How does the graph of the catenary y = a cosh x/a change as a \u003e 0 increases?

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

문제 7.3.10

7장, 문제 7.3.10

How does the graph of the catenary y = a cosh x/a change as a > 0 increases?

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Recall the definition of the catenary function: \(y = a \cosh\left(\frac{x}{a}\right)\), where \(a > 0\) is a parameter that affects the shape of the curve.

Understand that \(\cosh(z)\) is the hyperbolic cosine function, which is always positive and symmetric about the \(y\)-axis, with a minimum value of 1 at \(z=0\).

Analyze how changing \(a\) affects the graph: since \(a\) appears both as a vertical scaling factor and inside the argument of \(\cosh\), increasing \(a\) will stretch the graph vertically and horizontally.

More specifically, as \(a\) increases, the curve becomes wider (flatter) because the argument \(\frac{x}{a}\) changes more slowly with \(x\), and the minimum value at \(x=0\) increases proportionally to \(a\).

Summarize that increasing \(a\) results in a catenary that is less steep near the vertex and has a higher minimum point, making the curve appear broader and taller.

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Definition and Properties of the Catenary

The catenary is the curve formed by a hanging flexible chain or cable under its own weight, described by the equation y = a cosh(x/a). It is distinct from a parabola and characterized by its hyperbolic cosine shape, symmetric about the y-axis, with the parameter 'a' controlling its steepness and width.

Effect of the Parameter 'a' on the Graph

In the equation y = a cosh(x/a), the parameter 'a' scales both the vertical and horizontal dimensions of the curve. As 'a' increases, the catenary becomes wider and flatter near the vertex, meaning the curve stretches horizontally and the minimum point rises, reflecting a less steep shape.

Hyperbolic Cosine Function Behavior

The hyperbolic cosine function, cosh(t), grows exponentially for large |t| and has a minimum value of 1 at t = 0. Understanding cosh's shape helps explain the catenary's form: it is always positive, symmetric, and smooth, with the parameter 'a' affecting the input scaling, thus altering the curve's width and height.

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Inverse identity Show that cosh⁻¹(cosh x) = |x| by using the formula cosh⁻¹ t = ln (t + √(t² – 1)) and considering the cases x ≥ 0 and x < 0.

108

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Evaluate the following derivatives.

d/dx ((1/x)ˣ)

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Catenary arch The portion of the curve y =17/15 - cosh x that lies above the x-axis forms a catenary arch. Find the average height of the arch above the x-axis.

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15–20. Designing exponential growth functions Complete the following steps for the given situation.

a. Find the rate constant k and use it to devise an exponential growth function that fits the given data.

b. Answer the accompanying question.

Population The population of Clark County, Nevada, was about 2.115 million in 2015. Assuming an annual growth rate of 1.5%/yr, what will the county population be in 2025?

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22–36. Derivatives Find the derivatives of the following functions.

f(x) = cosh²x

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Haitz’s law predicts that the cost per lumen of an LED bulb decreases by a factor of 10 every 10 years. This means that 10 years from now, the cost of an LED bulb will be 1/10 of its current cost. Predict the cost of a particular LED bulb in 2021 if it costs 4 dollars in 2018.

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