1. Suppose that f(z) is analytic for lz <1 and sa Show that there is a constant c < 1 such that |f(z)| < c isfies |f(2)] <1, f(0) 0, and lf (0) < 1.] Let r <1. for 리-r

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Q: 7- The function f(z) = z is: a) Analytic on C b) Analytic only at zo = 0 c) Not analytic

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Q: 1. Show that for all f e L'(R) lim dr = 0. %3D

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Q: 1. Use the Cauchy-Riemann conditions to find out whether the following functions are analytic. a.…

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Q: (5) Use the limit definition of derivative to prove the following: (a) For any a e R, (f(ax))' =…

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Q: 4.1.7. Show that if f : R" + R" satisfies |f(x)| < |r|² for all r € R" then f is differentiable at…

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Q: 16. Suppose that f is differentiable at c and that f'(c) + 0. Show that there exists a ð > 0 such…

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Q: Let f(x) = V13 – x. Compute f'(4) using the limit definition f'(4) =| Find an equation of the…

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Q: Let f(x) = 2x2 - 3x - 5. Show that the secant line through (2, f(2)) and (2 + h, f (2 + h)) has…

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Q: 1- Discuss the continuity of the following functions at given poin { V9r4+x² 5x2+3x+1 (a) f(x) = if…

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Q: 4 Given that f(2) is analytic at z = 0 and that f G) =, n = 1,2, 3..,find f(2). VE

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Q: Define f:R → R by f (x) = 5x if x is rational and f (x) = x² + 6 if x is irrational. Prove that f is…

A: Given, f:R→R by f(x)=5x if f(x) is rational and f(x)=x2+6 if x is irrational.To prove that f is…

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Q: 7. Suppose that f: R R is differentiable at c and that f(c) = 0. Show that g(x):= |f(x)| is…

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Q: Define f:R → R by ƒ (x) = 5x if x is rational and f (x) = x² + 6 if x is irrational. Prove that f is…

A: Given :  f:ℝ→ℝ  defined by ;         f(x) = 5x,if x is rational.x2+6if x is irrational To prove :…

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Q: 9. Consider the function f: R → R, defined by (x²sin(1/x) ifx # 0 f(x) = if x = 0 Then a. f is not…

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Q: 20. Use the following information to sketch the graph of the continuous func- tion y = f(x) whose…

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Q: Use Taylor's Theorem to find an interval [−z,z][-z,z] where the Maclaurin polynomial p3(x)=x−x^3/6…

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Q: Show that the converses of parts (a) and (b) of Theorem 6.2.8 are false by finding counterexamples.…

A: Theorem: Let f be a differentiable function on the interval I. Then, If f'x&gt;0 for all x∈I, then…

Q: (3 Let f(x) = 2x2 01 waluate limits and continuity of f(x) at 1 and at 0.

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Q: (c) Suppose that f : [-1,1] →R is a continuous function such that r²+ f(x)² = 1 for every rE[-1, 1].…

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Q: 1. Determine whether k(x) is continuous at r = 3, where %3D x + 3, x 2.

A: Given  k(x)=x+3,  x&lt;0x2+1, 0≤x≤22x,   x&gt;2 To check the continuity we can use the graph as…

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Q: (8) Suppose that f(x) and g(x) are differentiable and one-to-one and f(2) = –1, f'(2) = -3, f(-1) =…

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Q: 7.14. Show that if f = u+iv is analytic in a region S and u is a constant function (i.e.,…

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Q: 1. Use the definition of the derivative of complex function to show that f(z) is not differentiable…

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Q: a. Let f(x) = { 2 sin(2 – x) – 4, x 2. Is f differentiable at r = 2? b. Let (2h(x) + x)² f(x) = x +2…

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Q: 2.23 Let 1 - x2 f(x): if x € Q, if r ¢ Q. Find all points p at which f is continuous. Prove your…

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Q: For f(x) = √ √2-4 4- ™ Of is not continuous at x = 4 Of is continuous at a Of is continuous at a 5…

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Q: 1. Consider the function z = f(x, y). If fa(a, b) and fy(a, b) both exist, and they are equal, then…

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Q: yLet f(2) be analytic in {z: |2| < 1} with f(0) that |f"(0)| < 2. = 0 and |f(2)| < 1 for all z in…

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Q: Assume thatf is differentiable in (-oo, x) and af'(r)N 0 for all a E (-00, 00). Prove that f has the…

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Q: 3. Let f)=x+x+2x-1E2,lx). Then x= 1 is a root for r() of mmltiplicity 3D1 None 4 2

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Q: 9. Show that if the analytic function f(z) has a zero of order N at zo, then f(z) satisfying g'(zo)…

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