Let f : [0, 1]→R be defined by f(x) :=x3 and let P :={0, 0.1, 0.4, 1}. Compute L(P, f) and U(P, f).

Q: Let f € L'(0,1). Assume that f(t)t2"dt = 0 for all n f = 0 a.e. 1,2,3,.... Prove that %3D

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Q: Let f = zy + yx, v = [y, z, 4z - x], w Find - [vª, zª, x®]. (a) grad f and f grad f at (3, 4, 0) (b)…

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Q: Suppose that F(x, y, z) = 0 defines z as a function of x and y and F is differentiable. Prove that…

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Q: Let fn(x) = E s;(x) Y; be a linear smoother. Find a condition on the weights s;(x) so that fn(x) is…

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Q: Verify that the given functions are linearly independent by evaluating the Wronskian at x = = 0.…

A: Given a set of functions e2x ,  xe2x , cosx , sinx . To verify that the functions are linearly…

Q: Consider F(x,y)=(4x^3+4y)i+(4x+6y^2)j Show that F is conservative, find a function f such that F=…

A: Given F(x,y)=(4x^3+4y)i+(4x+6y^2)j F is conservative as shown below

Q: For the function f(x, y) show that fx(0, 0) and fy(0, 0) both exist but that f is not differentiable…

A: We have given;

Q: For the function f(x,y,z) = In (5x + 7y + 1Oz), find f,, f, and f,.

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Q: Find a function v(x, y) such that, given u = 2x(1– y), f(z) = u + jv is analytic in z.

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Q: Suppose that z = f(x, y) is differentiable at the point (1024, 256) withf(1024, 256) = 2 and f,…

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Q: Find the local linearization of f(z) =z² at 9. lo(z) = %3D

A: Given : f(x) = x2   x0 =  9

Q: Let f(x, y, z) = ln(x² + z) + y²e®z – cos(2yz). Find fzyn. Zyx ·

A: Given function is  f(x, y, z)=ln(x2+z)+y2exz-cos(2yz)

Q: Let H(x) integral ,for -2 ≤ x ≤2.Use geometry to evaluate H(2).

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: 2. Let f(x)=√(2-x)². Use the local linear approximation of f(x) at xo = 1 to estimate √0.99 (1.01)².

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Q: 5) Determine the entire function f(2) = u+ jv satisfying Re(f(2)) = u(x, y) = ae" cos(y) + r? – y²,…

A: That's easy. Thumb up. Have a good day!!! We use Milne Thomsan's theorem

Q: Let F(x, y, z) = z* +x+y sin x +x In z 0, where z =f(x, y), then at (0,0,1) and %3D are equal to:

A: Given,          Fx,y,z=z4+x+ysinx+xlnz=0 , where z=fx,y

Q: Let F(x, y) = G , y 1 )- Calculate the work against F required to move an object from (1, 1) to (3,…

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Q: Let g(x) V1+ 8t dt. Use the Fundamental Theorem of Calculus to find g'(x). g'(z) =

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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: Let Y be a function of the RV X as follows: YA SX, x², x Compute fy (y) in terms of fx (x). Assume…

A: Given that fX(x)=12πe-12x2   If Y=X, then  fY(y)=12πe-12y2         =12πe-12x2

Q: Calculate of of and for the function f(x, y) defined as : ax' dy sin (xy) 1 + y f(x, y) =

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Q: If f(x, y, z) = In(x7 + yº + z®) + (x° + yº + z®), then find fr, fy and fz at (1, 1, 1).

A: →fx=∂f∂x, During derivative of ∂f∂x, consider 'y' and 'z' as a constant→fy=∂f∂y, During derivative…

Q: Let D, be the linear transformation from C'(a, b] into C[a, b]. Find the preimage of the function.…

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Q: Q = find of るf and るこ for the Ganction f(xz)-ディズ メ+Z

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Q: Find dwdt (a) by using the appropriate Chain Rule and (b) by converting w to a function of t before…

A: w=x2+y2+z2,  x=cost,  y=sint,  z=et a).…

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

A: See the attachment

Q: If f:R⇒Rf:R⇒R is defined by f(x)=e−cosxsin(x)f(x)=e−cos⁡xsin⁡(x) for every x∈Rx∈R then…

A: The given function is: f(x)=e-cos x sin x To find the derivative first. f'(x)=ddxe-cos x sin x…

Q: B. Use Green's theorem to evaluate f (x²y)dx + y²dy Where C is close path formed y = x and y = x³…

A: Above first question is solved. Solving more than one question is against the policy of bartleby.

Q: 10.Show that F(x, y) = (1 + xy)e*yi + (ev + x?e*y)j is conservative and find a function "f" such…

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Q: 45. Let f(x, y) = mxy², where m is a constant. Find a valuc of m such f(x, y)dA = 1, where R = [0,…

A: It is given that, fx,y=mxy2, where m is a constant.We have to find the value of m such that,∫∫Rfx,y…

Q: Let F(x,y,z) = z* + x+ y sin x + x In z 0, where z fx.y), then at (0,0,1) and are dy equal to: ax

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Q: Show that F = (2x - 3)i - z j + (cos z)k is not conservative.

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Q: Evaluate F.dr, where F(x,y)=[2y-3cos(x)]i + [4x +6sin(y)]j and C goes counterclockwise around the…

A: The objective is to evaluate ∫CF·dr where Fx,y=2y-3cosxi+4x+6sinyj and C goes counterclockwise…

Q: Let {yk } be the DFT of {x; }, where x = [0, –1, 0, 1] y3 = O i -1 -i 1

A: Here, x0=0, x1=−1, x2=0 and x3=1 We have to find y3 Now, y3 is given by…

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: For a function h and an interval I, the oscillation of h on I is defined by w(h, I) = sup |h(æ) –…

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Q: Suppose that z = f(x, y) is differentiable at the point (64, 1024) withf, (64, 1024) = 3 and dz…

A: Topic:- partial derivative

Q: Find that F = 12x²yi + (4x + 8yz²)j + 8y² zk is conservative, and find a function f such that F =…

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Q: Let V=P(R) and for j≥1 define Tj(f(x))=f(j)(x), where f(j)(x) is the jth derivative of f(x). Prove…

A: let n be any integer and a1,a2,...an be scalars such that  a1T1+a2T2+a3T3+...+anTn=0 where T is the…

Q: Let g(x) 8 In(t) dt. Use the Fundamental Theorem of Calculus tc g'(x) =

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Q: Let Ly = y" + py' + qy. Suppose that yi and y2 are two functions such that Lyi = f(x) and Ly2 =…

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Q: Let w f(t,0) where t and 0 are functions ofr and s, %3D dw is equal to Or then according to the…

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Q: Show that F(x, y, z) = (y² – e* sin z)i + (2xy + z)j + (y – e* cos z)k is conservative and hence…

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Q: If f(z) = sin(z 2) = U(x, y) + iV (x, y), determine U and V . Verify that the C-R conditions are met…

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Q: x + Y f(x, y, z) = x sin + e*% – 1

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Q: Let z = f(x, y) and r =r cos 0, y = r sin 0. Use the chain rule to express: af %3D 11. af dy Te and…

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Q: Use Green's theorem to evaluate f (x²y) dx + y²dy here C is close path formed y = x and y = x³ from…

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Q: Let c(t) Use the chain rule to compute (foc)'(t). (cos(t), sin(t), t²) and f(x, y, z) = x² + y² +…

A: The objective is to find the derivative.

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: