Find the critical numbers of the function. g(x)=√(1-x^2) |...

Name: Problem 38, Chapter 4: Calculus Early Transcendentals
Uploaded: 2020-05-14T05:12:48-08:00
Duration: 5 min 27 s
Channel: Swetha Mohandas
Description: Find the critical numbers of the function. g(x)=√(1-x^2)

Key Concepts

Chain Rule

The chain rule is a fundamental technique in differentiation used to compute the derivative of composite functions. It allows one to differentiate a function that is applied to another function by multiplying the derivative of the outer function evaluated at the inner function by the derivative of the inner function.

Critical Numbers

Critical numbers are values in the domain of a function where its derivative is either zero or does not exist. These points are important in determining local extrema and understanding the behavior of functions, as they indicate potential maximums, minimums, or points of inflection.

Domain of a Function

The domain of a function is the complete set of input values for which the function is defined. When finding critical numbers, it is essential to consider the domain, as points outside of it, even if they yield an undefined derivative, are not considered critical numbers.

Differentiation

Differentiation is the process of computing the derivative of a function, which measures the rate at which the function's value changes with respect to changes in its input. The derivative is key in identifying where functions increase, decrease, or have horizontal tangents, which in turn helps locate critical numbers.

Transcript

00:01 So for this problem, we're going to find the critical points of this equation, square root of 1 minus x squared.

00:08 So to find the critical point, you have to find the derivative of this equation and find out when the derivative equals zero.

00:17 So, for example, the critical point would be the derivative of, so c equals zero.

00:27 So first, so first to find the derivative, we're going to have to do change.

00:34 Rule.

00:36 So to start that out, we're going to have to, i'm going to rewrite this to make it a little bit easier to see.

00:52 So we're going to do it to the one half.

00:59 So when we're doing this, first step in the chain rule, we bring the power down.

01:05 So it'd be one half.

01:09 And then we subtract it by 1 the power so it would be 1 minus x squared and 1 half minus 1 is negative 1 half and then the last step of the chain rule is to find the derivative of what's inside the parentheses and then multiply it by that so we multiply this whole thing by negative 2x so this is the first step of our chain rules now we just have to simplify the equation.

01:48 So as we can see, we can cancel the twos, and that leaves us with a negative x.

01:57 And since this is to the negative one -half, we can move this to the denominator, and we can do the square root of 1 minus x squared.

02:12 So now this is the derivative of our original function and now we're trying to find out when equals zero.

02:21 So now when we're looking at this, we can look at the beginning, we can look at the numerator, and when does negative x equals zero? well, this one's easy.

02:33 So negative x equals zero when x is zero...

Please give Ace some feedback