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(a) Given $f(x)=x^{2}-4,$ and $g(x)=\left(x^{2}-4\right)^{10} .$ Find the points of horizontal tangency for $f(x)$ and $g(x) .$ (b) Repeat for $f(x)=x^{2}-3 x+2$ and $g(x)=\left(x^{2}-3 x+2\right)^{15} .(\mathrm{c})$ What can you deduce about the relationship of the zeroes of $f(x)$ and $g(x)$ and the zeroes of their derivatives? (d) In general, if $f(x)$ is any function, what can you say about the points of horizontal tangency of $f(x)$ and $[f(x)]^{N} ?$

(a) (0,-4)$;\left(0,4^{10}\right),(\pm 2,0)$(b) $(3 / 2,-1 / 4) ;\left(3 / 2,-1 / 4^{15}\right),(1,0),(2,0)$(c) $[f(x)]^{N}$ has horizontal tangents at the zeros of $f$ and at the horizontal tangents of $f$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Missouri State University

University of Nottingham

Idaho State University

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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for the following problem, we want to take our function F of X equals x squared minus four. And then GFX which is expert -4 to the 10th. And we want to find the points of horizontal tangent C for fx and gx. And what this is going to look like is taking the derivative. So if we take the derivative of this will end up getting two X. So we know that the point of horizontal can agency will be when X equals zero right here. Which makes sense here. On the other hand, what we can do is we're gonna have to do the chain rule. So it's gonna be 10 times X squared minus four the ninth and that can be times two X. So we find where this equals zero. Um we see that there's going to be a spot of horizontal tangent C at X equals zero. But then there's also going to be one plus two and minus two which is expected from this term right here.

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