Let f be differentiable function then then the quantity Vf · û where û is a unit vector is maximized if û =

Q: sketch the space curve and find its length over the given interval. Vector- valued function:…

A:

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

A: the vector field is  g(x,y,z)=(x+4y+3z)32  point (2,1,1) to find the directional derivative in the…

Q: Find a unit vector in the direction of maximum increase of f(x, y, z) = x2 + xz + 2yze* + z? %3D at…

A: Given f(x,y,z)=x2+xz+2yzex+z2

Q: Find a function f(x, y, z) such that Vf is the constant vector (3,2, 8). (Use symbolic notation and…

A:

Q: Assume that f : R^3 → R is differentiable. Prove that the directional derivative at x in the…

A: Solution of given problem is followed by

Q: Use the parameter x = t to find a vector-valued function for the space curve represented by the…

A:

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: Q3: ( (a) Prove that div(u x w)= w•curl u-u curl w functions. where u and w are vector

A: As per our guideline we are supposed to answer only one asked question.kindly repost other question.…

Q: 6.Consider a function f: R2 R, where f(xy) = cos(xy). Let u = be a unit vector. Use the definition…

A: Here we will use Definition of Directional derivative of f(x , y) in the direction of vector <u1…

Q: Find a function f(x, y, z) such that Vf is the constant vector (4, 5, 3). (Use symbolic notation and…

A:

Q: Let f(x,y,z) = x2 y−xy2+z3, P(1,1,2), u is the vector from P to Q(1, −2, −2). Determine the maximum…

A:

Q: - (a x r) = a x r' for any constant vector a. dt Show that

A: To prove: ddt(a×r)=a×r'. Consider L.H.S. Differentiate with respect to t using product rule.…

Q: Sketch the space curve. Vector-Valued Function r(t) = -ti + 6tj + 5tk 2 -2 5 0 y Z Z 2 5 Find its…

A:

Q: Integrate the function f(r,y, z) = Over the entire 3D coordinate system, i.e., all of R'.

A: We have to integrate the function fx, y, z=1x2+y2+z21+x2+y2+z2 over the entire 3D coordinate system,…

Q: (b) Identify the unit vector u pointing in the direction of maximum increase of the function f(x, y)…

A:

Q: Find the directional derivative of f at point p in the direction of vector u. Where Z-X f(x, y, z) =…

A:

Q: If r(t) is a vector-valued function that is continuous on the interval [a, b), then r(t) dt is a…

A:

Q: Find a vector-valued function whose graph is the indicated surface. The plane x + y + z = 2 r(u, v)…

A: Given plane    x+y+z = 2                 this gives   z = 2-x-y            Now, let  x=u    and…

Q: Find a function f(x, y, z) such that Vf is the constant vector (7,5, 4). (Use symbolic notation and…

A:

Q: Find a function f (x, y, z) such that ∇f is the constant vector〈1, 3, 1〉.

A: It is given that, ∇f=<1,3,1>   It is known that a vector ai^+bj^+ck^ can also be represented…

Q: Suppose an extremal value of f occurs when x = x*, y = y*, 1 = X* (we'll call (x*, y*, A*) an…

A: In the question it is given an extremal value of the function

Q: Let R be a differentiable vector-valued function such that R'(t) = (3t2, 2tvt2 + 1, 2t + 3) and R(1)…

A:

Q: Find a function f(x, y, z) such that Vƒ is the constant vector (9,5, 2). (Use symbolic notation and…

A:

Q: Sketch the space curve. Vector-Valued Function r(t) = -ti + 2tj + 6tk ** kk Find its length over the…

A: Given problem:- rt=-ti+2tj+6tk , [0,1]

Q: Consider the vector-valued function. (a) Show that is continuous at t = -1. (b) Evaluate (R. RY(2).…

A: Like a scalar function, features like continuity and differentiability are also demonstrated by…

Q: Determine the interval(s) on which the vector-valued function is continuous. (Enter your answer…

A: Given problem:- rt=13t+1i+1tj Determine interval(s) on which given vector valued function continuous…

Q: use the parameter x=t to find a vector-valued function for the space curve represented by the…

A: Given surface x2 + z2=4 and x - y=0   and the parameter x = t

Q: Let f : R2 → R is a differentiable function at point a. (a) Let w, v ∈ S1:= {(x, y) ∈ R2 | x2 + y2=…

A:

Q: Find the domain of the vector-valued function. (Enter your answer using interval notat r(t) = F(t) +…

A:

Q: Let f : R" → R be a continuously differentiable function, while æ € R" is its non-stationary point…

A:

Q: 2. (a) Prove that the function f(x, y) = xy is differentiable everywhere in its do- %3D x² + y?…

A: TO solve the question, we use the following facts (a) A functions is differentiable at a point if…

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

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

Q: Sketch the space curve. Vector-Valued Function r(t) = -ti + 2tj + 6tk O 5 5 5 * * kk 5 Z Find its…

A: We have to sketch the vector and find the length of curve.

Q: Let z = f(x, y) be a function whose graph passes through the point (a, b, c). There is fi(a, b) = ||…

A: Use directional derivative  concept to solve

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

A:

Q: If r and u are differentiable vector-valued functions of t, then r(t) - u(t)) = r'(t) - u'(t). O…

A:

Q: Find a function f(x, y, z) such that Vƒ is the constant vector (9,7,7). (Use symbolic notation and…

A:

Q: ATA, B curi(A x B) = A (div B) – B (div A) + (B-V) A - (A.V) B, are two differentiable vector point…

A:

Q: (1) Give an example of a function z = f(x, y) and a unit vector i such that f(0,0) = 2, fy(0,0) = 1…

A:

Q: a) Find the directional derivative f (x, y, z) = x In(y+z) at point P (2, e, e) in the direction of…

A: We have to find directional derivative of vector f in the direction of vector v:

Q: Determine the interval(s) on which the vector-valued function is continuous. (Enter your answer…

A: Here ,              We have given a vector valued function now we have to determine the intervals on…

Q: 3. (True/False) If f(x, y) has a local minimum and is differentiable at (a, b), then f(a, b) = 0 for…

A: We have to check given statement true or false

Q: Assume that f : R³ → R is differentiable. Prove that the directional derivative at x in the…

A:

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

A:

Q: Exercise I f(1, y)yz (a) Let f(r,y) = rỷ - . Then deduce that f(1,y)zy (Clairaut's theorem). %3D

A:

Q: Let f (x,y,z) = x2 + 2y - z and let P be the point (x , y , z) = (1 , 1 , 1). (i) Find the Minimum…

A:

Q: xY (a) Prove that the function f(x,y) = ,2+ y? is differentiable everywhere in its do- main. (b)…

A: Here we use the facts that (a) A function of two variable is differentiable at a point if its first…

Q: Find the directional derivative of the function f(x, y, z) = xYz in the (5, –3, 2). direction of…

A:

Q: find the open interval(s) on which the curve given by the vector-valued function is smooth.…

A: given   r(θ)=(2cos3θ)i+(3sin3θ)j

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.