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Problem 23 Easy Difficulty

Use Formula 11 to find the curvature.
$$y=x^{4}$$

Answer

$$
\frac{12 x^{2}}{\left[1+16 x^{6}\right]^{3 / 2}}
$$

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Video Transcript

all right. In this problem, we want to use Formula 11 which I have in the top right corner of the screen to find the curvature Kappa of why equals X to the fourth power. So we're gonna use the formula in the top right corner of the screen. So we know that we need both our first and second derivative of the function to be able to do so. So I'm gonna take the first derivative of Why So why prime, which is gonna be four x to the third power. Then I'm gonna take second derivative. Why? Double prime, which is equal, Teoh 12 x squared too. Um, then I can look at Kappa of X in the formula as the absolute value of the second derivative, which is going to be 12 x squared. It's all over one plus the first derivative squared. So four x cubed squared that is raised to the three halfs power so we can simplify this just a little bit. Uh, in the numerator, X squared is always going to be a positive number. So 12 x squared will always be positive so we can drop the absolute value signs and then in our denominator, we can simplify things just a little bit, Um, and write it as one plus for X Cubed Quantity Squared, which ends up being 16 x to the sixth Power and that is raised to the three halfs power. And that is the curvature Kappa of the function y equals X to the fourth power.

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