How is the graph of a function related to the graph of its derivative?

A concave down interval on the graph of a function corresponds to a decreasing interval on the graph of its derivative (intervals A, B, and D in the figure). And a concave up interval on the function corresponds to an increasing interval on the derivative (intervals C, E, and F).

What does the graph of a derivative tell you about the original function?

The differences between the graphs come from whether the derivative is increasing or decreasing. The derivative of a function f is a function that gives information about the slope of f. The derivative tells us if the original function is increasing or decreasing. Because f′ is a function, we can take its derivative.

What are the inflection points on a derivative graph?

Inflection points are points where the first derivative changes from increasing to decreasing or vice versa. Equivalently we can view them as local minimums/maximums of f′(x). From the graph we can then see that the inflection points are B,E,G,H.

What is the relationship of the second derivative of a function with that of its graph?

On the graph of a function, the second derivative corresponds to the curvature or concavity of the graph. The graph of a function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the opposite way.

What does the first derivative tell you in a word problem?

The first interpretation of a derivative is rate of change. This was not the first problem that we looked at in the Limits chapter, but it is the most important interpretation of the derivative. If f(x) represents a quantity at any x then the derivative f′(a) represents the instantaneous rate of change of f(x) at x=a .

How do you find the point of inflection?

To verify that this point is a true inflection point we need to plug in a value that is less than the point and one that is greater than the point into the second derivative. If there is a sign change between the two numbers than the point in question is an inflection point.

How does second derivative work?

The second derivative may be used to determine local extrema of a function under certain conditions. If a function has a critical point for which f′(x) = 0 and the second derivative is positive at this point, then f has a local minimum here.