The hyperbolic cosine function, denoted cosh⁡(x)\cosh\left(x\right)cosh(x), is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as cosh⁡(x)=ex+e−x2\cosh\left(x\right)=\frac{$$e^{x}$+e^{-x}$$}{2}cosh(x)=2ex+e−x.b. Evaluate cosh⁡(0)\cosh\left(0\right). Use symmetry and part (a) to sketch a plausible graph for y=cosh⁡(x)y=\cosh\left(x\right).

Ch. 2 - Limits

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Briggs 3rd Edition

Ch. 2 - Limits

문제 2.5.84b

2장, 문제 2.5.84b

The hyperbolic cosine function, denoted $$\cosh$$\left$(x$\right$)$ , is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as $$\cosh$$\left$(x$\right$)=$\frac{e^{x}$+e^{-x}}{2}$ .

b. Evaluate $hyperbolic cosine of 0$ . Use symmetry and part (a) to sketch a plausible graph for $y = \cosh (x)$ .

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To evaluate $ \cosh(0) $, substitute $ x = 0 $ into the definition of the hyperbolic cosine function: $ \cosh(x) = \frac{e^x + e^{-x}}{2} $.

Calculate $ e^0 $ and $ e^{-0} $. Since any number to the power of 0 is 1, both $ e^0 $ and $ e^{-0} $ equal 1.

Substitute these values into the expression: $ \cosh(0) = \frac{1 + 1}{2} $.

To sketch the graph of $ y = \cosh(x) $, note that $ \cosh(x) $ is an even function, meaning it is symmetric about the y-axis. This is because $ \cosh(-x) = \cosh(x) $.

The graph of $ y = \cosh(x) $ resembles a U-shape, similar to a parabola, but it is not a parabola. It has a minimum value at $ x = 0 $ and increases exponentially as $ x $ moves away from zero in both directions.

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주요 개념

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Hyperbolic Functions

Hyperbolic functions, such as hyperbolic cosine (cosh), are analogs of trigonometric functions but for a hyperbola instead of a circle. They are defined using exponential functions, with cosh(x) = (e^x + e^(-x))/2. These functions are useful in various applications, including modeling shapes like hanging cables and in solving certain differential equations.

Symmetry in Functions

Symmetry in functions refers to the property where a function exhibits a certain balance around a point or axis. For example, the hyperbolic cosine function is even, meaning cosh(-x) = cosh(x). This symmetry can be used to simplify calculations and sketch graphs, as it indicates that the graph will mirror itself across the y-axis.

Graphing Techniques

Graphing techniques involve methods to visually represent mathematical functions. For the hyperbolic cosine function, understanding its key points, such as cosh(0) = 1, and its symmetry helps in sketching its graph accurately. Recognizing the shape of the graph, which resembles a parabola opening upwards, is essential for visualizing the behavior of the function across different values of x.