In Problems 3 to 12, find the average value of the function on the given interval. Use equation (4.8) if it applies. If an average value is zero, you may be able to decide this from a quick sketch which shows you that the areas above and below the r axis are the same.
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Q: please provide detailed solution for a to c, thank you
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- Use a computer to reproduce the table and graph in Figure 2.4: two Einstein solids, each containing three harmonic oscillators, with a total of six units of energy. Then modify the table and graph to show the case where one Einstein solid contains six harmonic oscillators and the other contains four harmonic oscillators (with the total number of energy units still equal to six). Assuming that all microstates are equally likely, what is the most probable macrostate, and what is its probability? What is the least probable macrostate, and what is its probability?I am confused with part (e). I don't understand the steps. How is 1.427 obtained? How is the 20t moved to the left side of the equation, since it is inside the COS() function? I am just not understanding the math. Can you step it through with an explanation at each individual step?Sketch f(y) versus y. Show that y is increasing as a function of t for y < 1 and also for y > 1. The phase line has upward-pointing arrows both below and above y = 1. Thus solutions below the equilibrium solution approach it, and those above it grow farther away. Therefore, ϕ(t) = 1 is semistable.
- 1 W:0E *Problem 1.3 Consider the gaussian distribution p(x) = Ae¬^(x-a)² %3D where A, a, and A are positive real constants. (Look up any integrals you need.) (a) Use Equation 1.16 to determine A. (b) Find (x), (x²), and ơ. (c) Sketch the graph of p(x).Example of numerical instability: Take y′ = −5y, y(0) = 1. We know that the solution should decay to zero as x grows. Using Euler’s method, start with h= 1 and compute y1, y2, y3, y4 to try to approximate y(4). What happened? Now halve the interval. Keep halving the interval and approximating y(4) until the numbers you are getting start to stabilize (that is, until they start going towards zero).Problem 2.34 Consider the "step" potential:53 V (x) = [0, x ≤0, Vo, x > 0. (a) Calculate the reflection coefficient, for the case E Vo
- The Newton–Raphson method for finding the stationary point(s) Rsp of a potential energy surface V is based on a Taylor-series expansion around a guess, R0. Similarly, the velocity Verlet algorithm for integrating molecular dynamics trajectories [xt, vt] is based on a Taylor-series expansion around the initial conditions, [x0, v0]. Both of these methods rely strictly on local information about the system. How many derivatives do we need to compute in order to apply them?Consider the functions f(x) = x and g(x) = sin x on the interval (0, ). (a) Complete the table and make a conjecture about which is the greater function on the interval (0, ). (b) Use a graphing utility to graph the functions and use the graphs to make a conjecture about which is the greater function on the interval (0, ). (c) Prove that f(x) > g(x) on the interval (0, ). [Hint: Show that h′(x) > 0, where h = f − g.]Suppose function fhas the graph as shown below