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The figure shows a lamp located three units to the right of the y-axis and a shadow created by the elliptical region $ x^2 + 4y^2 \le 5. $ If the point (-5, 0) is on the edge of the shadow, how far above far above the x-axis is the lamp located?
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Calculus 1 / AB
Harvey Mudd College
University of Michigan - Ann Arbor
Idaho State University
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.
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. We're giving it a lips and were given to tension lines. And we know that there is a life and it is located. Um, so this is between the Y Axis and Len is three, and we're supplying the hide off slim. So basically, the game plan here is to find the question off this tension line so that we could block the actually unity. Give us divide Valley. So we're trying to find the question of attention. All right. No vino de question of Philip's thatis ex scraped off your wife's crazy five. Let's take derivatives to find the slope of times when we have two eggs waas eight white on white frenzy zero. From this we see that why prime physical threats or eight while, and that is laid eggs over or what? Now we drive this equation using the question of the ellipse, and we do not know. Actually, we do know that this given point Elsner passes true these, um, alums. So we first need to find this point that is right on the lips. And that is also on this change in lying. Um, you know that any question off the line Bobbie or my minus or not is evil, Times explains. That's not, let's say that. Why money this zero is why not? And negative five are excellent for a woman. Zero is equal And where for them We're just gonna leave that ISS so negative x or y times X What's fine? Promise if you multiply both sides but for why we got four wives scraped to be eclipsed Indicative of expert My ex wife ex. So that is or wants Craig plus expert is ignited by thanks we got left aside we have or westward plus xx And look at the question off. Um, an ellipse be known that then the left inside should be five negative five thanks is equal five sort of x will be going to war. All right, we know that X is negative one. So let's just play that in into the question of the ellipse. Be a beehive. Negative one spread plus four wives heard He's a good fire s 04 wives. Credit should be. Or it means that what spread will be warm. So it means that why could be was for writers one. We are interested in this point, right here. Now we know that X Gordon off. It is NATO one, and we can see that. Why should be on a positive side that will see that from this? People want to be one. So this point right here is negative or and were all right. Now, since we know X and y quarters, we can talk like slopes of slope as they get to explore. Or why we said that will be negative. Negative one before. So that'll be one Ford. So then the question of why is it good for Times X? What's five? I am using this one here. Since we know the question of this line, all we need to do is to plug this given in and out quick. Why? So why will be won over four times? Three plus fire. That is eight or four. And that is to so they hide up. This lamb is too
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