Please write neatly. 10.30 Let T E L(En, Em). Prove that T is differentiable at each...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem) 2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
Let U be an open subset of R". Let f:UCR"-R be differentiable at a E U. In this exercise you will prove that if ▽f(a) 0, then at the point a, the function f increases fastest in the direction of V f(a), and the maximum rate of increase is Vf(a)l (a) Prove that for each unit vector u e R" (b) Prove that if ▽/(a)メ0, and u = ▽f(a)/IV/(a) 11, then Let U be an open subset of R". Let...
Let a continuously differentiable function f: Rn → R and a point x E Rn be given. For d E Rn we define Prove the following statements: (i) If f is convex and gd has a local minimum at t-0 for every d E R", then x is a minimiser of f. (ii) In general, the statement in (i) does not hold without assuming f to be convex. Hint: For) consider the function f: R2-»R given by Let a continuously...
(a) Suppose the equation defines a differentiable function y-f(z) (0) Find the derivatint ()-(e,l) (ii) Write the linear approximation for f(x) around a = e and use this to approx- dy Hence, e T,y 5 marks imate f(3). markS (b) Evaluate the following limits. Simplify your results if possible. 5 marks 5 marks] lim cot 5x sin 6x cos 7a (i) (ii) limIn (a) Suppose the equation defines a differentiable function y-f(z) (0) Find the derivatint ()-(e,l) (ii) Write the...
Please write neatly Let f(x) be a Cl(R) function and u(t, x) be a solution to the inviscid Burger's IVP Ut + uuz = 0, 4(0,x) = f(x). (a) Use the formula ult, x) = f (x – ut) to derive the following explicit identity for the partial derivative Ou du Ət f($)f'(5) 1+tf'(E) with $ = X – ut. (b) Assume that there exists a number wo E R such that f'(20) < 0 and f(x) > 0. Use...
Theorem 10.1.15 (Chain rule). Let X, Y be subsets of R, let xo e X be a limit point of X, and let yo e Y be a limit point of Y. Let f : X+Y be a function such that f(xo) = yo, and such that f is differentiable at Xo. Suppose that g:Y + R is a function which is differentiable at yo. Then the function gof:X + R is differentiable at xo, and .. (gºf)'(xo) = g'(yo)...
Let V and W be a vector spaces over F and T ∈ L(V, W) be invertible. Prove that T-1 is also linear map from W to V . Please show all steps, thank you
Please write neatly! 22. Let S denote the plane 2x +y+ 3z = 6 in the first octant with the upward normal, and C denote its triangular boundary. Use Stokes' Theorem to evaluate the line integral F dr where F = <2z - x, x +y +z, 2y-x>. 22. Let S denote the plane 2x +y+ 3z = 6 in the first octant with the upward normal, and C denote its triangular boundary. Use Stokes' Theorem to evaluate the line...
real analysis 1,2,3,4,8please 5.1.5a Thus iff: I→R is differentiable on n E N. is differentiable on / with g'(e) ()ain tained from Theorem 5.1.5(b) using mathematical induction, TOu the interal 1i then by the cho 174 Chapter s Differentiation ■ EXERCISES 5.1 the definition to find the derivative of each of the following functions. I. Use r+ 1 2. "Prove that for all integers n, O if n is negative). 3. "a. Prove that (cosx)--sinx. -- b. Find the derivative...
Let V be a finite-dimensional vector space and let T L(V) be an operator. In this problem you show that there is a nonzero polynomial such that p(T) = 0. (a) What is 0 in this context? A polynomial? A linear map? An element of V? (b) Define by . Prove that is a linear map. (c) Prove that if where V is infinite-dimensional and W is finite-dimensional, then S cannot be injective. (d) Use the preceding parts to prove...