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Fanciful shapes can be created by using the implicit plotting capabilities of computer algebra systems, (a) Graph the curve with equation$ y(y^2 - 1)( y -2) = x(x -1)(x -2) $At how many points does this curve have horizontal tangents? Estimate the $ x- $ coordinates of these points. (b) Find equations of the tangent lines at the points (0, 1) and (0, 2).(c) Find the exact r-coordinates of the points in part (a).(d) Create even more fanciful curves by modifying the equation in part (a).

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(a) $$x \approx 0.42265$$(b) $$y=-x+1 \text { and } y=\frac{1}{3} x+2$$(c) $$x=1 \pm \frac{1}{3} \sqrt{3}$$(d)

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

Derivatives

Differentiation

Missouri State University

Campbell University

Oregon 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.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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Fanciful shapes can be cre…

01:03

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04:21

Dynamically Constructing a…

so for this probably expanded sign should be easy to take. Impressive derivative. Then computers exploding off the point. Where's a horizontal tension? The tension lines horizontal, which means that the relative zero So that's something which is have the sob numerator. Zero. So three X Square minus six x plus cheery ho zero and we probably just fact arises. Actually, that's that would probably have to use a former so ex evos one over six six prosper minus square of sixties thirty six miners twenty four twelve. So that's one prosper on minus. Um, so that's one over three square root of three. So those are the point where you have horizontal tension line and for from the point zero one actually, I'm just answering posse People's Party. They asked you to ask me which I will leave you to you. Ah, for the to use a graphing calculator and the party are also leave it to you for part B. Ah, we just have to figure out a point and a slope you're planning an exit was zero. Wife wants one. We have slow because Shiyu over negative two, which is never if one. So the equation should be why minus one equals negative X. And for zero two, the slope, It's reply it. Actually, it was. You know why I was two, Saul. Numerous ways too. Denominator is, ah, eight times for his thirty two minors. Twenty five twenty four. So is eight minus four is four plus two is six. So Slope is one of our history. So the equation should be why minus two horse won over three X and and those are the equation of tension alive.

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