Let f: A>B and frove/disprove whe ther d g:B>6 both be Surgec hive funa gof is Surgec tiven

Q: Let f(z, y) = zy-y+a?. D.((1,2) =? where u=. OA- O B. -37 Oc- 37 OD.

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Q: (c) Suppose that o is differentiable. Prove the identity | øVø •n ds : (OV²ø+ V¢ • V¢) dA

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Q: Let f(æ, y) axy + az?y+ y³. (a) div(grad(f)) = (b) Find a so that div(grad(f)) = 0 for all æ, y. %3|…

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Q: (b) f(x) = where A; E S™ – (det(Ao + x1A1 + ...+ xn An))/m, on {x | Ao + ¤1A1 ++xnAn > ► 0},

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Q: Let а d f e h i Assume that det(A) = –11, compute the following. NOTE: Enter all values exactly. (a)…

A: det(A)=det(AT) det(kA)=kndet(A) ; where 'n' is the order of the matrix. det(A-1)=1det(A)

Q: Let F(t) = * F(e)dt (dont include the «C) Fine Question Help: D Video Add Work

A: We know, ∫tndt=tn+1n+1 ∫-sin-tdt=∫ddt-cos-t=-cos(-t)

Q: If f'Cx) ="e find all wrical auedy or honilantal Ao

A: Here we have to find the vertical and horizontal asymptote of the function f'(x)=-e1xx2

Q: Ql: Let f(x, y) = x + y,0 < x < 1,0 < y < 1. Find fi (x), f2(y), Cov (X,Y).

A: As per our guidelines we are suppose to do only one question  Given function isf(x+y)=x+y ;…

Q: 3. Let F lt) be the antider¡vatPve of f (t) the most general antiderivative of flt>= 8 sec?(t) -92

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Q: Evaluate f(x, y, z) dV for the specified function f and box B. f(x, y, z) = z3; 13 x < 8, 1 S ys 5,…

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Q: Let A = {x E Z: 8x} and let B={rEZ:12lx}. Then * ACB ВСА None of these A=B

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Q: Suppose that the only Införmation we have ab8ut a fuRcEiOH FIS y= f'(x) (a) Use a linear…

A: Given: f(1)=-7

Q: a)-What is fof g ofh 3)? fari= gal= bprove f:R¬R fer )=4lis not Surgective and is not injective.

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Q: EX: Verify that z= I+ V3i satis fies the 2. eq.

A: the given equation is z2-z+1=0 given the value of z is z=1+3i2 substitute the value of z in the…

Q: Use the convolution theorem to obtain f(t) if : 3.1) F'{F(jw)} = %3D 7+ 8iw-w?

A: Note: Here, we shall find  F-1Fjω=F-167+8iω-ω2=ft using the convolution theorem. (∵If…

Q: dy let y = xtan z then is da elect

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Q: Use the Fundamental Theorem of Calculus Part I to compute F'(1) given •8 Fle) - [V V7+t2 dt.

A: We use fundamental theorem of calculus

Q: a) Show that F is conservative while G is not. b) Evaluate GdR, where C3 is defined by R(t) = tî +…

A: This is a problem of vector calculus.

Q: we Gtokes theos em. to pvaluate Fel8 whes F=Z?E t y?; +XK and cf1,0,0) (0,1,0) andlogo, i) nr…

A: By using the stokes theorem we have to evaluate ∫CF·dr where F=z2i+y2j+xk and C:1,0,0,0,1,0, and…

Q: a ppose det = 1. Com и [a+x а — х let b+y b — у c+z с — 2 ) 1 א

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Q: f(z) =z2 is discontinuous in C

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Q: f(z) dz where a inside r= Z-a 2ni O 2nif(a) O 10

A: Using Cauchy integral Formula:

Q: Compute (g(x)), where f(x) and g(x) are the folla opute (g(x)), where f(x) and g(x) are the tell dx…

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Q: Q: If f(x)= ax* + withE a, > 0, b >0 +b,x + b, then show that Lt f(x)=o ifn>m. %3D

