f (x, y, z) dS. H

Q: If f(x) is an integrable function defined on the closed interval [a, b] ], ƒ (x) dx represents the…

A:

Q: Let H be the hemisphere x2 + y² + z² = 30, z 2 0, and suppose f is a continuous function with f(5,…

A: Let H be the hemisphere x2+y2+z2=30, z≥0 and suppose f is continuous with f(-5,1,2)=14, f(5,1,2)=19,…

Q: Evaluate f(x, y, z) dV for the function f(x, y, z) = xz² and region B = [0, 3] × [5, 7] × [4, 5].

A:

Q: Let H be the hemisphere x2 + y2 + z2 = 11, z 2 0, and suppose f is a continuous function with f(1,…

A: Given,       x2+y2+z2=11,   z≥0     f1, 3, 1=17,   f1, -3, 1=17,   f-1, 3, 1=17 and    f-1,-3, 1=18

Q: If a continuous RV X follows N(0, 2), find P{1 1}.

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Q: Find all values of c such that f is continuous on (-0, o0). S1-x², {. f(r) %3D x, C = (smaller…

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Q: For a continuous function f(x) on the set 2 [- 5,2], the net area Tx)dx must be -5 always positive…

A: The given question has been solved in detail

Q: Show that a uniformly continuous function on (a, b) is bounded.

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Q: x² - y² z = g(x, y) = x² + y² 0 for for (x, y) = (0,0) (x, y) = (0,0)

A: Given function is z=gx,y=x5−y5x4+y4 for x,y≠0,00            for x,y=0,0. We have to check whether…

Q: Show that the function f defined by (x, y) = (1, –1) f(x, y) = r? + y (x, y) # (1, –1) x + y is not…

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Q: Let f:XT, → XT, such that f({a}) = {b, c} į Is fa continuous function? i yes,no, or can't you tell?…

A: Let (X, ΤX) and (Y, ΤY)be topological spaces A function f : (X, ΤX) → (Y, ΤY) is continuous  if…

Q: Let N := {(x, y) e R² | 1 < x² + y? < 4} be a punctured disc and let f : N → R be a continuous…

A: As per the question, we are given a punctured disc (annulus) Ω , on which a locally convex function…

Q: 2. Let f : [a, b] R be a bounded function. Prove that if f is continuous at ro € [a, b), then L(|f])…

A: Here we use the property of continuty.

Q: Let f be a continuous function over an interval I containing a critical point e and f is…

A:

Q: Prove that the function f(x) = Root3(x) is uniformly continuous on [−1, ∞). (Hint: consider the sets…

A: Given that fx=3x we have to prove that fx is continuous in interval [-1,∞) lets consider the set…

Q: Let f(z, y) be a continuous function such that f2, Su Jay and fuz are all continuous everywhere. If…

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Q: Show that the function f defined by (x, y) = (1, –1) f(x, y) = { x2 + y (x, y) # (1, –1) x + y is…

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Q: • Let X be a continuous r.v; with pdf f(x) : 0 < x < 1, x, f(x) = {2 – x, 1< x < 2, 0, elsewhere.…

A: We have given that, Let X be a continuous random variable with pdf: f(X) = {  x,   0&lt;x&lt;1…

Q: Let D := [0, 1] and let f: D→ R be the function defined by f(x) = VT. Show that f is uniformly…

A: According to the given information,

Q: Let f:X→R be a Lipschitz continuous function defined on X. Show that f is uniformly continuous on X.

A: Given : function f:X -&gt; R is a Lipschitz continuous function defined on X . To show : f is…

Q: What value of c makes the following function continuous at (0,0)? f(z, y) = C, x+ y³ +2, (x,y) +…

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Q: Is f(x) = x² uniformly continuous on (0,1)? Why or why not?

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Q: Use the results of Exercises 15 and 16 to find the area of the region Ω shown in Exercise 13.

A: In questions 15 and 16, we can find the area of the bounded region by using the definite integral…

Q: Determine the set of points at which the function is continuous. if (x, y) # (0, 0) 5x? + y? f(x, y)…

A: We know that given function is continuous in its domain so we need to check only at point where…

Q: Show that the function f defined by (r, y) = (1, –1) (r, y) # (1, –1) f(r, y) = 1² + y is not…

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Q: (3) Determine the values of a for which x4 + y* , if(x, y) # (0, 0), (x² + y?)ª f(x, y) = ,if(x, y)…

A: solution.

