a) Show that the function defined as follows 1

Q: Let $c$ be a critical number of a function $f$ that is continuousinterval, except possibly at $c$,…

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Q: Define f(x) = x if x is rational and f(x) = 0 if x is irrational. Compute So f dx and ſ f dx. Is f…

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Q: (3) that кете Use Riemann's Criterion to prove the constant function f. R > R = k given by f(x) 4…

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Q: Iff: [a,b] R is bounded and has only a finit number of discontinuous points, then f is Riemann…

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Q: 1. Consider the function: f on [0, 1] by 7 if r1 f(x) |0 if 1 0, 1]. Prove that f is Riemann…

A: To prove that f is not continuous but Riemann integrable

Q: f is not integrable on [0, 1].

A: It is given that f:[0, 1]→ℝ by f(x)=5x  if x∈ℚ0    if x∉ℚ We know, if lower integral and upper…

Q: 5. If f is continuously differentiable on (0, 1] and f(1) = f (0) = -1/6, prove that dr 22 2 Hint:…

A: Continuously Differentiable Function: A function f is continuously differentiable on [0, 1] if its…

Q: Suppose that g is a Riemann integrable function on [0.3, 31.5] and f(x) = g(x) except for the values…

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Q: 3. If f : [a,b] → R is a bounded function such that f(x) = 0 except for x in {c1, C2, .., Cn} in [a,…

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Q: 2. If f is continuous on an open interval containing x, and fchanges its direction of concavity at…

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Q: Derive a finite difference approximation to of(x,y) + f(x,y) using the points f(x, y), ƒ (x+h, y),…

A: Here we use taylor series to find this.

Q: 2. a) Compute the integral fa f(x) dx using 5 point composite rectangle, trapezoid and Simpson rules…

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Q: 9 Prove that if m(x) is differentiable on (-∞, 0) and its derivative m'(x) is bounded then m is…

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Q: 1. Consider the function f defined on [0, 0), f (x) = {x* sin- x + 0 x = 0 where r > 0. Determine…

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Q: Find the Wronskian for {x, -sin x, cos x}

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Q: 4 A function fx) = 1- x² + xr is defined in the closed interval [-1, 1]. The value of x in i onen…

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Q: Show that the function f defined on [0, 1]'by the conditions 1 f(x)=2rx when 7=1, 2, 3,...... És…

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Q: y = sech*x = In(_+ V1- x², sechx = In(- 1+ V1-x² Prove that .... ..

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Q: Suppose that f and g are integrable and that -1 f(x) dx = 5 h(x) dx = 4 7, dx = %3D Use the rules to…

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Q: To prove this function to be Reiman Integrable * 4/4 { 0 ifx= 1 3 if x + 1 Let f : [0, 3] → R be…

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Q: 2. Express the following limits as a definite integral of a suitable function over a finite…

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Q: Use Riemann sums and a limit to compute the exact area under the curve y = f(x) = 2x2 + 1 on the…

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Q: 7. Prove that if the functions f and o are bounded and integrable in [a, b], then 6. 9. f(x) $(x) dx…

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Q: Prove Fermat’s Theorem for the case in which f has a localminimum at c .

A: It is given that f has a local minimum value at c. Thus, by the concept of local minimum, fc≤fx when…

Q: If f: [a, b] - R is bounded and has only a finit number of discontinuous points, then f is Riemann…

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Q: 3. Suppose that f and g are integrable and that f(x) dx = -1; , f(x) dx = 5; and f, h(x) dx ='4,…

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Q: 2. a) Let u(x, y) be a continous function on a closed and bounded, simply connected region R. Also…

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Q: 6. Suppose that f: R -R is uniformly continuous so that for some 6 > 0, and for all 1, y € R I/(=) –…

A: The objective here is to prove: if f(0)=0then for every x>0 fx<1+xδ Given: f:R→R is uniformly…

Q: Suppose f> 0 is integrable on [a, b], continuous at xo € [a, b], and f(xo) > 0. Prove that f>0.

A: Given f≥0 is integral on a,b continuous at x0∈a,b and fx0>0

Q: 5. Determine the interval on which f(x) = x+2 is integrable. %3D (-00, 2) [-2, co0) (-00,-2) U (-2,…

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Q: Show that the function f defined on [0, 1] by the conditions 1 f(x)=2rx when 7=1, 2, 3,...... És…

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Q: A set A C [0,1] is dense in [0, 1] iff every open interval that intersects [0, 1] contains a point…

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Q: Find aland U so that the function S 7z³ – 3x² + 7, z -2 is both continuous and differentiable. al =

A: To Determine: fins a and b so that the function is continuous and differentiable given: we have a…

Q: (1) Give an example of a function f such that | f |is Riemann integrable but f is not.

A: Consider the function f on the interval 0,1 defined by:  fx=1, if x is rational-1, if x is…

Q: (b) Let f be a function defined by 1 if x e Qn [a,b] f(x) = - |-1 if x€ (÷) n [a, b]. Show that fis…

A: we are given that f(x)=1   if x∈Q∩[a,b]-1 if x∈(RQ)∩[a,b] To prove f is not Riemann-integrable  on…

Q: 2,- Show Continuous at x =0 me the follewing functron rot fex) = Ix/ or he

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Q: Let f(x) be a continuous function on [0, 1] and consider the limit 1 Xj +... f + In · drn. lim dx1…

A: As per the question we have to interpret the given integral as an expectation value and find the…

Q: Consider the sequence of functions {f} defined on the interval [0, 1], where fn(r)= nx 1+n²r²¹ Vx €…

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Q: Suppose that f and h are integrable and that: i fx) = -1, f(x) dx = 5, h (x) dx = 4 ,Find: a. S–2…

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Q: 1. Express the following limits as definite integrals. (a) lim )V3(x;)² + In(x;) Ax, [2,7], where Ax…

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Q: 11. Show that the definite integral of f(x) = c over [a, b] is c(b-a) by applying the definition of…

A: Here given function as fx=c where the function fx is defined on the interval a, b.

Q: Show that the square integrable function f(x) = sin( πk log x/ log 2 )for k ≥ 1 are orthogonal over…

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Q: 4. Express the following limit of the Riemann sum n lim (x + x² cos x; ) Ax, n→∞ i=1 ㅠ where x₁ =…

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Q: Assume that J is continuous on [0, 3 and differentiable on (0, 3) with f (z) > 6 for each z in (0,…

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Q: A set of real numbers D is called dense if and only if every open interval contains at least one…

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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: 5. Consider the function if x € Q f(x)= if x # Q. Show that f is not Riemann integrable on [0, 1]. n…

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Q: Consider the nonsmooth function: 1 f(x) = max x; +;|x||², 1<i<K where x E R", x; is the i-th…

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Q: 11. Verify Rolle's theorem for: c) f(x) = In () on [1,3] %3D

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