2. Let T: P₁ (R) → R2 be the map given by T(f) = (f(1),f" (1)) where P₁ (R) is the set of polynomials of degree 4 or less and f"(1) is the evaluation of the third derivative of f at 1, i.e.. d.r.³ =1 (a) Show that T is a linear map.

Q: 1) y%= (ans.: max.(1,0.5); min.(-1,-0.5)) 1+ x in (2 3-50,8))

A:

Q: (2) Given the following level curves for an unknown function f(x,y), 9 25 Estimate the following.…

A: (a) Estimate f(3,0): To estimate (3,0)f(3,0), we need to look at the level curve that passes through…

Q: Calculate the gradient vector Vf of the function f(x, y) = x² – x + y- x²y – 2y2 at the point (2,1)…

A: According to the given information, it is required to determine the gradient vector of the given…

Q: Find the gradient of f(m, n) = m³ +n³. grad f = i +

A:

Q: Determine which of the following pairs of functions are linearly independent. NO_ANSWER 1. f(0) =…

A: The linear dependency of functions can be evaluated using the value of the Wronskian of those…

Q: 6. Find the gradient of f(x,y) = ,x' - u°. grad f i +

A: We will find the first order partial derivatives with respect to x & y and use the definition of…

Q: Explain why, when u = ⟨1, 0⟩ in the definition of the directional derivative, the result is ƒ(1a,…

A: Formula:  Du f(a) =limh→0 f(a+hu)-f(a)hu=1, 0…