Find f, and f, and evaluate each at the given point. f(x, y) = arctan (2, -2) f,(x, y) = f,(x, V) = f,(2, -2) = f,(2, -2) =

Q: Consider F and C below. F(x, y) = x² i + y² j C is the arc of the parabola y = 4x² from (-1, 4) to…

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Q: Integrate f(x,y) = x + y over the curve C: x2 + y? = 9 in the first quadrant from (3,0) to (0,3).

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Q: Show that each function in satisfies a Laplaceequation. ƒ(x, y, z) = 2z3 - 3(x2 + y2)z

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Q: (a) f"(x) = e* – 2sin(x), f(0) = 3, f(T/2) = 0 (b) f"(t) = Vt – cos(t), f(0) = 2, f(1) = 2 -

A: The formula used: (i)∫ex dx=ex+c(ii)∫xn dx=xn+1n+1+c(iii)∫sin x dx=-cos x+c(iv)∫cos x dx=sin x+c

Q: A) Use the chain rule to find where w'sy',x= sint + cost y = cost - sint. B) If f(x. y) - xy?+ x'y ,…

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Q: Show that each function in Exercise satisfies a Laplaceequation.ƒ(x, y) = e^-2ycos 2x

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Q: Find the linearizations of the functionsat the given points. ƒ(x, y, z) = √2 cos x sin ( y + z) at…

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Q: Evaluate | f(x,y) ds where Cis the portion of the curve y = In x from the point (1,0) to the point…

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Q: Let f(x) = x³/2. Determine the length of the arc swept over x E la, b).

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Q: Find f, and f,, and evaluate each at the given point. f(x, y) = arctan A, (2, -2) | fx(x, y) = %3D…

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Q: Find the arclength of y = 2x3/2 on the interval x is greater than or equal to 1 and less than or…

A: Equation of given curve is y=2x32 Differentiate with respect to x. dydx=232x32-1=3x

Q: Given: vector-valued function R(t) whose derivative R'(t) = (4t, 6t, —12t) (c) Determine the…

A: Here is the solution of the given problem.  To solve this problem we use simple concept of…

Q: Let C be a portion of a curve from (2,0) to (2,2). Let F(r,y) = (tan(x²) + e²v¸ 2re²v

A: Introduction: A vector field is conservative if ∂P∂y=∂Q∂x, Where f(x,y)=Pi+Qj.

Q: Find the arclength of f(x)=x on [0,1].

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Q: Find the path of a heat-seeking particle placed at point P on a metal plate whose temperature at (x,…

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Q: Use the Chain Rule to find az/as and az/at. as az at || || z = arcsin(x - y), x= s² + t², y = 3 -…

A: Given z = arcsin(x-y)x = s2 + t2y = 3 - 4st

Q: Find the arc length of y = -6x + 2 on [2,5] (use calculus). %3D L = 1 + f'(x)² dx

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Q: Show that each function satisfies a Laplace equation. 1.ƒ(x, y, z) = 2z3 - 3(x2 + y2 )z 2. ƒ(x, y) =…

A: For a function f(x,y,z) to  satisfy a Laplace equation, it has to fulfill the following criteria.…

Q: Using an appropriate linearization, the local linear approximation of f(x)%3D+ cos r at is 2.

A: To determine the linear approximation of the function.

Q: Find the linearization L(x,y) of the function f(x,y) = e (X cos (5y) at the points (0,0) and 0,- 2

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Q: Evaluate d:/dx on the surface x: + + x²y®=² = 3 at the point (1, 1, 1).

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Q: Express the curve by an equation in x and y given x(t) = sin(t) and y(t) = 8 + cos² (t). a) O-x² +y…

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Q: Find the Laplace of f(t) = (e-3t - e5t )2

A: To Find L(e-3t-e5t)2: Expand (e-3t - e5t )2: (e-3t-e5t)2=e-3t2-2(e-3t·e5t)+e5t2…

Q: Determine whether the functions y, and y, are linearly dependent on the interval (0,1) y =1-2 sint,…

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Q: alu,v at dx,yl Q2// B :- If u+v=e"cos y and u- v=e" sin y find the Jacobain of the function (х,у)…

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Q: Let f(x) = x3/2. Determine the length of the arc swept over r E (a, b).

