What is the significance of the second derivative

We can say that f is increasing or decreasing at an increasing rate. We can say that f is increasing or decreasing at a decreasing rate. At each of these times the velocity is positive and the particle is moving upward, increasing in height.

An inflection point is a point on the graph of a function where the concavity of the function changes from concave up to down or from concave down to up. The concavity changes at points b and g. At points a and h, the graph is concave up on either side, so the concavity does not change. At points c and f, the graph is concave down on either side. And at point e, even though the graph looks strange there, the graph is concave down on both sides — the concavity does not change.

Inflection points happen when the concavity changes. Because we know the connection between the concavity of a function and the sign of its second derivative, we can use this to find inflection points. An inflection point is a point on the graph where the second derivative changes sign. In order for the second derivative to change signs, it must either be zero or be undefined.

This is of course not a straight line, but rather a parabola. This, I think you will agree is a straight tangent line. In this sense, the derivative is not a tangent line, but rather a function to generate the tangent line at each point along a curve.

Similarly, one can think of the second derivative as the function which generates the rate of change of the first derivative at every point along the function. As stated above, if the second derivative is positive, it implies that the derivative, or slope is increasing, while if it is negative, implies that the slope is decreasing.

We can then read off the slope at any point along the original curve. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.

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The simplest physical example is probably distance, speed and acceleration Garf Garf 1, 1 1 gold badge 6 6 silver badges 21 21 bronze badges. Featured on Meta. Now live: A fully responsive profile. Linked Related