Suppose a particle travels along the surface z = f(x, y), where f: R² →R is a differentiable function. The (x, y)-coordinates of the particle isgiven by a differentiable path c: [0, 1] - → R². Let to € (0,1) andsupposec(to) = (1,3), d' (to) = (-1,4), Vf(1,3) = (3,1), Vf(-1,4)= (0, 1)Find the velocity vector of the particle at time to.Show Transcribed TextMultivariable CalculusPlease, I need step by step, thank you.

Suppose that the particle travels along the surface where be a differentiable function.The…

Answered: Suppose a particle travels along the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Given a function 0(r, y, 2) = 3x + In (3y + 1) + e" cos z. i. Find the directional derivative of o at (0, 1, 7) in the direction of a = 4i – 2j + 4k. ii. Find the minimum directional derivative of o and its direction at (0, 1, 7).
35. Evaluate F.dr: (a) F = (x+z)i +zj+yk. C is the line from (2,4,4) to (1,5,2). (b) F= x²i+zsin(yz)j + y sin(yz)k. C is the curve from A4(0,0,1) to B(3,1,2) as shown below. B A (c) F = yi-xj+zk. C is the circle of radius 3 centered on the z-axis in the plane z=4 oriented clockwise when viewed from above. (d) F = 4x³i + (x+y)j. C is the curve y=sin(2x) from (0,0) to (7/2,0). (e) F = (-y³ + sin(x²))ī + (x³ − ln(y² +1)) j . C is the circle of radius 5 centered at (0,0) in the .xy- plane oriented counterclockwise.
13. Find an equation for the tangent plane of the graph of f at the point (xo, yo, f(xo, yo)) for: (a) ƒ: R²2 → R, (x, y) → x − y +2, (xo, yo) = (1, 1) (b) (c) ƒ: R² → R, (x, y) ↔ x² + 4y², (xo, yo) = (2, -1) f: ƒ: R² R, (x, y) → xy, → (xo, yo) = (-1, −1) (d) f(x, y) = log (x + y) + x cos y + arctan(x + y), (xo, yo) = (1, 0) -2 (e) f(x, y) = √√√x² + y², (f) f(x, y) = xy, (xo, yo) = (1, 1) (xo, yo) = (2, 1)
Let f(x, y, z) = 36xz - 14y² and let C be the curve given by r(f1) = (1, (Use symbolic notation and fractions where needed.) [ f ds = , f) for 0 ≤ 1 ≤ 2. Compute fcf ds.
Let f(x,y)=(x^2+y^2 +1)^1/2. Find fx(0,0) and fy(0,0)(if they exist),and decide if f(x,y) is differentiable at the origin.
Find that F = 12x²yi + (4x³ + 8yz?)j+ 8y² zk is conservative, and find a function f such that F = Vf, and use the result to evaluate , F •dr, where C is any curve from A = (0, 0,0) to B = (-3,2, 3).
Suppose F(x, y) = (x² + 3y, 7x − 7y²). Use Green's Theorem to calculate the circulation of F around the perimeter of the triangle C oriented counter-clockwise with vertices (6, 0), (0, 3), and (-6, 0). Jo F. dr =
3. (b) Let f(x, y) = log((ry - 2)²), u =(³, 4). i. Calculate the directional derivative Vaf (3, 1). ii. Determine the unit vector (direction) u for which Vaf (3, 1) is maximal.
Which of the following is the equation of the tangent plane to the surface z=x2 - x+3y² +y at the point P(1, 1,4) ? (a) x+7y =6 (b) x+7y-z=4 (c) 3x+5y=8 (d) 3x+5y-z= 4 (e) x+7y+z=12
5. Let f(x, y) = xln(x/y)+ xy² a. Calculate f and fx b. Determine the directional derivative off at the point (2, 2) in the direction of the vector a=i-2j c. Suppose that x = ste' and y=2se¹. Calculate af Ət d. Determine an equation for the tangent plane to the surface z = f(x, y) at the point (2, 2, 8) on the surface.
a) Compute the work done by F = (y², sin(z), x) along a straight line from (0,3,0) to (1, 0, 1). b) Compute the work done by F = (y², sin(z), x) along a triangular path from (0, 3, 0) to (1, 0, 1) to (0, 0, 2) and back to (0, 3, 0). c) Compute the work done by = (x, y², sin(z)) along a circular path of radius 3, centered at (0, 1, 0), in the plane y = 1, counterclockwise when viewed from the origin.
Let A- (a, a2 a3} and B= (b, bz b; be bases for a vector space V, and suppose b, = 2a, - 4az, b,= -a, + 4az, by =a, +az + 3a,. %3D %3D a Find the change of-coordinates matrix from B to A b. Find (x, for x=b, - 2b, + 2b,

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