Let f: R→ R given by f(x) = 2a² and A = |-3, 1] U [2, 4]. Then f(A) is O 12, 36] O [0,32] O [1, 16] O [4, 32] o [1, 32]

Q: 7. Let S = [0, 1] × [0, 1] and ƒ: S → R be defined by f(x, y) (2x³+y², 10, Show that f is integrable…

A: Follow the steps.

Q: Let z=x+iy ∈ C where x∈R\{1}, y∈R, and |z|=1. If w = (z+1)/(z-1), then show that Im(w) = y/(x-1).

A: Use the following formulae, to obtain the required result. Real part of a+ib=aImaginary part of…

Q: Two functions u and v of x and y are said to satisfy the Cauchy- Riemann eguations if u, = v, and u,…

A: Thje given functions are,u(x,y)=y(x2+y2) , v(x,y)=x(x2+y2)

Q: Let W = F(x , y , z) be a function such that Fx(0 , 0 , 0) = -1 , Fy(0 , 0 , O) = 3 , F2(0 ,0,0) = -…

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Q: Find the Wronskian for the set of functions.{1, x, ex}

A:

Q: Calculus

A: see 2nd and 3rd step.

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 X₁, X₂,..., X be a r.s. from N(,0) and Y₁, Y₂...., be another independent r.s. from m…

A: Solution

Q: 2) Use the Wronskian to show the three functions are linearly independent. f(x) = x, g(x) = cosx,…

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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° :…

A:

Q: Let f:R→R be defined by f(x) : [ 2x – 1, z<2 | 2x +1, 22 , then f-'(4, 7() is None of the choices O…

A: Hello. Since your question has multiple parts, we will solve the first part for you. If you want…

Q: Theorem (6) If f is bounded and R – I on [a, b], f(x) > 0, Vx € [a, b] then ſ f 2 0

A: Prove that: If f is bounded and R-I on [a, b], fx≥0 for all x∈a, b , then ∫f≥0,

Q: (a) Let f(r, y) = -2r*y² – In(y²). Compute fyzz (b) Let g(r, y, 2) = 1(3y* – 52²). Compute

A: (a) Given function is fx,y=−2x5y2−lny2 We have to find the fyxx. First differentiate…

Q: 1. Let f(x,y, z) = vx + /y- Vz +In(49 – x² – y² – z²). Evaluate f(4, 4, 4).

A: Substitute the values

Q: 2. (a) Let f be defined for all x, y by f(x, y) x-y - x. Show that f is concave (i) by using Theorem…

A: Given: a) f(x, y) =x - y - x2 Theorem 2.3.1: This theorem says that if f is twice continuously…

Q: 3. Let C[0, 1] be equipped with the inner product (f. g) = f(x)g(x)dx, where f = f(x) and g = g(x).…

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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: of at (x, y, z) = (2, 3, 4). ay Let f(x, y, z) = (x + y)z. Evaluate

A: Given :fx,y,z = x+yz

Q: Exercise 3. Show that f : Z → N so that f (z) = z² + 1 for all z E Z is neither one-to-one %3D nor…

A: One-One Function: A function is said to be One-One if for every element in the domain there exists…

Q: prove that F(r) = 4z-3z is chaotic on the interval [-1,1]. (Hint: consider g(6) = 30 on S'.)

A: The given function: F(x)=4x3-3x The given interval is [-1,1] To prove that F(x) is chaotic: Chaotic…

Q: Find f. f "(x) = x-2, x > 0, f(1) = 0, f(6) = 0 %3D %3D In 6 (In 6) f(x) = -In(u) + u 5

A:

Q: Show that the Dirichlet function f defined on [0 , 1] by f(x) = { 1 if x is rational 0 if x is…

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Q: Prove that the function f(x1,x2) = In (e*i+= + eV=i+1) is convex over R².

A:

Q: Let f(x, y, z) = x²y°z – xyz², and compute: (a)

A: In this question we will use basic of vector  calculus i.e gradient of any function and curl of…

Q: Let A: C [0,1]→C [0,1]. Prove that operator A is continuous reversible, find A!, 2.11. Ax(1) = x(1)…

A: We have given operator A such that A: C[0,1]-C[0,1].Also we must know that the inverse operator…

Q: A function f(x,y) is homogeneous of degree 7 in x and y if and only if, __________.

