Question 3. Let a € (0,1) and define f : T→R by 1 f(x) = sin(nx) na n=1 for all r E (-T, 7]. Prove that f converges for all x E T. (Hint: For r + 0, show E-1 sin(kx)< TanET: Then use Question 3 from the Midterm.) =D1

Q: Decide on the function f(x)  is uniformly continuous on the interval (0,∞).

A: The objective of the question is to determine whether the function f(x) = cos(x) - 1/x is uniformly…

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A: First we have to show that v-u1+v2&lt;tan-1v-tan-1u&lt;v-u1+u2 where 0&lt;u&lt;v. Then we will show…

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Q: 14. Discuss the differentiability at x = 0 of the function 1 x sin -, x = 0, x f: x→ 0, x = 0.

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Q: 6. Say f(x) = 2x2² + cos x. Determine which of the statements below are true. (More than one may be…

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Q: 6. Find the intervals on [0,2r)where f(0) = sin²(0) is concave upward/ concave downw %3D

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Q: 2.6. Let ƒ be continuous on [0, a]. If x e [0, a], define fo(x) = f(x) and let %3D 1 fa+1(x) = -…

A: We will use the basic knowledge of integral calculus to answer this question. Note that we can only…

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A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

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A: Using Fourier series and putting some particular value to get our required series.

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A: we have to find dydx for the given equation.1. y=6z2+18x4-12x8differentiate w.r.t…

Q: 1- Calculate R6 for f(x) = 9 10x on interval [2, 4].

A:

Q: cos x - 1 f(x) = x

A: f(x)=A function y = f(x) is uniformly continuous at an open interval (a, b) if f(x) is continuous on…

Q: 2. Let r and y be real numbers be such x, y E {6i 3 2 2 3弧,T} T T T T (a) If 4 cos (2t) dt = find x.…

A: Here integral is already solved and answer is given in term of f(x) and g(y). Which is true because…

Q: Use the prové thát 1 x| < ]x| fór áll * # 0. onclü X| S x| för åll x € R. Hint. Apply MVT to f(x) =…

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Q: 89. Assume that f (0) = f(1) = 0 and that f" exists. Prove I s"(x)f(x) dx = –[ s(x)² dx 10 Use this…

A: We will apply integration by parts: Let u = f(x) and dv = f''(x)dx hence, v = f'(x) and du =…

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A: As per the question we are given an improper integral and we have to cheak its convergence using…

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A: The function is given by fx = tankxx ,x&lt;04x+3k2 ,x≥0 To evaluate : The value of k so that f is…

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A: Given that fx = sinπx . Also given that h and k are two functions such that h'x = cosπx , k'x = fx+1…

Q: Show if x, is convergent or divergent T.3 'n sin(x) dx 1+x S'

A: Introduction: The convergence or divergence of an infinite series depends upon the sum of the…

Q: 14. Discuss the differentiability at x = 0 of the function 1 x sin -, x = 0, x 4.1. The Derivative…

A:

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A: Given,          y=sinxx+2x+π

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A: Fourier series

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A: a.  Now put

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Q: Show how the Newton-Cotes formula f f(x) dx = f(0) + f(1/2) + f(1) can be used for f(x) dx. Apply…

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