Detail solution required 3. (Analytic branches of multiple-valued functions)(a) Let w + Vw denote the principal branch. Describe the domain of analyticity of the functionf(2) = Ve + I and find its derivative.(b) Let w + Jw denote the principal branch. Describe the domain of analyticity of the functiong(2) = Vz2 – 4 and find its derivative. Is there a branch of vz² – 4 which is analytic on C\[-2, 2]?

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Answered: 3. (Analytic branches of…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.5: Rational Functions

Problem 54E

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Decay of Litter Litter such as leaves falls to the forest floor, where the action of insects and bacteria initiates the decay process. Let A be the amount of litter present, in grams per square meter, as a function of time t in years. If the litter falls at a constant rate of L grams per square meter per year, and if it decays at a constant proportional rate of k per year, then the limiting value of A is R=L/k. For this exercise and the next, we suppose that at time t=0, the forest floor is clear of litter. a. If D is the difference between the limiting value and A, so that D=RA, then D is an exponential function of time. Find the initial value of D in terms of R. b. The yearly decay factor for D is ek. Find a formula for D in term of R and k. Reminder:(ab)c=abc. c. Explain why A=RRekt.
Consider a differentiable function f with domain R and derivativesf'(x)=-aebx(1+bx) and f"(x)=-abebx(2+bx) , with a and b nonzero real numbers.The function has only one critical point x=-1/b and a local maximum at x=-1/bUse the Second Derivative test to find the value(s) of a and b
1. Use the Cauchy-Riemann equation to determine if the function f(z) = x3 - i(2 - y)3 is analytic or not. Provide all the sufficient conditions and the domain of analyticity and then find the derivative if it exists.
Find the critical numbers of f(x) = x4(x-1)3 A. What does the second derivative test tell you about the behavior of f at these critical numbers? B. What does the first derivative test tell you?
11. Let h(x) = x^4 − 2x^2 + 3.(a) Find all critical points and use the First Derivative Test to characterize each as a local maximum, local minimum, or neither.(b) Use the Second Derivative Test to check your answers.(c) Find all inflection points.
a. Locate the critical points of ƒ.b. Use the First Derivative Test to locate the local maximum and minimum values.c. Identify the absolute maximum and minimum values of the functionon the given interval (when they exist). ƒ(x) = x2 + 3 on ⌈-3, 2⌉
a) Calculate the derivative of f1 (x) = 2x4 - 3x3 + 1/2x2 + 3. b) Study the sign of the derivative f1'(x) at x ∈ ℝ by factoring f1' (x). Construct a table with the sign of the derivative according to the value of x, the variations of f1(x) (increasing or decreasing) and the limits of f1(x) at x = ± ∞. Enter in the table the values of x that cancel out f1'(x), are they local or absolute minimums or maximums? c) Calculate the equation of the line tangent to the curve: y = f1(x) at x = -1.
a) Calculate the derivative of f1 (x) = 2x4 - 3x3 + 1/2x2 + 3. b) Study the sign of the derivative f1'(x) at x ∈ ℝ by factoring f1' (x). Construct a table with the sign of the derivative according to the value of x, the variations of f1(x) (increasing or decreasing) and the limits of f1(x) at x = ± ∞. Enter in the table the values of x that cancel out f1'(x), are they local or absolute minimums or maximums?
Linearizations at inflection points Show that if the graph of a twice-dierentiable function ƒ(x) has an inflection point at x = a, then the linearization of ƒ at x = a is also the quadratic approximation of ƒ at x = a. This explains why tangent lines fit so well at inflection points.

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