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Q: Given: vector-valued function R(t) whose derivative R'(t) = (4t, 6t, —12t) (c) Determine the…

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Q: Find f, and f, and evaluate each at the given point. y' f(x, y) ), (8, -8) = arctan fx(x, y) = f,(x,…

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Q: Using an appropriate linearization, the local linear approximation of f(x) n+ cos a at is %3D

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Q: Let c be the path c(t) = (2t, t, log(t)), defined for t > 0. Find the arc length of c between the…

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Q: 3 4. Find the arc length of the graph of the function f(x) =;(x² + 1)ī over the interval [0, 1]. =…

A: y =f(x) =23(x2+1)32f(x)=23(x2+1)32f'(x)=23*32(x2+1)12(2x)f'(x)=(2x)(x2+1)12 The Formula for arc…

Q: 2. Let F(x,y) = (cos(x – 2y), –2 cos(x – 2y)). Find curves C and C2 so | that, F. dr = 0 F. dr = 1

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Q: Consider a curve given by Ř(t) = (2 cos t, 12t + 3, -2 sin t). Let f(t) be a function such that f(1)…

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Q: Find the linearizations L(x, y, z) of the functions at the given points. ƒ(x, y, z) = tan-1 (xyz) at…

A: To find the linearization Lx,y,z of the function fx,y,z=tan-1xyz at 1,0,0

Q: Find fa, fy, and fz. f(x, y, z) = 5z ln(x³y³ cos(z)) fx = fy fz f₂ = =

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Q: Given: vector-valued function R(t) whose derivative R'(t) = (4t, 6t, -12t) (a) If R(0) = (-1,4,-5),…

A: We will use integral to find original position vector and arc length of given curve

Q: Find f, and f, and evaluate each at the given point. y' f(x, у) ). (8, -8) = arctan fx(x, y) = f,(x,…

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Q: x = Asin(wt+B); w a parameter, not to be eliminated %3D

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Q: 50. w = sin x + y²z + 2z, arcsin(t – 1), y = t³, z = 3 %3D

A: Since you have posted multiple question in this question which are not interlinked so I have solved…

Q: The length of the curve y = In(cos x), 0< x <:

A: To determine the length of the curve, y=lncos x, 0≤x≤π4.

Q: Let f(x, y) = 4e + arctan () Match the following.

A: Given,  fx,y=4exy+arctanyx…

Q: Find the linearizations L(x, y, z) of the functions at the given points. ƒ(x, y, z) = tan-1 (xyz) at…

A: Consider a function: fx, y, z=tan-1xyz……1   Given point, say A, is: A=1, 1, 1   On comparing A with…

Q: Evaluate fy and f, at the given point. f(x, y) = e³Y sin(x), (T, 0) fx(T, 0) fy(T, 0) %3D

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Q: x + Y f(x, y, z) = x sin + e*% – 1

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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: The function w=xycosz, x=t,y=t2,c=arccost. Then we have to find dwdt(a) by using the appropriate…

Q: 24. Evaluate . (x – y) dx + (x + y) dy counterclockwise around the triangle with vertices (0, 0),…

A: Given integral is, It can be represented using Green's Theorem as,

Q: Find the length of one arch of the cycloid x= 6(t-sin(t)) y= 6(1-cos(t))

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Q: Consider the given curve: R(f) = (2 cos f, 12f +3,-2 sin f) Reparametrize the curve using arclength…

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Q: Find the arc length of the graph from A to B. f(r) = r/2 +1 24 22 20 18 16+ 14+ 12+ 10 8. (4, 9) 6.…

A: the arc length L=integration A to B of √(1+f'(x)^2)dx.

Q: Given: vector-valued function R(t) whose derivative R' (t) = (4t, 6t, -12t) (b) Reparametrize R…

A: Let's reparameterize vector value function R(t).