2. (a) Let M > 0 and RR satisfy the condition f(x)/(v) ≤ May for all x, y E R. Show that is uniformly continuous on R. [5 marks]

Q: 'he function f: E → R is uniformly continuous if

A: We have to find the answer using the definition of uniform continuity.

Q: An RC circuit with a 100 Q resister and a 10^(-2) F capacitor is driven by a voltage…

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Q: 2. Consider the function / with domain (-2, o0): 3|2 – 1| 2 + 27 - 4 if – 2 <z<1 f(z) = if r = 1…

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Q: 2. Let f: RR be defined by x EnQ f(x) = { 4x, x² +4, x EQ Use the sequential criterion to show that…

A: f:ℝ→ℝfx=4x, x∈nℚx2+4, x∈ℚ

Q: Prove: Let the function f:X →Y and g:Y →Z be continuous. Then the composition function g•f:X→Z is…

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Q: Suppose f is defined on R and f satisfies |f(x) − f (y)| ≤ C\x − y|ª 1 x, y € R. Here C and a are…

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Q: b) Give an example to show that the product fg need not be uniformly continuous.

A: Given: f:A→B, g:B→C are uniformly continuous.To show: b) The product fg need not be uniformly…

Q: [1+2p(x-1)+(x-1)² x≤1 x>1 [qx+p -) -) Find the values of q, in terms of p, for which fis continuous…

A: Given:f be a function defined by:a. To find the value of q in terms of&amp;nbsp;p, for which f is…

Q: 10. Prove the following change of variables formula: Let p be a real-valued differentiable function…

A: Introduction: The department of calculus wherein we look at integrals and their houses is called…

Q: 2. Let f be defined on an open interval (a, b) and assume that for some e e (a, b) we have f'(c) > 0…

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Q: A function is continuous everywhere. It is known that f"(x) > 0 on the interval (0, 3) and f"(x) <0…

A: Given that:                          f(x) is continuous everywhere.f''(x)&gt;0 on the interval (0,…

Q: Is there a continuous function such that - f: [1,2] U [3,4] → R - f(1) * f(4) < 0 - f(x) not egual 0…

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Q: 6. Suppose that ƒ is differentiable on (a,b). interval [a, b]? Justify your response for find a…

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Q: 11. with f(-1)=6, f(1) = 8 and f(6) = 3. Intermediate Value Theorem? Let f be a function that is…

A: Given:f be a function that is differentiable on the closed interval -1, 6 such…

Q: 3c. Let 0 < a < 1 and ƒ : (a, 2) → R be given by ƒ (x) = ½. Show that ƒ is uniformly continuous.

A: Given: Let  and  be given by We have to show that  is uniformly continuous.

Q: (a) Assume f: S → R is uniformly continuous and S is bounded. Prove that the range of f is bounded.…

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Q: Suppose that X P(X < 3Y). unif(-2, 2) and Y ~ unif(-1,2) are independent continuous RVs. Find P (|X…

A: Given: X ~ U (-2, 2) Y ~ U (-1, 2) X and Y are independent.

Q: f. Is the domain of f(x.y)= In (6x + 5y“) bounded or unbounded? O Unbounded O Bounded

A: the domain will be Unbounded because

Q: 2: If f(x) and g(x) are continuous on (a,b]and if f =[s . Prove that 3cela,b] such that f(c)=g(c).…

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Q: 15) Prove that if ffx) is continuous at x=0 then so is V f(x) provided that fix) zo for all x in the…

A: Step 1: Step 2: Step 3: Step 4:

Q: 4. Prove that the function f: [0, ∞)→ R; f(x) continuous. = √ is uniformly

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Q: 55. Let F be a function given by Sx2 + 1, for x s 2, 2x + 1, for x >2. F(x) = a) Verify that F is…

A: We have Fx=x2+1 ,  for x≤22x+1 , for x&gt;2 We have to verify   (a) Function is continuous at x=2…

Q: 3. For m₁ and m₂ numbers, with m₁ { Prove that the function f:R → R is continuous but not…

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Q: A boy has a collection of 20 pennies, 8 nickels, 10 dimes and 15 quarters. How many coins must the…

A: ANSWER :

Q: Prove that f(2) = z/(z* + 1) is continuous at all points inside and on the unit circle Iz] = 1…

A: we need to find those points where the denominator ( z^4+1) equals to 0

Q: Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum…

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