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Find the dimensions of the largest rectangle with lower base on the $x$ -axis and upper vertices on the curve whose equation is (a) $x^{2}+y^{2}=4$(b) $9 x^{2}+4 y^{2}=36$.

(a) $2 \sqrt{2} \times \sqrt{2}$(b) $2 \sqrt{2} \times \frac{3 \sqrt{2}}{2}$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 4

Applications I - Geometric Optimization Problems

Derivatives

Harvey Mudd College

Baylor University

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.

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Find the largest possible …

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Largest Rectangle A rectan…

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A rectangle with sides par…

uh well we have a rectangle inscribed in the hemisphere here for hemispherical. And so this point here is on the circle. And so at this point and I want to find the point on the circle um you know X. And Y. That maximizes the area of this rectangle. So we know X squared plus Y squared. And since I asked us to use to find um used two different radio here, I was just going to leave it as our square. And so then a the area of it here is this is two X. And this is why so we get to xy solving for X. Plugging it into here. We get really we only need the positive sign because we're gonna it's gonna be included. Um We could get this solution to, but then that's going to give us um the same answer. So we get this take the derivative respect to Y said why you gotta wipe cry Y one and uh my ones in here and then set that equal to zero. And we clearly see here from here that why why won the optimal solution for why is plus or minus? R over a squared to Well, we only need the plus because we're only looking at the upper half here. So it's this are over square too. And so we plug that in to um do our area here. We find that in fact the area is just our square. So whatever this is is going to be our area. And so in the first case they give us the radio um a radius of two. So this is four star areas, for in the second case they give us uh radius of six. So, you know, this was 36, and so this is going to be 36. And so that's always generally case the area of this rectangle is this radius squared? Not sure. I'm trying to think of a geometric interpretation for this. You know, these triangles here. Um I'm not sure exactly, it's probably some way of interpreting this and showing why geometrically that should be.

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