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The area of a square with side length $s$ is $A=s^{2}$. Consider a square whose side has been extended to length $s+\Delta s$.(a) Find the exact change in area.(b) Find the differential approximation to the increase in area.(c) Draw a picture of the original and the expanded square.(d) Give a geometric interpretation for the approximation and for the error.

(a) $2 s \Delta s+(\Delta s)^{2}$(b) $2 s \Delta s$(c) and(d)

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 6

Linearization and Differentials

Derivatives

Harvey Mudd College

Baylor University

University of Nottingham

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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The length of one side of …

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A square of side length s …

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A square of side length $s…

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Area The measurement of th…

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Expanding square The sides…

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The area A of a square dep…

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(a) Write the area A of an…

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The side of a square is me…

a square off side length s is inscribed in a circle. Off radios are in part A. We want to express the area A off the square. As a function off, the radios are off the circle. In party, we're going to find instantaneous rate of change of the area. A, With respect to the radios are party. We will evil wait. The rate of change of a at R equals one and r equals eight. Any party we're going to determine the unit of measure off the variation or the rate of change off the area. Respecter of radius if we measure the radius and inches and the area in square inches, So we start with bar A. Yeah, And for that, we're gonna make it some sketch here. Uh huh. Off the situation. So we have square or less, which is inscribed or a circle. Something like this, Right? And if this is the case, then we know that the center off the A circle is the Samos intersection of the diagonals of T square. Okay. And with that information, we know that the radios off the circle is just a half off one day ago. So we can't say that we draw another radius here. Yeah, we'll have. Here are right. Triangle where we have the hype. Opportunity is equal to two are And then we have decide of tea. Ah, square is s. So if we draw that, let me put the center here to identify like this. And if we draw the right triangle here separately, the situation is this We have S s and two are here. Ifthis red triangle on the right is the same as this one, which is Mm here in the inside the circle and which is determined by these two sides. This one here. He's one here. And the opportunity is the diameter the average er of T circle. So identifying this situation here, we can use a fight. Secker and serum. Yeah, yeah, yeah. To say that to our square, The high potential square is equal to the song off the squares off the legs of strength. That is in this case, two s square. That is four. Our square is to as square and as we want to write the area or express the area of the square as a function of our we know the area of square is s square. So we solved this last equation for us. Square right, Which is for over to that is to our square. Yeah. Uh huh. Okay. That is if we call The function are A, which is dependent on our is equal to two are square and this is expression of the area of the square in terms of the radius of the circle. Okay, so in party, you will find thes centennial's rate of change of the area of respect to art. There is a derivative off a respect to our at any radius R which is the same as a private are we know this is relative is four times are and, uh, this can be found if you want, by definition, the limit. When age goes to zero off a of R plus age minus. I have a of ours over age, and this is just derivative off a are Yeah, mhm. So deserve a tive at any radius. R is the limit When age goes to zero off to R plus R plus age square. We're using this oppression we obtained in part A When it's to our square over age. That's the limit when age goes to zero off to our square plus two our age plus age square minus two are square over age. This is the limit. When age goes to zero off Thio are square My plus four our age plus to age square managed to our square over h We canceled out his two terms and this is equal to the limit. When age goes to zero off or our age plus to age, square over age will take common factor age in the numerator and we get this is the limit. When age goes to zero off age times for are plus to age over age can cancel out age numerator denominator Because age goes to zero where is never equal to zero and this limit is the same as the limit off For our plus to age, the expression to age goes to zero when age goes to zero. So this is for our which is a constant in this case because the variable in this limited stage and so we get to four are which is the same we obtained here by using some derivative algebra. Okay, so in part C, we evil Lloyd that rate of change of the area with Specter is circle at one and at eight. So that one is simply four times one, which is four and at eight is to put our equals a in the formula. Got four times are there is four time eight is 32 and finally, Bardi, we're saying that we measured are in inches and a in square inches, right? So when we calculate is rate of change off the area, we respecter of radius. In the calculations, we do in the limit to calculate, we do the difference off the images off the functions in the Liberator and the independent. Variable in the denominator, that is, we are doing inches square inches, uh, in the numerator because we are using the function area, which is given in square inches and were divided by the units of the independent Variable R, which is inches, which is equal to inch square over inch, which is inch so the variations off the rate of change of the area. Respect to rate is got to be measured in inches. So did derivative of favors. Victor are must be man assured and inches if the area is measuring inch square inches. On the radio seem this measure in inches

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