What is special about the derivatives of vector functions of constant length? Give an example.

Q: What is the gradient vector of a differentiable function ƒ(x, y)? How is it related to the…

A: The objective is to write what is the gradient vector of a differentiable function ƒ(x, y) and how…

Q: Find the maximum rate of change of f(x, y) = In(x² + y?) at the point (-5, -5) and the direction in…

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Q: What is meant by a tangent vector?

A: To Explain:  a tangent vector

Q: What is special about the derivatives of vector functions of con-stant length? Give an example.

A: What is special about the derivatives of vector functions of constant length

Q: 2. Suppose that f (1,2) = 3 and that the directional derivative of ƒat point (1,2) in the direction…

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Q: Explain why the family of vector-valued functions that are the antiderivatives of a vector-valued…

A: The family of vector-valued functions that are antiderivatives of a vector-valued function will…

Q: Calculate the instantaneous rate of change of g at the point (4, 1,2) in the direction of the vector…

A: Instantaneous rate of change of a function g in the direction of a vector v is  =∇g.vv Maximum rate…

Q: Suppose that f (1,2) = 3 and that the directional derivative of f at point (1,2) in the direction of…

A: Given : f1,2=3 and the directional derivative of f at 1,2 in the direction of the vector i+j is…

Q: 6. Use a property of derivatives of vector valued functions to show that # [F(t) × F'(t)] = F(t) ×…

A: to show ddtrt→×rt→'=rt→×rt→'' we will use  ddtu→×v→=u→ddtv→+v→ddtu→

Q: What is the derivative of a function ƒ(x, y) at a point P0 in the direction of a unit vector u? What…

A: For a function z=f(x,y): the partial derivative with respect to x gives the rate of change of f in…

Q: Evaluate the directional derivative Duf (3, 4) if ƒ (x,y) = /æ² + y² and u is the unit V vector in…

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Q: Determine whether the statement is true or false "If a particle moves along a sphere centered at the…

A: The given statements is “If a particle moves along a sphere centered at the origin, then its…

Q: An object moves with velocity vector (t,t?,-t) starting at (1,2,3) when t=0. Find the function r…

A: Since you have asked multiple questions so as per guidelines we will solve the first question for…

Q: Represent the plane curve by a vector-valued function. y = (x − 2)2

A: y=x-22 Now let xt=t Now put xt=t in y=x-22, we get y=t-22

Q: Find the directional derivative of f at P in the direction of a vector making the counterclockwise…

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Q: 3. A unit vector tangent to the curve y x2 at the point (2, 4).

A: We are entitled to solve only one question at a time so we will answer only the first question.…

Q: How do you find the tangent vector to a smooth curve at a point? How do you find the tangent line?…

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Q: Find the direction in which the maximum rate of change occurs for the function f(x, y) : 5x sin(xy)…

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Q: Suppose ƒ is differentiable at (9, 9), ∇ƒ(9, 9) = ⟨3, 1⟩, and w = (1, -1). Compute the directional…

A: Given that ∇ƒ(9, 9) = ⟨3, 1⟩. Note that w = (1, -1) is not a unit vector. Its unit vector along same…

Q: Represent the plane curve by a vector-valued function y=1-x^2

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Q: an object is moving in 3-dimensional space with velocity vector v(t)=(sin(t),t,t^2) starting at…

A: Consider the given velocity vector: vt=sint,t,t2 Find the position vector: Integrate velocity vector…

Q: a)What is the slope of the tangent line to the curve y=2sin4x  at x=⊼(pie)/4? b) Find a unit vector…

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Q: Find the maximum rate of change of f(x, y) = ln(x² + y²) at the point (3, 5) and the direction in…

A: Here is the solution of the given problem. Unit Direction:

Q: Find the derivative of f at the point P( – 3, – 2) in the direction of = ( – 1, – 1) | -(– 3, – 2) =…

A: Given, f(x,y)=-0.02x2y3   (b) The directional derivative for a function f(x,y) in the direction of…

Q: Find the directional derivative of the function f(, y) = In(x +y°) at the point (- 1,3) in the…

A: As per our guideline we are allowed to do 1 question at a time. Please post other questions next…

Q: Find the direction in which the maximum rate of change occurs for the function f(z,9)2zsin(ay) at…

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Q: What formulas are available for writing a moving object’s acceleration as a sum of its tangential…

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Q: Suppose that f(I,2) =3 and that the directional derivative of fat point (1,2) in the direction of…

A: Thr given problem is to estimate the value of f(1.1,2.1) with given values f(1,2)=3 and directional…

Q: Find the maximum rate of change of f(x, y) Maximum rate of change: = In(x² + y?) at the point (3, 1)…

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Q: Suppose that the directional derivative of a function f at a point P in the direction of the vector…

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Q: Find the two-unit vectors that are parallel to the tangent line to the curve y =x2sin(x) at the…

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Q: give two example of tangent vector on manifold

A: Manifold is a topological space that which is almost similar to a euclidean space near to each of…

Q: How do I find the domain and the derivative of the vector function

A: Domain is the set of all possible inputs for the function. So to find the domain of the given vector…

Q: The directional derivative Df is a vector parallel to u. OTrue False Reset Selection

A: We have to find that the given statement " The directional derivative Duf is a vector parallel to…

Q: Find the maximum rate of change of f(x, y) = ln(x² + y²) at the point (5, 2) and the direction in…

A: As per bartleby guidelines, for more than one questions only 1st one is to be answered. Please…

Q: (a) What is the maximum directional derivative of g(x, y) = y²e2¤ at (2, – 1) and in the direction…

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Q: Find the directional derivative of the given function at the given point in the direction of the…

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Q: The directional derivative of f(x, y, z) = xyz at point (-1, 1, 3) in the direction of vector î - 2ĵ…

A: Remark: We know that the directional gives the rate of change of a function at a specific point in…

Q: find the directional derivative of the function in the direction of the vector v

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Q: Find the unit vector u along which the rate of change of the function f(x, y) = x 2 + xy3 at the…

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Q: Find the directional derivative of the function f(x, y) = In(x +y°) at the point (2, 3) in the…

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Q: (B) At point (1,5), Find the Vector that is tangent and Normal to the curve: X' = 2y + 10

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Q: What is the physical significa

A: What is the physical significance of line integral of vector field?

Q: Find a unit vector pointing in the direction in which the maximum rate of change occurs for the…

A: Consider the formula for vector ? f (x, y) in the direction of the maximum rate of change for the…

Q: Find the directional derivative Duf(1, 1), where f(x, y) = x2 + y2 and u is the unit vector at an…

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Q: Find the directional derivative of f at P in the direction of a vector making the counterclockwise…

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Q: Represent the plane curve by a vector-valued function. y = x + 5

A: To Determine: represent the plane curve by a vector function . Given: we have a function y=x+5…

Q: Calculate the directional derivatives of the function f(x,y) = x² – y? in the direction of the…

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Q: Find the directional derivative of f(x,y,z)=yz+x^4 at the point (1,3,2) in the direction of a vector…

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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: