The position vector r describes the path of an object moving in the xy-plane. Position Vector Point r(t) = ti + (-t2 + 8)j (1, 7) (a) Find the velocity vector v(t), speed s(t), and acceleration vector a(t) of the object. v(t) = || s(t) a(t) (b) Evaluate the velocity vector and acceleration vector of the object at the given point. v(1) = a(1)

Q: Represent the plane curve by a vector-valued function. x2 /16−y2/ 4= 1

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A:

Q: .3 At which point P does the curve r(t) = ( 2t have the tangent vector (24, -8, 3) ? (a) P(2, 2, 2)…

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Q: 2. Find the directional derivative of o = yx2 +x²z + 3xyz in the direction of vector n = î + 2 ŷ -2·

A:

Q: The directional derivative of x,y) - 16/x - 13y at the point (11,8) in the direction of the given…

A: The direction derivative  of f(x,y) in the direction of unit vector, is defined as follows.

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Q: : Find the directional derivative of the following function, f(x, y, 2) = xy cos(10yz) at the point…

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Q: Determine the directional derivative of the function f (x, y, z) = x2 + xy + y + z2 in point P = (1;…

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Q: Show that the graph of the vector-valued function r(t) = is a circle, and find its radius and…

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

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Q: Consider the function f (x, y) = 3 + 5x2 + 2y2, which is differentiable in an environment D from…

A: Given f (x, y) = 3 + 5x2 + 2y2

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Q: Find the directional derivative of f(x,y) = x² + y² – 2x – 4y at the point (-2,3) in the direction…

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Q: 2. An object moving along a curve in the xy-plane has position (x(t),y(1)) at time t with the…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

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Q: 1. (a) Find the rate of change of the function f(x,y) = x² + 3ry at the point (3,2) in the direction…

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A: Use the gradient of function for finding the directional derivative.

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Q: =) Find the directional derivative at the point P = (2, 1, 1) of the scalar function in the…

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Q: Find the directional derivative of the function f(x, Y, z) = x + ln(y² + z²) at the point A(2,1, 1)…

A: To find the directional derivative of the function f(x,y,z)=x+ln(y2+z2) at point A(2,1,1)  in the…

Q: Find the directional derivative of f(x, y, z) = x²y−yz³ + z at the point (1,−2, 0) in the direction…

A: Solution:f(x, y, z) = x²y−yz³ + zat the point (1,−2, 0) in the direction of the vector a = 2i +…

Q: Find the directional derivative of f(x, y) = 10 − 3x^2 + (y^4)/4 at P(2, −3) in the direction of the…

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Q: #3. Consider the function xy2 if (x, y)# (0,0), f (x, y) = {x2+y2 if (x, y) = (0,0). Calculate the…

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A: Given:

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Q: Represent the plane curve by a vector-valued function. x2 + y2 = 25

A: It is given that, x2 + y2 = 25. Let x=5cost and y=sint , Then 5cost2+5sint2=25cos2t+sin2t=25

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A: This question is related to vectors, we will use gradient of function to solve this.

Q: a) What is the rate of change of f(x, y) = 3xy + y^2  at the point (2, 4) in the direction of vector…

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Q: Find a vector equation for the curve of intersection of the surfaces x^2 + y^2 = 4 and z = xy.

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Q: Find the derivative of the vector function r(t) = ta x (b+ tc), where a = (4, -1,5), b = (-3,-3,-1),…

A: Given, the vector function      r(t)=ta×(b+tc),where       a=4,-1, 5, b=-3,-3, -1, and  c=1, 2, 2,…

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A: Use the given information carefully.

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A: Solution-

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Q: Find the directional derivative of f(x,y,z) = x²z – xz³ + zy at the point (1, -2,0)in the direction…

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Q: a) Determine the gradient for the vector A = 4x'yz+ x²y²z² at point (2,-2,3) travelling %3D along in…

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Q: 9. Find the directional derivative of f(x , y , z) = 3x° +y° +z° - 9x + 9y – 9z + 4 at the point 9.…

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Q: 6. Suppose that the temperature T at point (x, y,z) is given by the function T = 2x2 – y? + 4z², and…

A: Given temperature function is: T=2x2-y2+4z2 The tiny bug is at point 1,-2,1 Now, to find in which…