Q: Use the definition to prove f =1 is uniformly continuous on [0, 0).

A:

Q: If fx and fy are each continuous in an open region R, is f(x, y) continuous in R? Explain.

A: If fx and fy are each continuous in an open region R, is f(x, y) continuous in R? Explain.

Q: Determine if f is continuous at (0, 0) where: 3.x?y x4 + 2y2' (x, y) # (0,0) f (x, y) = (x, y) =…

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Q: Show that the function f defined by 1, (x, y) = (1, –1) %3D f(x, y) = { a2 + y (x, y) # (1, –1) x +…

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Q: Show that the function f defined by 1, (x, y) = (1,–1) f(r, y) = { x² + y (x, y) # (1,–1) r+ y is…

A: In this question, the concept of limit is applied. Limit Limits are the points at which a function…

Q: If f, g: R → R s.t. f and g are uniformly continuous. and bounded, then show that f.g is also…

A: Let f ,g are uniformly continuous And f and g are bounded

Q: Let Y be any topological space and y: [0, 1] → Y be a continuous function so that r(0) = r(1) = xp.…

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Q: Is g(x) = ln( x2+2 ) uniformly continuous on the interval (−∞,∞)?

A: given function is                                                        g(x)=lnx2+2 log is defined…

Q: Show that the function f defined by (x, y) = (1, – 1) f (x, y) : x2 + Y (x, y) + (1, – 1) x + y is…

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Q: Use the definition to prove f = (x-1)/(x+1) is uniformly continuous on [0,∞).

A: Given function is fx=x−1x+1. We have to show that fx=x−1x+1 is uniformly continuous on 0,∞. Let us…

Q: Let T: (X,d) (Y,d) be a continuous function for all n. If {T,} converges uniformly to T on X, then T…

A: Given Tn:X,d→Y,d^ is a continuous function for all n and Tn converges uniformly to T on X. We have…

Q: Let (s n(x-y) |x|+ly\ = 0 f(x, y)={ |x|+|y]| (0, (x, y) = (0,0) Is f continuous at the origin? Why?

A: The continuity of the function can be checked by checking the limits of the function at the point.…

Q: Show that the function f defined by 1, f(r, y) = { r2 + y (x, y) = (1, –1) (x, y) # (1, –1) is not…

A: We have to show that  f(x, y)  is not continuous at  (1, -1)

Q: If |fn(x) – fn(y)l 0 and all x, y in a compact interval, show that {fn} is uniformly…

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Q: If f(z) is continuous at z then it must be differenttiable at z, 0.

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Q: Let f: X→Y, g:Y-Z be uniformly continuous. Then gof:X-Z is uniformly continuous.

A: The main objective is to proof that the function g∘f is uniformly continuous, i.e. for every ε&gt;0,…

Q: Let f : [0,∞) → R. Assume that f is uniformly continuous on [0, 1] and on [1,∞). Show that it is…

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Q: Let H be the hemisphere x2 + y2 + z2 = 50, z ≥ 0,and suppose f is acontinuous function withf (3,4,…

A: Let H be the hemisphere x2 + y2 + z2 = 50, z ≥ 0,and suppose f is acontinuous function with f (3,4,…

Q: Let H be the hemisphere x? + y2 + z2 = 41, z 2 0, and suppose f is a continuous function with f(4,…

A:

Q: Let f be a continuous function over an interval I containing a critical point c and f is…

A:

Q: Let H be the hemisphere x2 + y2 + z2 = 30, z 2 0, and suppose f is a continuous function with f(5,…

A:

Q: Let x = s + t and y = s - t. Show that for any differentiable functionf (x, y),

A: Given: x=s+t and y=s-t and a differentiable function fx,y To show: ∂f∂x2-∂f∂y2=∂f∂s∂f∂t,

Q: Show that if u = x – y and v = x + y, then for any continuous f, -2sx * f(x – y, x + y)dydx = / |…

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Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0). The multivariable limit of this function as it approaches the points (x0, y0) and is denoted as: