Question

61–62. Points of intersection and areaa. Sketch the graphs of the functions f and g and find the x-coordinate of the points at which they intersect.f(x) = sech x, g(x) = tanh x; the region bounded by the graphs of f, g, and the y-axis

Accepted Answer

First, recall the definitions of the hyperbolic functions involved: $$f(x) = \operatorname{sech} x = \frac{1}{\cosh x}$$ and $$g(x) = 	anh x = \frac{\sinh x}{\cosh x}. $$Understanding their shapes will help in sketching the graphs. Sketch the graph of $$f(x) = \operatorname{sech} x$$, which is an even function with a maximum at $$x=0$$ where $$f(0) = 1$$, and it approaches 0 as $$x 	o \pm \infty. $$Then sketch $$g(x) = 	anh x$$, an odd function that passes through the origin, increasing from $$-1 to$$ $$1 as$$ $$x$$ goes from $$-\infty to$$ $$\infty. To $$find the points of intersection, set $$f(x) = g(x)$$, which means solving the equation $$\operatorname{sech} x = 	anh x. $$Rewrite this as $$\frac{1}{\cosh x} = \frac{\sinh x}{\cosh x}$$, and simplify to find the values of $$x$$ where this holds true. Simplify the equation to $$1 = \sinh x. $$Solve for $$x by $$taking the inverse hyperbolic sine: $$x = \$$sinh^{-1}$$(1). $$This gives the x-coordinate of the intersection point(s). To find the area bounded by the graphs of $$f$$, $$g$$, and the y-axis, identify the interval of integration from $$x=0 ($$the y-axis) to the intersection point found. Set up the integral of the difference between the upper and lower functions over this interval: $$	ext{Area} = \$$int_0$$^{x_{intersection}} (f(x) - g(x)) \, dx. $$This integral will give the bounded area.

61–62. 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.

f(x) = sech x, g(x) = tanh x; the region bounded by the graphs of f, g, and the y-axis

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Key Concepts

Hyperbolic Functions

Points of Intersection

Area Between Curves