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Determine where the function is concave upward and downward, and list all inflection points.$$f(x)=\sqrt{9-x^{2}}$$

$$\mathrm{CD} \text { on }-3<x<3$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Missouri 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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Determine where the functi…

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mhm. Question 35 is asking you to determine where the function F of X equals square root of nine minus X squared is concave up or down and list all inflection points. So to do so, you need to get to the second derivative of F of X. So taking first derivative F prime of X would just be one half nine minus x squared to the negative one half, Um, and then using the chain rule multiplied by negative two X that can be simplified, then to negative X over nine minus x squared to the one half. Now using the question rule to get the second derivative F double prime affects is going to be equal to negative nine minus X squared to the one half minus one half times nine minus x squared to the negative, one half multiplied by negative two x because the quotient rule and then negative x all divided by nine minus X squared to the one half squared. Now I'm starting to simplify this. You have negative nine minus X square to the one half minus X squared times nine minus X squared to the negative one half divided by nine minus X squared From here, you can now divide each of these by the denominator separately, so you have F Still prime of X is equal to negative nine minus X squared to the negative one half minus X squared, divided by nine minus X squared. Do the 3/2. This is just equivalent to negative one divided by nine minus X squared to the one half. Therefore, if we multiply this side by nine minus X squared over nine minus X squared, you would have a common denominator of nine minus X squared to the 3/2. So this becomes f prime of X is equal to negative nine minus X squared minus X squared all over nine minus X squared to the 3/2. And lastly, you have nine negative nine divided by nine minus X squared to the 3/2. So this function now looking at it, you can find an inflection point when F double prime of X is equal to zero. This function will not ever be equal to zero, but it is undefined when X is greater than three. Well, when X is less than negative three. Therefore, we are looking at interval from negative three 23 where this function is defined So we can test con cavity with F double prime of X by testing a number within this range. So taking f double prime of zero is negative. 0.333 That means we are concave down, um, everywhere within this interval. Negative three, 23 and that's your answer to question 35.

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