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Using principles from physics it can be shown that when a cable is hung between two poles, it takes the shape of a curve $ y = f(x) $ that satisfies the differential equation$ \frac {d^2 y}{dx^2} = \frac {pg}{T} \sqrt {1 + (\frac {dy}{dx})^2} $where $ p $ is the linear density of the cable, $ g $ is the acceleration due to gravity, $ T $ is the tension in the cable at its lowest point, and the coordinates system is chosen appropriately. Verify that the function $ y = f(x) = \frac {T}{pg} \cosh (\frac {pgx}{T}) $is a solution of this differential equation.

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Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 11

Hyperbolic Functions

Derivatives

Differentiation

Campbell University

Oregon State University

Baylor University

University of Nottingham

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.

03:41

Using principles from phys…

01:33

02:00

01:47

02:17

To derive the equation of …

02:05

For the following exercise…

Why prime we know is gonna be t over. Sign age. Just remember that you're doing the chain rule. You just remember to cancel out what you can. We have signed h of Nation Axe T now the second derivative is gonna be co sign age times PG over tea times PG axe over tea. Therefore, remember that we know the identity co signed h scored Axe is one plus sign each scored X. Therefore we can write. This is PG over tea Cosan Age of PG acts over tea is PG over Tea co signed h PG Axe over Chief

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