4. Integrate the function f(z) = z² + 2z from z = -i to z = 1 in two ways, once along a straight line segment and once along a segment of the unit circle.

Q: Find the linear approximation at x = 0 to Write your answer in the form y = Ax + BỊ.

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Q: The function f defined by the rule f(z, y) = In(xy) increases fastest in the direction of T. F (1,…

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Q: 2. Show that the function u = p(xy) + Vry (y/x) provides the equation y². 0. dy?

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Q: Find the local linear approximation to f(x) = Vx at x = 1 and use it to approximate both 1.1 and…

A: To know which one is better we have to compare with calculator values of sqrt(1.1)  and sqrt(0.1)

Q: integrate ​f(x,y)=x2−y over the curve​ C: x2+y2=4 in the first quadrant from (0,2) to 2,2.

A: The solution are next step

Q: dy from t dx n for y as a function of x, and find two results are consistent by expressing the (a)…

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Q: Find the linearization of the function f(x,y) = V9x² + 16y² at the point (1, 1), and use it to…

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Q: - Sva cos 2t dt from x= 0 to x= Find the length of the curve y =

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Q: 2. Let f(x, y) = In(x² + y²). Show that fre + fyy = 0. %3D

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Q: A:- Find the orthogonal trajectories for functions In -1) = - COS X +a y

A: The given family of curves is: lny-1y=-cos x+a ,            where a is an arbitrary constant.

Q: Suppose that u and v are differentiable at x = 2 and that u(2) = 4, v(2) = -9, u'(2) = 5, and v'(2)…

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Q: Find the linear approximation L (x) to ƒ (x) = xcosx at a = T.

A: Given that, f(x) = x cosx and we have to find linear approximation L(x) at a = π

Q: Find the linear approximation to f(x) = cos(x) near a =- [Where L(x) = f(a) + f'(a)(x – a) ] 3

A: Solution: The objective is to find the linear approximation

Q: 2. Show that the function u = (xy) + JTY v(y/x) provides the equation y² : 0. dy?

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Q: Find the derivative of the given function at P. (3, 3, 3) in the direction of u = 3i + 4j f (x, y,…

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Q: Verify the final value theorem of f(t) =1+et[sint + cost]. O- 1 1

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Q: Find the linear approximation at x = 0 to Write your answer in the form y = Ax + B.

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Q: 4. Use the linearization of f(x, y) Vx + 2y to approximate f(3.01,2

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Q: For the function f(x, y) = (cos x)ey +v 1 + x + y. %3D Show the process of using the linearization…

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Q: Find the linear approximation of f(x) = In r at a = 1 and use it to estimate In(1.14). L(2) = %3D…

A: Please refer the attached image for complete solution.

Q: 3. Find the interval on which the curve of y= 2 1+3t+t² dt is concave up.

A: Second derivative is positive.

Q: Find the linearization of the function f (x) = x° 26 at æ = 1. a vā |a| sin (a) L (x) = Using this…

A: I am going to solve the given problem by using some calculus to get the required result.

Q: Consider the For at function y= In(x+1) . what valves" of x is y undefined ?

A: The given function is y=ln (x+1) We know that, for the function ln x, x>0

Q: Find the linear approximation of the given function at (0, 0). f(x,y) = sqrt(y+(cos(x))^2)

A: The given function fx, y=y+cos2x The linear approximation in 2 variables is given by Lx, y-fa,…

Q: Find the linear approximation of the function at the given point: f(x, y) = 2 sin x – 3 cos y at (0,…

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Q: (8) Find the linear approximation of the function f(x) = Va at a = 25 and use it to approximate the…

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Q: The tangent line to the curve y=f(x) is horizontal at where X = Xo f f(xx) = 00 A O f(xo) = 00 .8 O…

A: The tangent line to the curve y=f(x) is horizontal at where x=x0 if

Q: Q1: Use Green's theorem to evalute f. (xy + y2)dx + x²dy by the curve y = x and y = x2.

A: Given problem:- Use Green's theorem to evaluate ∫c (xy + y2 )dx + x²dy  by the curve y = x and y =…

Q: The function f(z) = 1 has a poles at z = %3| ez – 1

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Q: Find the linearization L(x) of the function at a. f(x) = sin(x), a =. 3 L(x) =

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Q: (b) Show that the function f(z) = ez, z = 0 and f(0) is not analytic at Z = = 0, although C-R…

A: The given function is fz=e-z-4=1e1z4 To show: f(z) is not analytic at z=0 and f satisfies C-R…

Q: 1. Integrate the function f(x, y) = x – y over the triangle T with vertices (0,0), (1,0), and (1,2).

A: Given triangular region is bound by the vertices 0,0, 1,0 and 1,2. The sides of the given triangular…

Q: Find the quadratic approximation at (0,0) for f(x,y) = ln(1+x^2 + y^2).

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Q: Supposed that z is defined implicitly as a function of x and y by the equation x2 + y2 +z2 = 4. Find…

A: The given data is: x2+y2+z2=4 To find the value of dzdx

Q: Let f(x,y)= 3+cos(x) sin(2y) a) Show that the function has a stationary point at (0,Pi/4). b) Is (0,…

A: To analyze the critical point (0,pi/4) of the given function of two variables

Q: 27. Find the length of the curve Vcos 2t dt y = from x = 0 to x = "/4.

A: The given curve is We know that by Second Fundamental theorem of calculus ,if F(x) is continuous on…

Q: find ∇f at the given point. ƒ(x, y, z) = x2 + y2 - 2z2 + z ln x, (1, 1, 1)

A: Given function is fx,y,z=x2+y2-2z2+zlnx and point=1,1,1

Q: Find the linear approximation 0 to Write your answer in the form y = Ax + B. at x =

A: Explanation: The Linear Approximation should be a line that can substitute (in a narrow interval)…

Q: 2. Show that the function u = p(xy) + Vxy v(y/x) provides the equation = 0. dy? -

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Q: Consider the function y defined implicitly by 3(x2 + y2)? = 100xy. Use the linear approximation at…

A: Given : 3(x² + y²) = 100xy

Q: Find the linear approximation of f(x) = Vx at xo = 27.

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Q: Integrate from f(x) = x4 from 0 to 1 by (Trapezoidal Rule) with h = 1, h = 0.5, h = 0.25 and…

A: Here, f(x)=x⁴ So, I=  Integrate from f(x) = x⁴ from 0 to 1. Now we can solve by Trapezoidal rule,…

Q: Find an equation of the tangent llne to the về at the given poln 6x y = (x + 1)2' (0, 0)

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Q: 2 Integrate the function f(2) = (z - 10i) from z = 0 to z = 5 + 5i along the line x = y.

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Q: 5) a) Compute Ko for the function f0=V+X by using the de finition of the deniratire, b) Find the…

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Q: Integrate the function f (z) = (z - 8i)² from z = 0 to z = 4 + 4i along the line x = y.

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Q: Find the linearization of the function f (x) = =° at z = 1. L (1) = Using this linearlization, ƒ…

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Q: Let W=x*yz.compute the differential dw.

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Q: 18. The equation of the tangent line to 2² sin T # 0 f(x) = 0 x = 0 at r = 0 is 1 (a) y = -x (b) y =…

A: Note:Since you have asked  multiple questions ,we will solve first question for you.To get remaining…

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: