Assume Y=1​+X+u​, where X​, Y​, and ​u=v+X are random​ variables, v is independent of X​; E(v​)=0, Var(v​)=1​, E(X​)=1, and Var(X​)=2.   Calculate ​Var(u​|X​=1), Var(Y|X​=1),​ Var(u|X​), Var(Y|X​).

Q: Evaluate the line integral. | (9x – 9y) ds C C : x² + y² = 64 | (9x – 9y) ds = C

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

Q: Find x, y e lº such that || x || ∞ = 1 = || y || = || x+y 2 || x #y.

A: As per the question we have to find two sequences x, y from the norm-space l∞ such that : ||x||∞ = 1…

Q: T(x,y, z) = x² + 2y² + z

A:

Q: If f(x,y) = x In(2y³ – 5) + y e* , find ! of and ôx ây

A:

Q: Show that ex and e−x are linearly independent in C(−∞,∞).

A:

Q: Find ∂ w/∂ x at the point (x, y, z) = (2, -1, 1) if w = x2 + y2 + z2, z3 - xy + yz + y3 = 1, and x…

A:

Q: Let f(x, y)= x'y+2x² and find f.(1,2) and f,(1,2) %3D 3x- y 1.

A: The given function is: f(x,y)=x2y+2x23x-y

Q: Solve for x, where u = (1, −1, 2), v = (0, 2, 3), and w = (0, 1, 1). 3u + 2x = w − v

A:

Q: Use the method of separation of variables to solve: (ôu ôu - = 0 in (0,1) × (0,+∞), (2) u(0,t) =…

A:

Q: If u =f(x² +2yz, y² + 2zx ).Prove that (v² – zx) + (x² - yz) ди + ди ху

A:

Q: , (х, у) (0,0) x+y f(x.y)%3D (х, у) (0,0)

A:

Q: Let u(x, t) be the solution of n. и = xt, -00 0, .2 ди (x, 0) = 0, –00 < x < ∞. u (х, 0) - %3| -…

A:

Q: The functions y1, Y2, . .. , Yn are linearly independent, if and only if W (y1, 32, ..., Yn)(to) #0…

A: If two functions are linearly dependent then the Wronskian is zero for every value of t.

Q: Show that Plx-1,y) = A(x, y) X+ y-1

A:

Q: Show that: (a) Cov(X,Y) = E[XY] – E[X]E[Y]. (b) If X and Y are independent, then fY|X(y|x) = fY(y)

A: Solution:a.Show that COV(X,Y)=E(XY)–E(X)E(Y)

Q: If (x, y) = (-V3, 1) , then find r and 0.

A:

Q: Show that the function u(x,t) = t/2e-*/4 (t > 0) satisfies u, = uxx.

A:

Q: Verify that ƒxy = ƒyx, for ƒ(x, y) = xey.

A: Given function is fx,y=xey We have to verify fxy=fyx Solution: fxy means that first we find partial…

Q: Use Cramer's rule to solve for y without solving for x, z, and w. 4x + y + z + = 7 5x + 3y z + 8 W…

A: Let's find.

Q: ) and f(x) = u(x, y) + iv(x, y) is analytic, %3D

A:

Q: Verify, if u = x + y+ z, v = x° + y° + z³ – 3ryz, w = x²+y² + z? – xy – yz – za are functionally…

A:

Q: Find the value of X and Y in given that; 1 - 2y 4 7 2 2 - 2x 4 - 5 I3D 3. 2.

A:

Q: If S(x, y)= x² +y , find f, and f, . %3D Vr

A: Fx is partial differentiation of F with respect to x.

Q: (b) If z =W, show that: x- y xy 0 =

A:

Q: verify that w xy = w yx . w= (3x - y)/(x+y)

A: We find w_x first using partial derivative with respect to x and use the quotient rule.

Q: verify that w xy = w yx . ω =(3x - y)/(x + y)

A: w=3x-yx+y Now, wxy=3x-yx+y·xy=3x-yxyx+y=3x·xy-y·xyx+y=3x2y-xy2x+y &amp;…

Q: (a) Vx E R, 3y E Z such that y > x + 110. (b) Vx E Z, 3y E N such that y > x + 110. (c) 3x E Z such…

A:

Q: Solve for x, where u = (1, −1, 2), v = (0, 2, 3), and w = (0, 1, 1). 2x − u + 3v + w = 0

A: Given u = (1, −1, 2), v = (0, 2, 3), and w = (0, 1, 1). And 2x-u+3v+w=0 Plugging the values,…

Q: im(z,v) →(0,0) F(x, y).

A:

Q: Find f, and f, for f(x,y)=y In (2x+7y).

A: F(x,y) = y ln(2x + 7y) To find : fx and fy

Q: Be z = (x-y)/(x+y), where x = (u/v), y = (v2/u), calculate ∂z/∂u and ∂z/∂v

A: The given equation is z=x-yx+y where, x=uv, y=v2u. Find the value of ∂z∂u and ∂z∂v. Here we use the…

Q: Let f (x,y) = e**ytan (x + y), find дудх

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

Q: Show that if w = f(u, v) satisfies the equation fuu + fov = 0 and Q3) (Chain Rule) a² – y? if u =…

A:

Q: Differentiate w.r.to "x" y = xsin-1x +x/(1-x3)

A:

Q: Express In x V in terms of in x, In y, and In z. 2mx-글 my-글 nz 2 Inx + 글 Iny-글 mz znx+my+nz In y 2nx…

A:

Q: Let z = 3x*y, find Vz.

A:

Q: If A = 2.x2i – 3yzj+xz²k and o = 2z – x°y, find A.Vø at the point (1, –1, 1). |

A: The gradient is calculated using the del operator and it can be represented as ∇=∂∂xi^+∂∂yj^+∂∂zk^.…

Q: 6. If u = given by x = +2, where x+y+z= 1, prove that the stationary value of u is y b a+b+c' a…

A:

Q: If f(x,v) = (y)²x ,Verify fy=fx ху

A: Given:fx,y=y2x Now, the value of fxy will be,fxy=∂2f∂y∂x=∂∂y∂f∂x=∂∂y∂y2x∂x Now, the value of ∂y2x∂x…

Q: Suppose Vf(x, y, z) = 2xyzexi + zex¯j + yex k. If f(0, 0, 0) = 1, find f(1, 3, 3). ,2 .2

A:

Q: Find f,, fy, and f, for f(x,y,z)= 5xy+ 3xz + yz.

A: It is partial derivative

Q: 8. xyzx +2y³ (z, = 1

A:

Q: Find those constants a, b, c, d such that f(z) = ax² + bxy + y² + i(x² + cxy + dy?)

A: The given function is : fz=ax2+bxy+y2+i(x2+cxy+dy2)

Q: If f(x,y,z)=exp(3x)cos(-4yz), find Vf. %3D

A:

Q: Consider f (x, y) = re63ry f-(x, y) = e63ry and fy(x, y) = %3D e63ry

A:

Q: Let f(x,y,z)=x^3 yz^2+4xy^2 z+y^2 z-2y^4 z^3 find fxyz

A: Use the following formulae, to obtain the solution.…

Q: Calculate az/ax, where xez + zeY = x + y.

A:

Q: If f = ux+v-y, g=u²+vy+ w and O(f.g,h) ô(11,v, w) h= zu -v+ vw compute

A:

Q: Determine if F(x, y) = (x – y)i + (x – 2)j is conservative.

A: The conservative nature of the vector field depends upon the curl of the given vector field.

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.