Graphs of functions and their derivatives

Front

What does it mean if a function's derivative is negative?

Back

The function is decreasing.

Front

What does it mean if a function's derivative is positive?

Back

The function is increasing.

Front

How can you determine where a function is increasing or decreasing using its derivative?

Back

A function is increasing where its derivative f'(x) > 0 and decreasing where f'(x) < 0.

Front

What is the relationship between a function and its derivative at a local maximum?

Back

At a local maximum, the derivative f'(x) = 0 and changes from positive to negative.

Front

What is the relationship between a function and its derivative at a local minimum?

Back

At a local minimum, the derivative f'(x) = 0 and changes from negative to positive.

Front

If f'(x) is positive and increasing, what can be said about f(x)?

Back

f(x) is increasing at an increasing rate.

Front

If f'(x) is negative and decreasing, what can be said about f(x)?

Back

f(x) is decreasing at an increasing rate.

Front

What does the second derivative test tell us about a function?

Back

The second derivative test helps determine the concavity of the function and can indicate local maxima or minima.

Front

What does it mean if the second derivative f''(x) is positive?

Back

The function f(x) is concave up.

Front

What does it mean if the second derivative f''(x) is negative?

Back

The function f(x) is concave down.