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The equation $ x^2 - xy + y^2 = 3 $ represents a "rotated ellipse," that is, an ellipse whose axes are not parallel to the coordinates axes. Find the points at which this ellipse crosses the $ a-axis $ and show that the tangent lines at these points are parallel.

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01:34

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

Derivatives

Differentiation

Mtro. C.

July 29, 2021

Your company produces certain gadgets at price of $5 each. Market research indicates that if those gadgets are sold at $15 each, there will be an averange of 200 sales a month, but each dollar increase (or decrease) in the price, there would be 8 fewer (o

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

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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In this problem were given an impression of an ellipse, we are first asked to show the points that this ellipse causes the x axis. So we are looking for x, intercepts x, intercepts mean that y should be 0 and if boys 0, if you plug 0 in you, have x, squared minus 0 plus 0 is equal to 3. It means that x square should be 3, so x would be positive. Negative plus minus root 3 and if that is the extent x intercepts, the coordinate them will be negative square root of 30 and positive square root of 3 and 0. Now, let's find the tangent length so to do that. We'Re going to take derivative of this art em curve with respect to x, we have 2 x minus product for the second term. We have y minus x times y prime plus 2 y times y prime is equal to 0 point from the west to y prime. Is equal to y minus 2 x, divided by 2 y minus x, and if we will get this derivative at the x intercepts, we have y prime, it will a negative to 3 and 0 and y prime. It will at that positive sure, roof. 3 and 0. Point if we look at negative square root of 3, we find derivative to be 2 square root of 3 square root of 3, so that is 2 and you will act at y prime square root of 3. We have negative 2 square root of 3, divided by negative square root of 3 point, so the answer is again 2. It means that if those students escept tangent lines are parallel, let's see that usually this is the ellipse that we would get with this function. This is the first tangent line at first x, intercept, and this is the second tangent line. These are onto intercessor root of 3 and 0 negative scrout of 3 and 0 point, and since we found the slopes to be equal, we know that those 2 are parallel.

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