The two-dimensional wave equation describing the vibrations of an infi- nite string is Θη Ot2 2021 მ2 where n = n(x,t), — -x< x < ∞ and c > 0 is a constant. Its general solution can be written as n(x,t) = F(x-ct) + G(x + ct), with F and G arbitrary smooth functions. If at time t = 0 the shape of the string is n(x, t = 0) at 9 = 5+x4 and the string is released with velocity (x, t = 0) = 0, determine n(x,t) for t > 0.

A particle moves along line segments from the origin to the points (2, 0, 0), (2, 5, 1), (0, 5, 1), and back to the origin under the influence of the force field F(x, y, z) = z2i+ 3xyj + 3y2k. Use Stokes' Theorem to find the work done.

If f'(x) = x5, what is f(x)? (Remember the constant of integration.) f(x) =

Please show all work ง 5. Evaluate the vector surface integral JJ (+2², x + 2y + 32², 5x − 2y + 2³) . Ndo. S Here, S is the sphere Sy²+2 1. = Since S is a closed surface, the outside of S is considered as the positive side on S. Hint: use the divergence theorem.

Find the derivative of the function f by using the rules of differentiation. 7 f(x) = -X417 4 f'(x) =

Evaluate the integral. Check your answer by differentiating. (Remember the constant of integration.)  (x4 − 9x2 + 2) dx

described above? Express your answer algebraically as a function of d, from the Black- Scholes model.

Let f(x) = √√x, g(x) = x + 8 and h(x) = 5. (f + g)(16) X Compute the following function value.

Suppose there is now friction and the equation of motion becomes x+2yx+w²x = 0, where 0 is some constant of friction. γ i. Find the general solution for x(t), under the assumption that w. You do not need to impose initial conditions here. ii. What is the range of for which the above solution is (1) heavily damped; (2) lightly damped. What happens to the above solution and the physical system when y = w?

Draw a graph of return on y-axis and standard devistion on x-axis