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Q: Let f(x ,y, z) = x²yª + z³ and x = 4s?, y = 2st?, and z = 3s?t. %3D %3D

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Q: suppore that f(x.v,Z) = x*y +x2 and x cu,v) = sincu)t(os (u) Y(u, v) = Sincu) - (oslW find of oflav…

A: Given: f(x,y,z)=x2y+xzx(u,v)=sinu+cosuy(u,v)=sinu-cosu To determine: ∂f∂u,∂f∂v

Q: Consider the function f(a) = 5x° + 7x7 An antiderivative of f(x) is F(x) 2x – 6. = Ax" + Bx" + Cx +…

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Q: Find Sea)ax Jf(x)dx if = 56 foR X<to (x fok xZ6 flx): %3D

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Q: Find and d?z and prove that 2 0²z IF 2 = f(xt ay) + F CX-ay) %3D

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Q: (i) Find af/Əx, and f/ayəx of the following functions. (a) f(xy) = x?y + x³y? (b) f(x,y) = 10x + In…

A: Partial differentiating f(x, y)  first order and second order

Q: Let f(x, y) = axy + ax²y + y³. (a) div(grad(f)) : = (b) Find a so that div(grad(ƒ)) = 0 for all x,…

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Q: 3 d The integral dxcan be rewritten as f f (u) du by making the substitution u = 4 2³ 4x³ + 3. o…

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Q: Let f(x, y) = axy + ax?y+ y. axy + ax'y+ y³. (a) div(grad(f)) (b) Find a so that div(grad(f)) = 0…

A: (a) Find the gradient of f.

Q: Exercise 3. sin(3z) Let f(z) and C: |z – n| = 1. Then S. f(z)dz = z²(z-n) а. О b. 2ni с. — 2пі d.…

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Q: 2. Determine if each of the statements below is true or false. If a statement is true then prove it.…

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Q: O: Given a r.v. Q O Compute VCY) using amig f of y (A(4),

A: Solution

Q: f 20f * f af If V²f(r,0) Find f(r,0) such that r dr v²S(r,0) = 0

A: Given:                          ∇2fr,θ=∂2f∂r2+2r∂f∂r+∂2f∂θ2-∂f∂θ To do:…

Q: If f(z) = 6tan(z) , then f'(r) = (A) In(6) 6tan(2) (B) sec2(z) 6tan(2) (C) In(6) sec2() 6tan(z) (D)…

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Q: Apply appre printe method to work ant the optimal oution f given moblem. Maximige GNo.5 z = 3x+22 +3…

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Q: Let f(x, y) = y? – x? and g(x, y) = e²+y. Verify that div(ƒVg) = fV²g+ VƒVg.

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Q: Consider the function f(z) = 728 + 10z5 – 2r2 – 6. An antiderivative of f(x) is F(x) = Ar" + Bx™ +…

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Q: Let f(x, y) = xe Ty", find fr, fy, fræ, fay and fyy-

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Q: 10. Let T: R → R² be define by T(x, y) = (x+2y,-3x+5y).

A: Recall: Let T:ℝn→ℝm is said to be one to one if u and v are the vectors in ℝn and T(u)=T(v) then…

Q: Let f: X Y and suppose S, {A;}ieI C P(X) and T, {B;}jeJ c P(Y). Show that cn. iel (i) f

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Q: . If z = f (x, y), where x = e" - e and y = e - e", show that TM UT az x: az az az dy ax UTM UTM UTM…

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Q: Q4 * If f(z) = z? ;z = x + iy Show me u(x, y) ;v(x, y) Determine ux; Vy|

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Q: Let z = f(x, y) = In(x² + y²), when x = 2, y = 3, and dx = 1, dy = 0.5, then %3D %3D %3D dz = Select…

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Q: (b) Compute f er+uds, where c(t) = (t, 2t), t e [1, 2]. %3D

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Q: Q/ z = x + f(a) and a = ry. Show that: 尝-端=

A: We use the definition of partial derivatives to prove the given result.

Q: f(z) =z? is discontinuous

A: f(z)= z2 is a polynomial function in z and polynomials are continuous everywhere.

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!