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The figure shows a circular arc of length $ s $ and $ a $ chord of length $ d, $ both subtended by a central angle $ \theta $. Find$ \displaystyle \lim_{\theta \to 0+} \frac {s}{d} $

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00:39

Frank Lin

02:16

Clarissa Noh

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 3

Derivatives of Trigonometric Functions

Derivatives

Differentiation

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

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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The figure shows a circula…

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$\begin{array}{c}{\text { …

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The figure shows circular …

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The accompanying figure sh…

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Circular Arcs Find the len…

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Find the length $s$ of the…

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Yeah. The figure shows a circular arc of like us and a quarter lengths. You want to find the limit as data approaches zero from the right of us divided by D. So this question is challenging the understanding of limits in particular system. An understanding of how to construct functions using things in the circles and then how to have a limit from there. So what this means is first we need to buy DNS has function the data if you want to be able to value the women above. So we can think of D. Using the data including so each of the two black legs in the figure. Either the or here are the radius of the circle are thus for data over to where this angle is bisected by the lines in particular, candy, we have that over to our side data over two. That's the equal to our idea that we know that this line bisects the angle and E. Over G. Giving us the original on either side simply because these two sides are equivalent length. Therefore these angles are equivalent. Therefore, we know the line connecting these 5 60 simply because of geometry principles with here. So now that we understand why how we constructed G equal to our side data over to we notice I think that some of the circumference of the circle is two pi R. As is simply the fractional, you know, over two pi times the circumference. Thus, C gives us equal our data. So from here we can put us over D. As data divided by two side data over to where the Rh equation cancel out one right? That's over D. That we have a clear given the black on the right. We see that getting towards the right hand limit data approaching zero for the right but the function approaches value want thus, from the graph, which is the limit, as data approaches zero for the right after everybody is equal to what we want. So

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