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A flexible cable always hangs in the shape of a centenary $ y = c + a \cosh (x/a), $ where $ c $ and $ a $ are constants and $ a > 0 $ (see Figure 4 and Exercise 52.) Graph several members of the family of functions $ y = a \cosh (x/a). $ How does the graph change as $ a $ varies?

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00:17

Amrita Bhasin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 11

Hyperbolic Functions

Derivatives

Differentiation

Missouri State University

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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All right, So let's go through this question. It basically is asking you if I have the expression. A Times coach off X divided by A as a changes. What kind of pattern do you see in the graph? Easy. We're just going to drop it and see what happens. Mhm. So first I am going to simply start with coach. Coach of X is a glass that looks like this. We know that it's an even function. It's a mixture off E to the X and E to the negative X, and it's split it in half. So that's why we know that it's It's a very nice curve that kind of looks like a parabola, and it's also usually modeled as things like wires handing. Uh huh. Let's see what happens when y is equal to to coach off X development. As you can see, the pattern is that now the height is instead of at one. It's now starting at two. It's being stretched out horizontally a little bit more. Let's see what happens if it's equal to five. Why equals two five? Coach off X of a five can see that now. The height is a little bit higher. It starts at five. It's being stretched out even farther. So long Story short, this pattern continues. If why equals a who's off X over a anything on the value of A, you're gonna be able to see what's gonna happen. So if it goes like this, when a is a fraction, it becomes very small. When a is a large large number, it just gets taller and taller and it gets stretched out horizontally. So I want you to pay attention to my cursor if you imagine that this is a point that I can pull to the left and to the right. That's basically what happens with wires. If you pull the wire sideways far apart from the center, the graph is going to be stretched out even more like this, and the very bottom part of the wire starts getting taller and taller. Right? So this is one of the reasons why this graph is very useful for representing how wires case okay? And that answers the question of how a coach X divided by a behaves

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