First and Second Derivative Practice

Front

What is the second derivative of a function used for?

Back

The second derivative is used to determine the concavity of a function and to identify points of inflection.

Front

Define concave up.

Back

A function is concave up on an interval if its second derivative is positive on that interval.

Front

Define concave down.

Back

A function is concave down on an interval if its second derivative is negative on that interval.

Front

What does a positive first derivative indicate about a function?

Back

A positive first derivative indicates that the function is increasing.

Front

What does a negative first derivative indicate about a function?

Back

A negative first derivative indicates that the function is decreasing.

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.

Front

How do you find the intervals of concavity for a function?

Back

To find intervals of concavity, take the second derivative of the function, set it to zero, and solve for critical points. Test intervals around these points.

Front

What is the maximum value of f(x) = x^3 - 3x^2 - 1 on the interval [-3, 2]?

Back

The maximum value is -1.

Front

On what interval(s) is the function y = x^3 + 6x^2 concave up?

Back

The function is concave up on the interval (-2, ∞).

Front

On what interval(s) is the function y = x^3 + 6x^2 concave down?

Back

The function is concave down on the interval (-∞, -2).