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Let $y=x^{3}-4 x$. (a) Find the equation of the line tangent to this curve at $x=1 .$ (b) Find the $x$ -intercept of the tangent line, that is, the point where the line crosses the $x$ -axis. (c) Write the expression $x^{3}-4 x$ in factored form and find the roots of $x^{3}-4 x=0 .$ Do you see any relationship between these roots and the intercept of the tangent line? [See Exercise 24 below.]

(a) $y=-x-2$(b) $x=-2$$(c) x(x-2)(x+2)$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

Derivatives

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University of Nottingham

Boston College

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.

08:46

Consider the equation $x^{…

04:59

Suppose the curve $y=x^{4}…

01:25

(a) Find the slope of the …

04:46

Suppose the curve $ y = x^…

05:01

03:42

Let $f(x)=\frac{1}{4} x^{4…

All right, so this is kind of a long problem, so we'll just jump right into, uh, in every part of it's a little bit different. So the first part is finding the tangent line at X equals one. Well, if you want to Tanja line first thing is, you better find the point, Which is when X is one will plug in One cubed is one minus four times. Once one minus four is negative. Three. And then you need the slope or the slope is finding, uh, you know why Prime tolerate it this way. Do y dx when X equals one? Well, do you? Why the X in this problem is equal to three X squared minus four. Well, if I plug in one here, I get three minus four, which is negative one. So the equation of the change of line will be Why equals the slope and then X minus the excrement, plus the like, would it? Now, if you distribute that in there, you'll get y equals negative. X B plus one minus three. So negative, too. So this is the first answer. The part a the tangent line and part B. Where does that cross the X axis well across the X axis when y zero I mean, equals zero right there. Eso Pretty straightforward. If I just add to start, add X to the left side. Zero plus X is X equals two. That's your ex intercept. So that's your answer. Department. Be your ex intercept is too well in part C. They want you to write the expression factored forms back to the original. Uh, this in factored Formby X times X squared minus four. That's greatest common factor. And then this is a difference of squares made for part C down here. All right. It as Ex X minus two. That's plus two in factored form. The reason why we like to write an effective form eyes because, um factored form tells you your ex intercepts. So the X intercept in this problem R 02 and negative, too. Um, and what's unique about that is that the X intercept in part B for the change of line is equal to one of the X intercept in the problem. This one specifically No, not that one. I put your that one. I should have been negative to. I had that earlier. Sorry about that. Eso specifically this one? Eso kind of interesting in this problem. So a B c and, uh, yeah, last part is just more of Ah, do you notice anything? But I'll leave that up to you.

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