(Section 17.3, 17.5) Consider F(a, y) = (e-" cos(y))i + (e-" sin(y))} (a) Show that F is conservative. (b) Find the potential function f(x,y) for the field F. (c) Evaluate F. dr = F Tds, where C is the line segment from (-1,1) to (1, 1).

Q: Let f (x) = f₁²² sin² (t)dt. Find ƒ' (π). ○ 2π sin² (π) π sin² (7²) ○ sin² (²) ○ 2π sin² (²)

A: The given function is

Q: Show that the directiona f(x, y, z) = z²x + y ² at ( X (1/15" + ) ; 255. them

A: fx,y,z=z2x+y3 P1,1,2v→=15i+25j We know directional derivative Dv→fP=∇fP.v→v→

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A: our objective is to find the required derivative

Q: 26) If y=sin(-0).find dy d0

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Q: • The Doman of f(x) = 4-Vx is 1. (2, co) 2. [-2, c0) 3. [2, 0) 4. [-2,2]

A: The domain is the set of all values of x for which the value of function f(x) exists.   For f(x)=4-x…

Q: Let f(x, y) = cos(xev*) + cos(x + ey²) , find дхду* O a. -2ye" (sin(xe") – x cos(xev*) + cos(x +…

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Q: 7. Set f(x) (a) (b) (c) (d) = 2 +4 8 cos 2x, 2sin2, + e-8/2 s² + 4 S s² +4 8 8² +4 (e) None of the…

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Q: Use the second derivative f"(x) = x(x+ 1)(x-3) to determine where the function f(x) is concave down.…

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Q: 2. Determine the directional derivative of o(x, y, z) = re" + y² + xyz at the point (2, 0, 3) in the…

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Q: Verify that fy = fx ху ух f (x, y) = 2x'y' -x'y b. f(x,y)=(x' + y²)° а.

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Q: Express the following function in the form f(z)=u(x,y)+iv(x,y) 1. cos(z^2)

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Q: 4) Let f(x) = - ax"+1 and a x) = 4X-3 5:) f(x) = Fxan? find the 3 first f,g and h. uch that (fo gohl…

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Q: 4/(A)Find the derivative of f(x, y) = xe + cosxy at the point (2,0) in the direction. дz əz (B)If Z…

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Q: √4x+4-√√5x+1 x-3 k B) 1 CO Q18) if f(x)= 2 700 116 D)- 1*3 point A) and if f(x) is continuous…

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Q: let it be 2= f(x,y) and Y=tan' () and X = === In (r²+5²) Prove that fxx+fxy= (y² + S) (fr+fso)

A: We have to prove

Q: 5. Find the gradient of cos(xy) – 3y* + 3 at the point (1,4)

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Q: Pron a. If u = f(x, y), where r = e' cost, and y = e' sint, show that (3)³ + (3) ² - - ² [(-)* +…

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Q: (1+x² +C B [2x (1+x) dx-A)- 11 10 11 f(x)=x+cosx, xe [0,7), then the function has an inflection…

A: from graph  stationary point: point of function is a point where the derivative is equal to zero .…

Q: a3 f 2- Find the third derivative for for f (x, y) = sin əx3 -2sin?cos()+2 sin) sin By'sini )+2…

A: The given function isf(x, y)=sinxy

Q: Q2// B :- If u+v3e"cos y and u- v=e' sin y find the Jacobain of the function alu,v at dx.yl (x.y)…

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Q: 1 1 1 1 ++* 1 2 1 1 -2 1 1 Which system of ODEs matches the velocity field above? Ox'= 1, y = x…

A: The velocity field is shown in the picture. Consider, x'=yy'=1 This means: y'x'=1yydy=dx Integrate…

Q: 4) dependent Show that the following set of functions is Linearly independent or linearly f₂(x) = 3,…

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Q: iv is analytic = -11..

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Q: • The Doman of f(x) = 4 – Vx is 1. (2, 0) 2. [-2,0) 3. [2, 00) 4. [-2,2]

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Q: Suppose that f'is continubus, f(o)-o and f(5)= e So f'(x) sin?Cf(x) dx 4) Calculate

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Q: Let F = (3t + 2) i + sin(2t) j+ 3ť²k Find F'(t), F"(t) and F"(t) 1) F'(t) = (2) F" (t) 3) F" (t) =

A: Solve for the equation for the derivative

Q: Determine each of the following derivative values that involve functions that are defined in terms…

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Q: 3 prove that sinh (2) isine

A: Since you have posted multiple questions, we will provide the  solution   only to the first question…

Q: f (x) = arctan(e“) ise ƒ '(-1) =?

A: To find the derivative of  f(x)=arctan(ex) at x=-1

Q: (x³ + bx + a, xs-1 lbx? - Let f(x) = .Then the values of a and b so that the %3D -a, x >-1…

A: The function is: f(x)=x3+bx+a, x≤-1bx2-a,      x>-1 So, the function is continuous at x=-1 if…