Compute / f ds over the curve specified. (1) f(x, y, z) = z², r(t) =< 2t, 3t, 4t > for 0 < t < 2

Q: The directional derivative of f(z, y,2) = r'y- v': at the point (3,2,1), in the direction of the…

A:

Q: dr. Let f(z) =z. Find from z-0 to Z-4+2í alongthe curve C given Z=21 to z=2i

A:

Q: Find the indicated partial derivative. f(r, y, :) = e"vz*; fzyz fxyz(x, Y, z) =

A: Given: f(x,y,z)=exyz3 To find: fxyz=?   Required formulas: Power rule: ddxxn=nxn-1 Product rule:…

Q: Obtain the gradient to a curve with parametnic equations given by and y= X= In 2 2t when t-1

A:

Q: Assume the second derivatives of ƒ are continuous throughout the xy-plane and ƒx(0, 0) = ƒy(0, 0) =…

A:

Q: Find the maximal value of the directional derivative of w = xy°z at the point (2, 1, 1)

A:

Q: Use Green's theorem to evaluate S. (5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

A:

Q: Compute the partial derivative of the function f(r, y, z) = e'-z cos(y) + z e=1/(1+y²) vith respect…

A:

Q: (b) A bounded, 3B > 0 such that |f(x)| < B Vx € (0,1), continuous function on (0,1) with unbounded…

A:

Q: 8. Find the directional derivative off at the p in the direction of v = () of f(x, y, z)=x²exx, p =…

A:

Q: (2) Let f(x, y) = x² – y². Find the tangent line to the level curve f(x, y) = -3 at (1,2).

A:

Q: Q4:- Find the derivative of the function at Po in the direction of the vector AC. f(x, y, z) = x² +…

A:

Q: Find the orthogonal trajectories of the family of curves: y = k (csc x + cotx) Ⓒy² = 2 (C- cos x) O…

A:

Q: 2/6 Consider the upper half of the ellipsoid f(x.y) = 1--n and the point P-- 3 on the level curve…

A:

Q: Prove that the function ((z*)2 Iz| > 0 f (z) = z 0, z = 0 is not analytic at z = 0 by calculating…

A: Given function is f(z)=z*2z, z&gt;00,        z=0 we have to prove, the above function is not…

Q: b) Find the directional derivative (D.) of the function at P in the direction of PQ f(z, y) = sin 2z…

A:

Q: Find the maximal directional derivative magnitude and direction for the function…

A: Consider the given function and point, f(x,y)=x3+2xy-cos(πy),(3,0) Now, find the derivative with…

Q: tion z2 – (x2 + y2)2 and let p = (0, 0, 0). ermine the set Tp(Zf ) of all tangent prs to Zf at p. Is…

A: Sol

Q: 8. Find the equations of the tangent plane and the normal line to y +z – x=2 at (-1, 1, 0).

A: Consider the given equation: y2+z2-x=2fx,y,z=y2+z2-x-2 Calculate the partial derivative with respect…

Q: Find the maxcimum rate of change of f at the given point and the direction in which it occurs. F(x,…

A:

Q: 1. Evaluate S. (2x+ y² – z) ds where C is the piecewise smooth curve below. (0, 1, 1) (0, 0, 0). (0,…

A:

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

A:

Q: Find the symmetric equations for the normal line LN to the surface S:f (x,y) = 5VT +y? at the point…

A: Given function is S:fx,y=5x2+y2. We have to find the symmetric equation for the normal line LN to…

Q: Find the orthogonal trajectories of the family of curves: y = k (csc x + cotx) Ⓒy²= 2 (C- cos x) O y…

A: This question is related to differential equations, we will solve it using given information.

Q: Evaluate S. (2xy – x²)dx + (x + y²) dy such that c is a curve bounded by y? – x , y – x² ?

