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A cubic equation of the form $y=a x^{3}+b x^{2}+c x+d$ having three distinct roots ( $x$ -intercepts) $x_{1}, x_{2}, x_{3},$ may be written as $y=a\left(x-x_{1}\right)$$\left(x-x_{2}\right)\left(x-x_{3}\right)$ (why?). Show that the tangent line to the curve at$\left(x_{0}, y_{0}\right),$ where $\left(x_{0}, y_{0}\right)$ is midway between two successive roots, has the third root as its $x$ -intercept.

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

Derivatives

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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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(a) The curve with equatio…

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You are given a polynomial…

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Cubic curves What can you …

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Show that the cubic polyno…

we have a cubic equation that's given in the form Y equals x cubed plus bx squared plus six plus D. Having three distinct routes. It can be written um as Y equals eight times Ackman X one Mm X -X two. Oh I'm X -X three. And the reason why is because those are the three values that make The function equal to zero. So if we know what makes the function zero, that means that when we plug in for X, for example, if we plug in X three, we know this whole thing goes to zero. Or if we plug in X two, we know this whole thing goes to zero. So that's where that ultimately comes from. We can also see that this is the case when we um take the derivative and look at the tangent line to the curve at why not? Where X not why not is between two successive roots. And we'll have the third route as the X intercept.

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