x+y If Z = x.n. f (***), then show that: X. Zx + y.Zy = z

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

A:

Q: 8. If f is continuous and f(x) dx 10, find xf(x²) dr. %3D Give proper reasoning.

A:

Q: Is the function continuous on the interval (0, T COS x O no O yes O cannot be determined

A: Given problem:- Is the function ex/cosx continuous on the interval [0, π] ?

Q: 3. If f is continuous and | f(x) dx = 10, find af(x*) dr. Give proper reasoning.

A:

Q: If f(x, y, z) = excosy + z, show that fxx + fyy + fzz = 0

A:

Q: Find Tn centered at x=8 for all n for the function f(x) = ex

A:

Q: If f(æ) = S Tm dt for x > 1, 1 | In t find f'(2).

A: Please refer the attached image for complete solution.

Q: Let f(z) - 4z-1 pnd alz) - ar+b, where a pnd b are positive constants. If the composite

A:

Q: Find fry for f(x, y) = In(x – y).

A:

Q: 25 If f is continuous and f(x) dx = 8, find xf(x²) dx.

A: given ∫025f(x)dx=8∫05xf(x2)dx=?

Q: If A(x) = f(t)dt is increasing on [0,10], then f(x) must be increasing on [O,10]. True False

A: Given

Q: 2. Let f(x, y) = In(x² + y²). Show that fre + fyy = 0. %3D

A:

Q: 4. Choose and work either a) or b), NOT BOTH. (a) Show that the function f (x) = sin z for x €R is…

A: In the question given that we have to solve any one of them. Let us solve the first one.

Q: As the fun ction Leay) xy js Continuous I+ x*+y? e (-1, 2)

A:

Q: Suppose that f(1) = 1, f(4) = 6, f (1) = 7, f '(4) = 5, and f " is continuous. Find the value of xf…

A:

Q: Suppose that f(1) = 4, f(4) = 5, f '(1) = 6, f '(4) = 3, and f " is continuous. Find the value of xf…

A:

Q: 43. Let f(x, y) = Is it possible to define f(0, 0) so x² + y2 * that f will be continuous at (0, 0)?…

A:

Q: Suppose f(x) is continuous and increasing on [a, b], and concave up on (a, b). Is Lo (the Left-Hand…

A:

Q: Verify Cauchy's MVT for f(x) and f'(x) in [4,9] where f(x) = Vx %3D

A: Cauchy's MVT : If two functions f(x) and g(x) are continues on [a, b] and differentiable on (a, b),…

Q: Let F (x) = %3D In tdt then F' (x) = et3

A: To find F'(t)

Q: Let f(x) = CX– 2, x > I lx+ 8, x < 1 For what value of c is f(x) continuous on (-00, 0)

A:

Q: Let g(x) = kf(x) for all x E/= {x: asxsb}. Show that Jglx)dx = kJ f(x)dx if k<0.

A: To prove the above result we use the definition of lower and upper Riemann integrals. For the proof…

Q: If f =x-e" and F=x'yi-2y'zj + xy'z'k, find: div(F x grad(yf )) at (4,1,-2)

A: The function                     Then, we have to find

Q: 25 If f is continuous and f(x) dx = 12, find xf(x?) dx.

A:

Q: Let f(x, y) = − sin2 (2x − y) Evaluate fx and fy at (π / 4, π / 3)

A:

Q: Suppose that f(1)=8, f(4)=−4, f′(1)=−3, f′(4)=2, and f′′ is continuous. Find the value of ∫from 1 to…

A:

Q: 5. Use Green's Theorem to evaluate f. $ (y² – 4x) dæ – (2+ x²y²) dy where C - - is shown below. y…

A:

Q: Suppose that a continuous function g satisfies: 9(z)dr = 42 and 9(2x)dz = 12. | z9(32°)dz. Find

A:

Q: 3. If f(x, y,z)= In *yz +e tar's="), find f,,fy and f. . хуz

A: we have to find the partial derivative

Q: Q4: integrate the following function f(x,y) for x from z to 2n and for y from 0 to f (x, y)=…

A:

Q: 12. Consider x²y x4 + y? f(x, y) : for (x, y) # (0, 0) and zero otherwise. Is f continuous at (x, y)…

A: For the solution follow the next step.

Q: Show that the function f(x) = csc(x) is not uniformly continuous on (0, pi/2]

A:

Q: Suppose f(r) = (In z)e. What is f'(x)?

A: d/dx (f(x)g(x))= f(x)g'(x)+g(x)f'(x)

Q: If ƒ is continuous on [a, b] and, what can you conclude about ƒ?

A: the integral given to us is ∫ab|f(x)|dx=0

Q: continuous at (C

A:

Q: If fCxl is continuous on [2,7],then Si f(u) dvz f (7)-(2) True e or false?

A: we have given that f(x) is continuous on[2,7].

Q: If f is continuous on [a, b] and cE[a, b], then f(x)dx does not exist. Select one: True False

A: Topic:- definite integration of continuous function

Q: If f is continuous on [0, 1], prove that f₁¹ f(x) dx = f₁¹ ƒ(1 − x)dx.

A: This question can be solved using properties of definite integrals.

Q: Suppose that  f(1) = 4, f(4) = 5, f '(1) = 6, f '(4) = 5, and f ''  is continuous. Find the value of…

A:

Q: Let f(x,y) = xcosy – ye*y. fry at (1,0)is equal to: -1 -5

A: fx is partial derivative with respect to x fy is partial derivative with respect to y

Q: Let f = f(x, y), x = s +t and y = s - t. Show that (9- ) - se se as at ax ду

A: It is given that, f=f(x,y), x=s+t and y=s-t. Show that, ∂f∂x2−∂f∂y2=∂f∂s·∂f∂t.…

Q: 1 dt, for x > 4. In(t) Let F(x) 4

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

A:

Q: - For what value of b is Sx, bx?, x2 -2 x<-2 х, g(x) = continuous at every x?

A:

Q: If z=f(x,y), x= r cos θ and y= sin θ, where f is a differentiable function expressing az/ax and…

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Q: The limit of f (x, y) : as (x,y) approaches (0,0): %3D O Infinity O Doesn't exist Exists and equal…

A:

Q: Suppose f is differentiable at (3, 4), V f(3, 4) = (-V3, 1), /V3 and u = Compute D„f(3, 4). 2

A: ∇f3, 4=-3, 1, u=32, -12

Q: If ƒ(x) is continuous [-00,b]then on ſf(x)dx= lim [ ƒ (x)dx а. a a

A: Explanation of the solution is given below...

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: