Jul 25, 2021 · Learn how to graph a derivative and how to analyze a derivative graph to find extrema, increasing/decreasing intervals and concavity.

Let f be an increasing, differentiable function on an open interval I, such as the one shown in Figure 3.3.2, and let a <b be given in I. The secant line on the graph of f from x = a to x = b is …

The derivative of the function f (x) is used to check the behavior of increasing and decreasing functions. The function is said to be increasing if the value of f (x) increases with an increase in …

We can test for concavity using the second derivative f00(x): Concavity Theorem: Let f(x) be a function. If f00(x) < 0 for all x 2 (a; b), then f(x) is concave down over (a; b). If f00(c) = 0 and …

In this section, we use the derivative to determine intervals on which a given function is increasing or decreasing. We will also determine the local extremes of the function.

Nov 16, 2022 · In this section we will discuss what the first derivative of a function can tell us about the graph of a function. The first derivative will allow us to identify the relative (or local) …

To put this in non-graphical terms, the first derivative tells us how whether a function is increasing or decreasing, and by how much it is increasing or decreasing. This information is reflected in …

By definition, a function f f is concave up if f ′ f ′ is increasing. From Corollary 3, we know that if f ′ f ′ is a differentiable function, then f ′ f ′ is increasing if its derivative f ′′(x)> 0 f ′ ′ (x)> 0.

Derivatives are used to describe the shapes of graphs of functions. x1 < x2, x1. derivative f′(x) = 0. A critical number, c, is one where f′(c) = 0 or f′(c) does not exist; a critical point is (c, f(c)).

Test the sign of f’(x) at an arbitrary number in each of the test intervals. Use the test for increasing and decreasing functions to decide whether f(x) is increasing or decreasing on each interval. …