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Sketch the graph of a function satisfying the following sign diagrams (Figure 28a and 28b).

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Missouri State University

Oregon State University

Baylor University

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

01:24

Sketch a function satisfyi…

01:02

Sketch the graph of a func…

01:03

'the graph of the fun…

01:09

For the final problem, we want to sketch the graph unsatisfied the sign diagrams. So what this is going to entail is looking at the graph of the function. And then considering um in this section we're considering con cavity in the second derivative. So it's important to note that when we take the second derivative, yes, it's like a derivative is greater than zero, then the graph is going to be concave up. And if the second derivative is less than zero and the graph is going to contact down okay. Using this knowledge, we can sketch the graph of the second derivative and it's going to relate to the slope of the tangent line or the slope of the um the slope of the derivative function.

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