Explanation Derivative of a function y = V ( x ) tells us the rate at which value of y changes in the function with respect to the change in value of x. Given: The given equation is V ( x ) = 2 ln ( tanh x 2 ) . Formula used: We shall use the chain rule and d ( x n ) = n . x n − 1 , d ( ln x ) d x = 1 x Calculations: we are given with the equation V ( x ) = 2 ln ( tanh x 2 ) Differentiate both sides with respect to x we get V ' ( x ) = 2. 1 tan x 2 . 1 2 sec h 2 ( x 2 ) = 1 sinh x 2 cosh x 2 Differentiating both sides with respect to x V " ( x ) = − 1 2 sinh 2 ( x 2 ) + cosh 2 ( x 2 ) sinh 2 ( x 2 ) cosh 2 ( x 2 ) V "

Loose-leaf Version for Calculus: Early Transcendentals Combo 3e & WebAssign for Calculus: Early Transcendentals 3e (Life of Edition)

3rd Edition

ISBN: 9781319019846

Author: Jon Rogawski, Colin Adams

Publisher: W. H. Freeman

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Differential Equations

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Question

Chapter 3.9, Problem 82E

To determine

To prove:

The double derivative of the function $V (x) = 2 \ln (\tanh \frac{x}{2})$ is given as $V " (x) = \sinh (V (x))$

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Chapter 3.9, Problem 81E

Chapter 3.9, Problem 83E

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The measured receptance data around two resonant picks of a structure are tabulated in the followings. Find the natural frequencies, damping ratios, and mode shapes of the structure. (30 points) (@)×10 m/N α₁₂ (@)×10 m/N w/2z (Hz) 99 0.1176 0.17531 0.1114 -0.1751i 101 -0.0302 0.2456i -0.0365 -0.2453i 103 -0.1216 0.1327i -0.1279-0.1324i 220 0.0353 0.0260i -0.0419+0.0259i 224 0.0210 0.0757i |-0.0273 +0.0756i 228 -0.0443 0.0474i 0.0382 +0.0474i

== 1. A separable differential equation can be written in the form hy) = g(a) where h(y) is a function of y only, and g(x) is a function of r only. All of the equations below are separable. Rewrite each of these in the form h(y) = g(x), then find a general solution by integrating both sides. Determine whether the solutions you found are explicit (functions) or implicit (curves but not functions) (a) 1' = — 1/3 (b) y' = = --- Y (c) y = x(1+ y²)

A circle of radius r centered at the point (0,r) in the plane will intersect the y-axis at the origin and the point A=(0,2r), as pictured below. A line passes through the point A and the point C=(11/2,0) on the x-axis. In this problem, we will investigate the coordinates of the intersection point B between the circle and the line, as 1 → ∞ A=(0,2r) B (0,0) (a) The line through A and C has equation: y= 2 117 x+27 (b) The x-coordinate of the point B is 4472 121,2 +4 40 (c) The y-coordinate of the point B is +27 121 44 (d) The limit as r→ ∞ of the x-coordinate of B is 121 (if your answer is oo, write infinity).

Chapter 3 Solutions

Loose-leaf Version for Calculus: Early Transcendentals Combo 3e & WebAssign for Calculus: Early Transcendentals 3e (Life of Edition)

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To prove: The double derivative of the function V ( x ) = 2 ln ( tanh x 2 ) is given as V " ( x ) = sinh ( V ( x ) )

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To prove: The double derivative of the function V(x)=2ln(tanhx2)is given as V"(x)=sinh(V(x))

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To prove:

The double derivative of the function $V (x) = 2 \ln (\tanh \frac{x}{2})$ is given as $V " (x) = \sinh (V (x))$