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Problem

Find the points on the lemniscate in Exercise 31 …

02:18

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Problem 42 Medium Difficulty

(a) The curve with equation
$ 2y^3 + y^2 - y^5 = x^4 - 2x^3 + x^2 $
has been likened to a bouncing wagon. Use a computer algebra system to graph this curve and discover why.
(b) At how many points does this curve have horizontal tangent lines? Find the $ x- $ coordinates of these points.


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Frank Lin

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Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

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Derivatives

Differentiation

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Derivatives - Intro

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.

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44:57

Differentiation Rules - Overview

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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Video Transcript

In this problem were given an equation and in part they were as to graph this equation, you could use any 1 function potter or even your calculators if you grab it. This is what the function looks like that is behaves in part. We were asked to find horizontal tensions, so basically we need to first differentiate this equation with respect to x, less to that we have 6 y squared times y prime plus 2 y times y prime minus 5 y to the prime is equal to 4 x. Cube minus 6 x, squared plus 2 x from this we see that then y prime is equal to 4 x cubed minus 6 x square of 2 x, divided by 6 y square plus 2 y minus 5 y poseur. Horizontal engines are pouring a 0, so we're going to say y prime to be 0. It means that numerator should be 0, so 4 x, cubed, minus 6 x, squared plus 2 x, should be 0, that is 2 x times 2 x, minus 1 times x. Minus 1 should be 0, so from this we see that x, coordinates of horizontal tangent are t x, 0, x, 1, half and x, 1.

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Top Calculus 1 / AB Educators
Grace He

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Catherine Ross

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Lectures

Video Thumbnail

04:40

Derivatives - Intro

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.

Video Thumbnail

44:57

Differentiation Rules - Overview

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.

Join Course
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