3.2 † Prove part (ii) of Theorem 3.1.2. Theorem 3.1.2 Let f and g be in R[a, 6} and c e R. Then i. f+g€ R[a, b) and (f + g)(x) dx = f(x) dr + | 9(x) dr. ii. cf e Rla, b] and | ef(2) dz = c f(2) dzr.

Q: Check it for f(x, y, z) = x²y°z*, that is, compute Vf first, then V x (Vf).

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Q: 4. If [æf (x) + 2] dæ = x³ – a² then find the function f (x). 3 A) 2 3 B) 2+ C) 3- 2 E) 2– D) 3+ --…

A: The given equation is x∫x fx+2dx=x3-x2.

Q: 2: find fr. fy, and fz. a f(x, y. 2) = I+ xy- 22 2

A: Given f(x,y,z)=1+xy2-2z2 To find fx, fy, fz

Q: true or false :  There exists a function F of two variables x, y such that dF = (x − y)dx + (x +…

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Q: 18. Prove that b b ²1. I 2 a 2 F(x) f(y) dy dx = ( [° F(x) dx) ². [HINT: Notice that 2 b (11(x) dx)²…

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Q: 3. Compute the value of F.dr if F = -yi + xj and C is a semicircle from (-1,1) to (1,-1).

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Q: 21. Let f(x) = 2x, x = [0, 1]. Take P = {0, 1, 1, 1, 1, 1} and set x₁=1, x₂ = 16, x3 = 3, x4={, x3 =…

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Q: Let C(x) = U{E CX: E is connected and x E E}. Show that C(x) is connected and that for all x, y EX…

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Q: Find the exact value of F. dr using any method. b. F(x, y, z)= C is the line segment from (1, 0,…

A: The given vector field is Fx, y, z=yz, xz, xy+2z. We have to find ∫CF·dr.

Q: ) Let f(x) = c) 1 x + 3 following: a) ƒ (-3) = = defined, write "undefined". Evaluate the Undefinec…

A: We have given function  fx=1x+3 (a) Put x=-3 f-3=1-3+3=10 f-3=Undefined (b) fx+h=1x+h+3

Q: The function z is defined by z(x, y) = x y-3 y Find A z and dz when x= 3, y= 5, Ax = - 0.04 and A y…

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Q: EX+ Prove that [X Ju cx)] = x'Jr-, cx) Solp 2. ヘャut 2m+2V (-1)" X 2m+V (m!)P(m+v+1). neno 2m+2U-1…

A: Property of gamma function: Γ(m+1) = mΓ(m) ............. (1)

Q: many

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Q: 8. Let g(r, 9) = (a) Show that g is harmonic in D- {(7,9) : + (y +3)2 < 4}. (b) How can the funetion…

A: Here given as gx, y=yx2+y2 ......(1)

Q: xyz 3. If f(x, y,z)= In| +e 2 find f,, f, and f..

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Q: xy* Let f (x, y) = if (x, y) # (0,0) and f (0,0) = 0. (a4+6y8)

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Q: Find the value of fææ + fyy at (1, –1) when 3 + x² + 2y?. xy f(x, y) 1. (frr + fyy)- -7 |(1,–1) 2.…

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Q: Suppose that y is a function of u, u is a function of x, and u = -1 when x = 2. If du/dx|x=2 = 3 and…

A: Solution

Q: Find div F if F(x, y, z) = (x2 + y) i + z2 j + (ey - z) k.           A . 2xy i + k          B. 2x…

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Q: QUESTION 6 Consider the function f(r, y, z) = 6x² + 4xy+ e4yz af Then is equal to: ду O A. 4x +…

A: we find out partial derivative with respect to y.   see 2nd step.

Q: af (8,9). Let f(x,y)=x?y*. Find ду

A: We are given: f(x,y)=x2y4 ⇒∂f∂y=x24y3=4x2y3

Q: Which of the following are functions from.16 ?the set of real numbers to itself fi = ((1,x): x E R)…

A: Given: The relations from set of real numbers to itself as f1=1,x : x∈Rf2=x3,x : x,y∈Rf3=x,x :…

Q: . A function f(x) has the following properties: (=) de = 4, f(2)dz = 6, f(2)dz S r(2) dz = -2, […

A: a) We divide the integral into three subintegrals

Q: Which of the following expressions are not valid? Why? (a) [ f* 1(x, y) dy dz [T (c) [[1(x, y) dy de…

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Q: Locate and classify the local extremum of f when f(x, y) 4.x2 – 6x – 6y² – 4y + 10xy. - - 1 1.…

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Q: 1 x+R+ 2)²(x+ x+ R+ 4) dx (Hint: Let u= x+R+2 x+R+4

A: ∫1x+R+22x+R+42dx

Q: Let z = 1+ iy. Determine whether f(2) is entire function or not? (Hint: Use Cauchy- Riemann…

A: Let  f(z) = u(x, y) + iv(x, y) Then the Cauchy-Riemann equations are ∂u∂x=∂v∂y and ∂u∂y=-∂v∂x

Q: dz dz and ду Suppose z is a function of r and y, and tan Vy? + x2 = z"e6y. Solve for

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Q: If f(x, y) = x²ye*xy, find fx and fyx.

A: Given function is

Q: 3. Find f. (g) f'(x) = 2x – e

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Q: dz (е) dz

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Q: (x +1)4 – y² Let f(x, y) = { (x + 1)4 + y? - if (x, y) # (–1,0) -1 if (x, y) = (-1,0) a. Show that f…

A: