(a) Use Cauchy's Residue Theorem to evaluate log-/22 dz. (2²+1)² Fr.R (b) Show in detail exactly one of the following. dz⇒0 dz • √ (2² + 1)² 6 log-/22 dz0 as R→∞ • Sm (2² + 1)² 0 log-/22 dz0 as r 0+ d
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- If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?Determine dimension x to 3 decimal places.Let z= x+iy and determine where the function f(z) = Log(z2 + 1) is holomorphic. What is its derivative in the region where it is holomorphic?
- is f(z) = Log_(pi/2)(1/z) holomorphic? on where?Assume that ƒ is continuous on [a, b] and differentiable on (a, b). Also assume that ƒ(a) and ƒ(b) have op-posite signs and that ƒ′ ≠ 0 between a and b. Show that ƒ(x) = 0 exactly once between a and b.Prove that integration of (x/sigma square ) times exponential of (-x square/2sigma square) for limit 0 to ♾ is 1
- Suppose that Xis a continuous unknown with -4< X< 4 having PDF denoted fsatisfying f ( x ) = c ( 16 − x^2 ) ,for -4< x< 4. Here c is a positive normalizing constant. What is the value of c?A manufacturing company owns a major piece of equip -ment that depreciates at the (continuous) rate f= f(t) ,where is the time measured in months since its last overhaul.Because a fixed cost is incurred each time themachine is overhauled, the company wants to determine theoptimal time T (in months) between overhauls.(a) Explain why ∫0t f(s) ds represents the loss in value ofthe machine over the period of time t since the lastoverhaul. (b) Let C =C(t) be given by C(t) =1/t[A + ∫0t f(s) ds]What does C represent and why would the companywant to minimize C ?(c) Show that has a minimum value at the numbers t =Twhere C(T) =f(T).Suppose that a nonnegativefunction y = ƒ(x) has a continuous first derivative on [a, b] . LetC be the boundary of the region in the xy-plane that is bounded below by the x-axis, above by the graph of ƒ, and on the sides by the lines x = a and x = b. Show that ∫ƒ(x) dx = - ∮C y dx.
- Let Q = Q(K, L, t), where K is capital, L is labor, and t is time. Both capital and labor as also functions of time, such that K=K(t) and L=L(t). Use total differentiation to find and expression for the total derivative dQ/dtWrite the limit as a definite integral on the interval [a,b], where c is any point in the ith subinterval.if h(x) = coth-1(x) if x>=1 [[x+1]] if -1 < x < 1 tan-1 (x2+2x+1) if x<= -1 Is h continuous at x=-1, 0, and 1? If not, Is it removable, jump essential, or infinite?