A:

Q: 2 - What is the directional derivative of the function f(x,y,z) = x3 - 2xy +z at point Ao= (1,1,0)…

A:

Q: Let (x, y) = 3y² - 2y – 3x² + 6xy. a. Find the equation of the tangent plane at the point (1, -1).…

A:

Q: Q1: Find the derivative of f(x,y,z)=x^3-xy^2-2z at point P_o (1,1,0) in the direction of vector…

A: Solution:-

Q: Find the gradient of the function w = 1/(√1 − x2 − y2 − z2),  and the maximum value of the…

A: The given vector is:

Q: Assume the second derivatives of ƒ are continuous throughout the xy-plane and ƒx(0, 0) = ƒy(0, 0) =…

A: Consider the provided information, fxx0,0=5,fyy0,0=3  and fxy0,0=-4 The Second Derivative Test to…

Q: Compute F. dr where F (x, y) : (7xy + x cos x) i+ (-yev + 6x²) j and C is a piecewise closed curve…

A:

Q: let f(x,y)= xexy  Compute all the first and and second partial derivatives of f.  Find the tangent…

A:

Q: Compute the partial derivative: f(x, y) = sin(x® – 5y) fy(0, T) =

A:

Q: Find the derivative of the function at Po in the direction of A. f(x, y, z) = xy + yz + zx, Po(1,…

A:

Q: 1 The partial derivative of f(x, y, z) = Arctan with respect to z is xy? z³

A:

Q: 5. (а) Find the directional derivative of f at P in the direction of ā if f(x,y) = y In(x) ; P(1,4);…

A: To find The directional derivative of f at p in the direction of a.

Q: 2. Let closed curve C consist of the line segment from (0, 0) to (2, 1), followed by the line…

A: We will solve both parts of this question using basic knowledge of line integrals and definite…

Q: 3. Evaluate the gradient of S(1, y, 2) = In(cosh z+ y) - arctan| at (1,y) = (7/2, 1).

A: Given that fx,y,z=lncosh x+y-z2 arctanx2y2+z2 To find the gradient of f(x,y,z) :…

Q: Compute the partial derivative: f(x, y) = sin (x* – 6y) f„(0, 7) =U

A: Given query is to find the partial derivative of the expression.

Q: Tangent to the curve z2 2xy + x² , z = 3 , at the point ( 1, 4 , 3 )

A: Consider the curve, z2=2xy+x2 ⇒x2+2xy-z2=0

Q: 8. Find the directional derivative of f (x, y, z) = xy²z³ at P(2,1,1) in the direction of Q(0, –3,5)

A:

Q: Find the maximum rate of change of f(x, y, z) = tan(2x + 2y + 3z) at the point (−9, 6, 2) and the…

A:

Q: Assume the second derivatives of ƒ are continuous throughout the xy-plane and ƒx(0, 0) = ƒy(0, 0) =…

A: Maxima or minima: Let f(x, y) is any function. 1) Evaluate fxx, y=0 and fy0,0 for critical point. 2)…

Q: Tangent to the curve z2 = 2xy + x² , z = 3 , at the point ( 1, 4 , 3

A:

Q: Assume the second derivatives of ƒ are continuous throughout the xy-plane and ƒx(0, 0) = ƒy(0, 0) =…

A: Given that, ƒxx(0, 0) = -6  ƒyy(0, 0) = -3 ƒxy(0, 0) = 4

Q: Find the maximum rate of change of f(x, y, z) = tan(7x + 3y + 6z) at the point (4, – 5, 2).

A:

Q: (2) Find an equation for the tangent plane to the graph of z = where z = f(x, y) is implicitly…

A: Given: The function is fx,y,z=xy+xz+yz-lnz2+1-4. Calculation: Obtain thethe value of  fxx,y, fyx,y…

Q: Let F(r, y, z) = (y² + 2z, 2ry + 4, 2a), and let the curve C be described by x = t, y = t, z = t3…

A:

Q: Use Green's theorem to evaluate S. (5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

A: According to the given information, it is required to use green’s theorem to evaluate the integral.

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: