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

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.5.84b

Chapter 2, Problem 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 $y = \cosh (x)$ .

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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