сервале) is a сontinuous Junction on lhe following mиst be true (то что из низ 0AM)? re is a number c in the open interval (a, b) such that f (c) re is a number c in the open interval (a, b) such that f' (c) re is a number c in the open interval (a, b) such that f (a)

Q: The weenay demand for petroeum of a 1ocal gas stanion in thourana itens ie given by: flo) = 48 ( x-)…

A: Given: X is the demand of gas per week in 000 fx=48×x-0.51-x    0.5<x<1

Q: of of Find and for the furction f(xy,z) = x y+ y + xỉ ôx 'ay

A: For the function f( x, y, z ) = x2y + y2 + xz2 . We need to find some partial derivatives.

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A: Given that z is a differentiable function of x and y and  x=ccosh u cos vy=csinh u sin v We will…

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A:

Q: Determine the largest interval (a,b) for which Theorem 1 guarantees the existence of a unique…

A: Determine the largest interval (a,b) for which theorem 1 guarantees the existence of a unique…

Q: Chain Rule with one independent variable Use Theorem 15.7 to find the following derivatives.

A: To find:

Q: Suppose that z = f(x, y) is differentiable at the point (1024, 64) withf, (1024, 64) = 4 and f,…

A:

Q: Prove that: cscz- cot z = 1 2. 2. %3D

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Q: 21 (important) Find the fixed point for F(x) = ax + s. When is it attracting?

A: Given Fx=ax+s

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Q: 10. What is the work done moving a particle from x = 0 to x = 1 m if the force acting on it is F =…

A:

Q: f(x) is increasing and bounded on [0, 1], then f(x) is Riemann-integrable on [0, 1].

A:

Q: Find a potential function for F or determine that F is not conservative.〈F = z, 1, x〉

A: F is conservative when curl F = 0 Given that f=(z, 1, x)=zi+j+xk Now evaluate curl F, curl F = ∇ × F…

Q: dz; |z – i| = 1 (z – i)3

A: We have to use the Cauchy integral formula to evaluate the following integral ∮ ez2z-ι3dz over the…

Q: Prove that af +y. -yof=nf (x, y). əx (Hint: Differentiate both sides of the definition of a…

A: Given  a) x∂f∂x+y∂f∂y=nf(x,y)b) x2∂2f∂x2+2xy∂2f∂x∂y+y2∂2f∂y2=n(n-1)f(x,y)

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A: a) y'=yx·lnx     We know that,   ln(1) = 0    Hence,  y'  is discontinous at x = 1  Hence,…

Q: find a potential function for F or determine that F is not conservative. F=(z,x,y)

A: F is conservative when curl F = 0 Given that f=(z, x, y)=zi+xj+yk

Q: 2. Let F = 3.xyz i – 4xyz j + 2xyz k . Find div F and curl F. -

A:

Q: Prove thut:f focK, 2) Y is a contimans findion then, g=XxX-> YxY 3 3 n K2) = (F(), PU=) Einction is…

A:

Q: Find the absolute extrema of the function over f(x, y) = x² + 2xy + y², R = {(x, y (x, y, z) = ( )…

A:

Q: It G acts set tdansikivey then show that y Salgdnap Hof 0n A by to the actioh af en A fusthus then…

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Q: dz - Prove that : Iz%3D2 %3D z%3D2 z+1 2in.

A:

Q: 4. UMNIA If the function h: [a, b] → R is bounded, Riemann integrable and satisfies h(x) > 0 for all…

A: Here given h:a, b→ℝ is bounded and Riemann integrable and hx≥0 for all x∈a, b, Then we have to show…

Q: Using the Cauchy's inequality find a bound on "(0), when flz) = e* and C |z| = 1.

A:

Q: Suppose that z is a differentiable function of x and y with dz (4, 7) = 4. ду dz (4, 7) = 60 and- ax…

A:

Q: Determine the set of points at which the function xyz f(x, y, z) 4x2 + 5y2 - is continuous. 1. { }…

A:

Q: Let f: [a, b]→R be a non-negative function and so that C ≤ f(x)for all x ∈ [a, b], where C ∈ R is a…

A: we have been given the mapping from f: [a,b]→R and is a non-negative function s.t C≤f(x) ∀x where  C…

Q: If fis integrable on [a, b], then a. f(x) dx = S-rx) dx c. - f(x) dx = S"f(x) dx b. F(x) dx = - ,…

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Q: whelher Y+ vist diverges caer Convavayes Determine 一 tby using. test ist grat 2y [Inv

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Q: Let F(x, y, =) = (6x° In(5y² + 9) + 4=5) i + ( 10yx + 7:)j + (20xz4 + 7y – 37 sin 72) k 5y2 + 9 and…

A: To evaluate

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A: Given that: A string is stretched between points x=0 and x=5. At a time t=0, the elastic medium is…

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Q: bot A de groutne conitant. est Cheafuntis uh that f cark3 and f'4)=-s. g(xl=In (f(x)) + A conx)

A:

Q: Exercise 5 3x + ax + b x² +1 Conşider the function f that is defined over IR as: f(x) = Where a and…

A:

Q: Solve x > 0, t> 0, x > 0, u (0, t) = f (t), t> 0, Ut = Urx; u (x, 0) = 0, u (x, t) is bounded for…

A: Given that,ut=uxxx>0, t>0u(x, 0)=0u(0, t)=f(t)u(x, t) is bounded for all x and t

Q: Osx 5. Use the Second Fundamental Theorem to find : S V4- t3 dt (7 pts)

A: follow next step

Q: Compute , F · dS for F = (sin(y), sin(z), yz), S being the rectangle 0 < y < 3,0<z< 6 in the…

A: Given :- ∬S F. dS for F = (sin y , sin z , yz) S being the rectangle 0≤y≤3 , 0≤z≤6 in the yz- plane…

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A:

Q: Suppose that z = f(x, y) is differentiable at the point (27, 81) withf,(27, 81) = 2 and f, (27, 81)…

A: Given that:                            z=f(x, y)fx(27, 81)=2fy(27, 81)=-6x=t3y=t4

Q: 3- Verify the properties (HINT Use integration by parts) F[ux] = iF[u] F[uxx] = -²F[u]

A:

Q: Consider the function f(x, y) = y? + x² – 3z + 8 on the closed triangular region with vertices (0,…

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Q: Let f:R^2 --> R^3 be given by f(x1,x2)=(e^(x1+x2^2, x2+cos(x1,x2), x1^2+x2^2+2) Compute the…

A: Using jacobian matrix,

Q: a) Consider IVP that is y' y/(x In x), y(1) = 0. What does the Picard-Lindelöf theorem imply for the…

A:

Q: Use Hölder inequality to prove |lulla < ||u||3||u||;-1 for u e L" (N) 1-2 Where p <q <r and p r

A: It is given that u∈LrΩ where, p≤q≤r and 1q=λp+1-λr.

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A: To find the work done

Q: Please just do 3(b).

A: (b) To prove that the given function is not Riemann integrable using Lebesgue;s theorem

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A:

Q: Show that f (z) = e? is not analytic using Cauchy-Riemann equations.

A:

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.