Question

Points of intersection and area. a. Sketch the graphs of the functions \(f\) and \(g\) and find the \(x\) -coordinate of the points at which they intersect. b. Compute the area of the region described. \(f(x)=\sinh x, g(x)=\tanh x ;\) the region bounded by the graphs of \(f, g,\) and \(x=\ln 3\)

Short answer

Question: Determine the area of the region bounded by the graphs of the functions \(f(x) = \sinh x\) and \(g(x) = \tanh x\) between \(x = 0\) and \(x = \ln 3\). Answer: The area of the region bounded by the graphs of the functions \(f(x) = \sinh x\) and \(g(x) = \tanh x\) between \(x = 0\) and \(x = \ln 3\) is approximately 1.52316.

Step-by-step solution

Sketch the Graphs of the Functions f and g

The first step is to have an idea of how the functions look like. The function \(f(x) = \sinh x\) is the hyperbolic sine function, and \(g(x) = \tanh x\) is the hyperbolic tangent function. It's important to notice that as the value of \(x\) increases, both functions tend to increase too. In order to obtain a visual representation, we highly recommend to use any graphing software to visualize the two functions. Knowing the overall behavior of thefunctions will help to visualize the region segment we have to find the area.

Find the Points of Intersection

Now, in order to find the points of intersection between the two graphs, we must find the x-coordinates when \(f(x) = g(x)\). Set \(\sinh x = \tanh x\) and simplify the expressions: \(\frac{e^x - e^{-x}}{2} = \frac{e^x - e^{-x}}{e^x + e^{-x}}\) Now divide both numerator and denominator by \(e^x\): \(\frac{1 - e^{-2x}}{2} = \frac{1 - e^{-2x}}{1 + e^{-2x}}\) Now, we want to find the x-coordinates: \(\frac{1 - e^{-2x}}{1 + e^{-2x}} = \frac{1 - e^{-2x}}{2}\) Cross multiply to get: \(2(1 - e^{-2x}) = (1 - e^{-2x})(1 + e^{-2x})\) Divide both sides by (1 - \(e^{-2x}\)): \(2 = 1 + e^{-2x}\) Subtract 1 from both sides: \(1 = e^{-2x}\) Taking the natural logarithm of both sides, we get: \(\ln 1 = -2x\) Rearranging, we have: \(x = 0\) So the two functions intersect at \(x = 0\).

Compute the Area of the Region

Now that we have determined the points of intersection to be at \(x = 0\), we can move on to find the area of the region bounded by the graphs of \(f, g\), and \(x = \ln 3\). To find the area, we need to find the integral of the difference between the two functions over the interval from 0 to \(\ln 3\): \(\int_{0}^{\ln 3} (\sinh x - \tanh x) dx\) We know that the integrals of \(\sinh x\) and \(\tanh x\) are \(\cosh x\) and \(\ln(\cosh x)\), respectively. So, we get: \([\cosh x - \ln(\cosh x)]_{0}^{\ln 3}\) Now, evaluate the expression at the limits and subtract: \(\left[\cosh (\ln 3) - \ln(\cosh (\ln 3))\right] - \left[\cosh (0) - \ln(\cosh(0))\right]\) Since \(\cosh (0) = 1\) and \(\ln 1 = 0\), the expression simplifies to: \(\cosh (\ln 3) - \ln(\cosh (\ln 3)) - 1\) Computing the values leads to the result: Area \(= \approx 1.52316.\)

Key concepts

Hyperbolic Functions

Hyperbolic functions are an important set of functions in mathematics, analogous to the trigonometric functions, but for a hyperbola instead of a circle. Where trigonometric functions like sine and cosine are defined based on the unit circle, hyperbolic functions such as hyperbolic sine \textbf{(sinh)} and hyperbolic tangent \textbf{(tanh)} are defined in terms of the unit hyperbola.

Specifically, for a real number x, the hyperbolic sine function is given by the formula \(\sinh x = \frac{e^x - e^{-x}}{2}\) where e is the base of the natural logarithm, approximately equal to 2.71828. The hyperbolic tangent function is defined as \(\tanh x = \frac{\sinh x}{\cosh x}\) where \(\cosh x = \frac{e^x + e^{-x}}{2}\) is the hyperbolic cosine function. In the context of the given exercise, understanding the shape and properties of hyperbolic functions is essential to visualizing the area that we aim to calculate.

These functions have unique features. The function \(\sinh x\) is odd, meaning it reflects symmetrically across the origin, much like the ordinary sine function, but it increases exponentially instead of oscillating. Meanwhile, \(\tanh x\), a scaled version of \(\sinh x\), grows rapidly near the origin and approaches one as x tends to infinity, reflecting its nature as a ratio of hyperbolic sine and cosine.

Intersection of Graphs

When two or more graphs share a common point on the coordinate plane, we say they intersect. The x-coordinates of these shared points can often uncover a deep connection between the involved functions or help specify a region in the plane.

To find the intersection of graphs mathematically, we set the equations of the functions equal to each other and solve for x. This often involves algebraic manipulation and might require solving systems of equations for more complex functions. In the context of hyperbolic functions, such as \(f(x) = \sinh x\) and \(g(x) = \tanh x\), finding their points of intersection involves equating the two functions and employing properties of exponents and logarithms to solve for the values of x that satisfy both equations.

In our exercise, the intersection occurs at x = 0, a result obtained by equating the functions and simplifying the resulting equation. It is worth noting that graphing calculators or software are immensely helpful in providing a visual confirmation of the intersection points obtained algebraically. Recognizing and calculating the intersection points is pivotal for determining the bounds of integration when computing the area between curves.

Definite Integral

The concept of the definite integral is central to calculus. It represents the accumulation of quantities and can be used to calculate the area under a curve, among many other applications. When dealing with the area between two curves, the definite integral measures the accumulated difference of the two functions across a specific interval.

To compute the area between the curves of the function \(f\) and another function \(g\) over an interval \([a, b]\), we would evaluate a definite integral of the form: \(\int_{a}^{b} (f(x) - g(x)) \, dx\) where the integrand \(f(x) - g(x)\) represents the vertical distance between the two functions at any point x within the interval. We then integrate this distance across the interval to find the total area.

In our exercise, the definite integral is used to determine the area between the graphs of the hyperbolic sine and hyperbolic tangent functions from x = 0 to \(x = \ln 3\). Evaluating the integral, we substitute the limits of integration and subtract the resulting values to get the required area. This calculation makes it clear how hyperbolic functions integrate and how their integration helps us solve practical problems involving the accumulation of differences between curves, a particularly useful notion in multiple areas including physics, engineering, and economics.