Multiple Choice
Evaluate the line integral of the vector field along the curve given by the vector function , where goes from to .
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0. Functions
Introduction to Functions
5:38 minutesProblem 7.3.61a
Textbook Question
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
Verified step by step guidance1
First, recall the definitions of the hyperbolic functions involved: \(f(x) = \operatorname{sech} x = \frac{1}{\cosh x}\) and \(g(x) = \tanh x = \frac{\sinh x}{\cosh x}\). 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 \(x \to \pm \infty\). Then sketch \(g(x) = \tanh x\), 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 \(\operatorname{sech} x = \tanh x\). 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: \(\text{Area} = \int_0^{x_{intersection}} (f(x) - g(x)) \, dx\). This integral will give the bounded area.
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Video duration:
5mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hyperbolic Functions
Hyperbolic functions like sech(x) and tanh(x) are analogs of trigonometric functions but based on hyperbolas. sech(x) = 1/cosh(x) and tanh(x) = sinh(x)/cosh(x). Understanding their shapes and properties is essential for sketching their graphs and analyzing intersections.
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Points of Intersection
Points of intersection occur where two functions have the same value for the same x-coordinate. To find these points, set f(x) equal to g(x) and solve for x. These points define the boundaries of the region enclosed by the graphs.
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Area Between Curves
The area bounded by two curves and the y-axis can be found by integrating the difference of the functions over the interval defined by their intersection points and the y-axis. This involves setting up definite integrals and understanding which function lies above the other.
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