A: Given problem is solved below.

Q: 4. Show that the map K: [1, o) → [1, ∞0) defined by 1 t dt K(x) = 3 e 2 is a contraction in the norm…

A:

Q: The function f(x) = x² + 1 is 2 Bijection from R to R Injection from R to R Onto and Injection from…

A:

Q: Let AC C([-1,1]) be defined by A = {f€c'(I-1, 1), f(0) = 0, f'(x)| < 1, V x€ (-1,1)} Prove that A is…

A: The set A is defined as A=f∈C1-1,1, f0=0, f'x≤1, ∀x∈-1,1. To prove that A is relatively compact.…

Q: • Exercise 2.5. Let V = C\{0} and define f : V - C by f(z) = 1/z. Then Sar f(z) dz = 0 for every…

A:

Q: az are Let F(x, y,z) = z +x+ysin x+ xIn z = 0, where z =f(x,y), then at (0,0,1) and %3D ax equal to:…

A: We have to find the partial derivatives of given function at point

Q: Let f(x,y,z) = x² e Vy*+ Compute f(3, - 5,3) and f(5,8, - 6).

A: Given that     f(x,y,z) = x2 e√(y^2+z^2)

Q: If f(x, y, z) = In(x7 + yº + z®) + (x° + yº + z®), then find fr, fy and fz at (1, 1, 1).

A: →fx=∂f∂x, During derivative of ∂f∂x, consider 'y' and 'z' as a constant→fy=∂f∂y, During derivative…

Q: Let f(x) = x² – 1 for -2 < x < 1, and let P = {-2,0, 1} be a partition of [-2, 1]. Find Uf(P), and…

A: From maximum value and minimum value in the corresponding intervals from that we can easily solve.

Q: Let f : (0, 1] → (0,1) be the function defined by f ) = „ for all n E N, and f(x) = x if æ E (0, 1]…

A:

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: Define f: Z x Z→ Z x Z by f((m, n)) = (3, 2n). Let A = {(n, 0): n ∈Z}. Fina f(A).

A: EXPLANATION : Given function  f:ℤ×ℤ→ℤ×ℤ defined by fm,n=(3,2n). given set A={(n,0):n∈ℤ}. putting the…

Q: If f (x, y, z) = x'z then f, is... Jika f (x, y, z) = x'z maka turunan parsial f, adalah... %3D…

A:

Q: Let C[0, 1] have inner product [f(x), g(x)] = ∫ f(x)g(x) dx. Find α so that (x + αx2 ) ⊥ (x + 1)

A:

Q: For x = 1, 2, 3 and y = 1, 2, let the joint pmf of X and Y be defined by f X,Y (x, y) = (x+y)/21…

A: Solution: The joint pmf of X and Y is  fX,Y(x,y)=(x+y)21     x=1,2,3    and y= 1,2  The pmf of X and…

Q: 22. (a) Prove that [0, x] = [0, y] for all x, y E Z, x + 0, y # 0. (b) Prove that [ax, ay] = [x, y]…

A: (a) we have to prove that  0, x = 0, y for all x, y∈Z, x≠ 0, y≠0 proof as x, y∈Z, x≠ 0, y≠0 Z is the…

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: A function F is defined on R² By: f (x, y) (x, y)# (0,0) F (x, y) = (x, y) = (0,0) y+2xy² x²+y2 Let…

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Q: Let F(x, y, z) - z* + x+y sin x + x In z- 0, where z- f(x, y), then at (0,0,1) and are az %3D %3D…

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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: The domain of the function f (x, y, z) = In(6 – 3xy) %3D is {(x, y) E R² : xy 2} {(x, y, 2) E R³ :…

A:

Q: 1 If f(x, y,z)= of then x- y- 3z ду

A: Differentiate wrt y.

Q: 5) Let z = f(x2 - y²). Show that y = 0 ду %3D ax

A: Here, ∂ represents the partial derivative, which means when differentiating with respect to x,…

Q: Let z = F(x , y) = xy2 + - , where x = u2 + v2 - 4 and y = uv . az Use the chain rule to find at (u…

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Q: To prove Theorem 1, (that If F is an antiderivative of f on an interval I, then the most general…

A: To prove the statement regarding the  antiderivatives of a function