

Back

Concavity refers to the direction in which a function curves. A function is concave up if its graph opens upwards (like a cup) and concave down if it opens downwards.

Front

How do you determine if a function is increasing or decreasing?

Back

A function is increasing on an interval if the derivative is positive throughout that interval. It is decreasing if the derivative is negative.

Front

What is the average rate of change of a function?

Back

The average rate of change of a function f(x) over an interval [a, b] is given by (f(b) - f(a)) / (b - a).

Front

What does it mean for a function to transition from concave down to concave up?

Back

This transition indicates a point of inflection, where the second derivative changes sign.

Front

What is the significance of the first derivative test?

Back

The first derivative test helps determine local maxima and minima of a function by analyzing the sign changes of the derivative.

Front

What is the second derivative test?

Back

The second derivative test is used to determine the concavity of a function and to identify local extrema based on the sign of the second derivative.

Front

What is an interval of increase?

Back

An interval of increase is a range of x-values where the function's output values are rising as x increases.

Front

What is an interval of decrease?

Back

An interval of decrease is a range of x-values where the function's output values are falling as x increases.

Front

How can you find intervals of concavity?

Back

Intervals of concavity can be found by analyzing the sign of the second derivative of the function.

Front

What is a point of inflection?

Back

A point of inflection is a point on the graph of a function where the concavity changes.