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Use the first and second derivatives to sketch the graph of the given equation. Also include the intercepts, whenever they are easily determined.$$f(x)=x^{2}-5 x+6$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Campbell University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

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Use the first and second d…

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we want to graph alphabet X equals X squared minus five X plus six. To do so we're going to deploy calculus clear sketching techniques. So we're gonna analyze F. F prime F double prime. First from F we get basic information about our graph. So from F we obtain information related to intercept and such. So that gives us domain all X, Y intercept X equals two and three. There are no assistance from F prime. We get information on where F is increasing or decreasing. So F prime is two X minus five. There are particular critical points at X equals 5/2. So to last five over to F prime is negative, decreasing to the right, prime is positive, increasing. So then we analyze the double prime. Ethical prime is too there are no petitioner critical points. So for all acceptable prime two is positive. Therefore it's concave up everywhere. Thus we can graph, we have our intercept market two or three Increasing rather decreasing, increasing until 5/2 and then concave up everywhere.

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