Let X, Y be sets and f : X → Y . Prove f-1(An B) = f¬1(A)nf-²(B). For any A, B CY.

Q: Let F(x, y, z) = y° z°i+ 2xyz³j+ 3xy° z²k. (a) Find the domain of F, and show that F is…

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Q: If f is continuous on A) True B False on [a,b], then Sº [ f(x a xf(x) dx = x f(x) dx

A: It will solve by part method integration  ∫u v dx = u∫v dx −∫u' (∫v dx) dx u is the function u(x) v…

Q: Let F(x, y, z) = vy-V %3D (a) Evaluate F(7, 9, 1) (b) Find the domain of

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Q: Show that if u =x - y and v =r+ y, then for any continuous f, e-2s* f(x - y,x + y)dydr = || (a+n)e-…

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Q: If f is continuous on [a, b], then there is cE[a, b] such that f(c) = av(f). Select one: True False

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Q: If f & R[a, b], then f is not continuous on [a,b].

A: Given statement statement:-  If  f∉Ra, b, then  f is not continuous on a, b.

Q: Show that a function f : R? → R defined by S (7, y) = ! if (x, y) + (0,0) 0, if x = y = 0 is…

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Q: Consider the functionf defined by f(x,y) = k. Use the definition to show that for fixed real numbers…

A: We are given that f(x, y) = k and a, b, c, and d are fixed real numbers with a<b and c<d.…

Q: If f(x,y)=x²y+y sin x, Prove that дхду дудх

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Q: Let f : [a, b] –→ R be twice differentiable and let f(a) = f(b) = 0 and f(c) > 0, where cE (a, b).…

A: Step:-1 Mean Value Theorem:-  Let a function f : a, b →ℝ be such that 1. f is continuous on a, b and…

Q: Let f: R" x R → R define as follows f(x,y) = a, then f is

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Q: (a) Prove the following relation: B(x + 1, y): B(x,y) x + y %3D

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Q: Let F(x, y,z)=xi + yj+zk, f(x,y,z)=||F(x,y,z)||. Prove that: Vf" = nf"-²F F v(In f) =

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Q: Let R represent Winnebago county in Illinois, and let f(x, y) represent the total annual snowfall at…

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Q: The domain of the function f (x, y, z) = In(6 - 3xy) %3D is {(x, y) E R? : xy 2} {(x, y, z) E R :…

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Q: The domain of the function f (x, y, z) = In(6 – 2æy) is O {(z, y) E R² : zy > 3} O {(z, y, 2) E R° :…

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Q: Let f and g represent differentiable functions such that f ″ ≠ 0 and g″ ≠ 0.

A: Concavity of a function can be determined using the second derivative of the function. If the second…

Q: For what value of b is

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Q: • Given the multivariable function f(x,y, z) = In(4x + 5y – 6z) + (2x – 3y + 4z)², determine the…

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Q: Let f(x, y) = xy - 4x on

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Q: Show in each case that f is differentiable at each a = (a,b) E R²: (a) f : R² → R with f(x, y) = x +…

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Q: (a) Prove the following relation: В (x + 1,у): B(x, y) x +y %3D

A: We have give Beta function relation, we have to show left hand side is equal to right hand side.

Q: if f (r) is differentiable prove that f(r)r is irrotational

A: The objective here is to prove that the vector field f(r)r is irrotational. Given: f(r) is…

Q: Let f:Z x Z+ Z x Z be the function given by f(z, y)= (2z + y,I- y). Find, with proof, an element af…

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Q: Compute f, for f(x,y)=xln(xy). %3D

A: We can find the answers as below.

Q: If f(x) and g(x)are continuous on [a,b]and if [f = [8 . Prove that 3ce[a,b] such that f(c)= g(c).

A: Here, we will going to use Intermediate Value  Theorem to prove this result

Q: Define f: R → [2, 0) by f(x) = x² +2,Vx € R.Prove that f is onto.

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Q: Assume that f : R → R is differentiable everywhere and satisfies for all h, x E R: f(x + h) – f(x) =…

A: Let's prove this.

Q: The domain of the function f(r, y, z) = In(6 2xy) is O {{z, y, z) E R³ ° : ay > 3} O {{z, y, 2) E R°…

A: see below the answer

Q: Prove: If f € R(a) on [a, b] and if |f(x)| < M on [a, b] , then | S. f da| < M[a(b) – a(a)].

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Q: If f is differentiable and non-negative on an interval I in R then /3f + 1 is differentiable on I.

A: The given statement is true. Because when f will be non negative then the derivative of the given…

Q: Let f(x, y, z) = xyt + z3 and x = st², y = s³t² , and

A: The given function is fx,y,z=xy4+z3 and the values of x, y, and z are st2, s3t2, and s2t2…

Q: The Fundamental Theorem of Calculus says that f(x)da = F(a) – F(b) provided f = F' Is continuous on…

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Q: Let F = (2y? + z)i+ 4ryj+ æk. (a) Show whether F is conservative.

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Q: Let f : R? → R² be defined by f(x, y) = (x – 2y, xy). Prove that f is continuous at (1, 1).

A: Let f:R2→R2 be defined by fx,y=x−2y,xy We have to prove that f is continuous at 1,1. For continuous…

Q: The domain of the function f (x, y, z) = In(6 – 3ry) is - o {(x, y) E R² : xy > 2} O {(x, y, z) E…

A: Please refer to the image below

Q: Let w = f(u), where u = x g(y), show that w,xx = f" (x g(y)). g²(y). %3D

A: Let w=f(u), where  u=x g(y). So using the chain rule, differentiating with respect to x, we get…

Q: Assume that f'' exists and f'' (x) = 0 for all x. Prove that f (x) = mx + b, where m = f'(0) and b =…

A: Given that  f'' exists and f'' (x) = 0 for all x. Integrating both sides ,we get By substituting…

Q: Let f be Continuous on [a,b] and diffrentiable on (a, b), Prave that if azo, then there are Xi, Xz,…

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Q: Evaluate f(322 + z)dz from z = 0 to z = i.

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Q: Use FTC I to prove that if |f '(x)| ≤ K for x ∈ [a, b], then |f (x) − f (a)| ≤ K|x − a| for x ∈ [a,…

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Q: Let f be in R[a, b]. Prove that -- f is in R[a, b] and ∫a b -- f(x)dx = -- ∫ab f(x)dx.

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Q: Let f: R" x R → R define as follows f(x, y) = then f is

A: Given that: f:Rn×Rn→R define as follows fx,y=∑xiyi¯i=1n. The objective is to determine the value of…

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is 0 a limit…

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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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