(8) Given a C1-function f : Rn->M, let M (x, z) E R#x R | z- f(x)) be the graph of f. Let TpM denote the tangent space to M at a point p = (xo, 20) E M. Find TİM and compute its dimension. Hint: draw a picture.
(Problem continued) 9 (b). Let F:R ([2) R be given by F In(z +1). Find its Tavlor series up to and inchading the degree 2 term (6 marks F give rise to an inner 2 (c). Referring to the function F in part (b) above, for which values of a does the matrix A (4 marks product on R2? Show how you obtained your answer. (Problem continued) 9 (b). Let F:R ([2) R be given by F In(z +1). Find...
5. Let A = P(R). Define f : R → A by the formula f(x) = {y E RIy2 < x). (a) Find f(2). (b) Is f injective, surjective, both (bijective), or neither? Z given by f(u)n+l, ifn is even n - 3, if n is odd 6. Consider the function f : Z → Z given by f(n) = (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer
(c) Let F be the vector field on R given by F(x, y, z) = (2x +3y, z, 3y + z). (i) Calculate the divergence of F and the curl of F (ii) Let V be the region in IR enclosed by the plane I +2y +z S denote the closed surface that is the boundary of this region V. Sketch a picture of V and S. Then, using the Divergence Theorem, or otherwise, calculate 3 and the XY, YZ...
Let f : R → R , f ( x ) = x^2 ( x − 3). (a) Given a real number b , find the number of elements in f ^(-1) [ { b } ]. (The answer will depend on b . It will be helpful to draw a rough graph of f , and you probably will need ideas from calculus to complete this exercise.) (b) Find three intervals whose union is R , such that f...
Let the function f R R be given by 1,)- f 1 z-1 Draw the graph of f versus the values of z. Is f a bijection (i.e., one-to-one and onto)? If yes then give a proof and derive a formula for f. If no then explain why not Let the function f R R be given by 1,)- f 1 z-1 Draw the graph of f versus the values of z. Is f a bijection (i.e., one-to-one and onto)?...
2. Let 6 marks (a) Find f(x),f"(x), and f"(x). (b) Find the second order Taylor expansion of f at 1, namely f(r) = ao + ala-1 ) + a2(z-1)2 + R2(x), where Ra is the remainder. You should find ao, a, a2, and R(p). 8 marks that the error in this estimation (i.e., R2(0.9)1) is at most 10-3. 6 marks (c) Use the Taylor expansion found above to estimate the value of f(0.9). Show Find f(x), f"(), and f" (b)...
1. Let f : [0, 1]2 → R be given by: 1 f(x,y) -»-< if x = 0 if x + y Show that f is integrable on [0, 1]2 and compute the value of the integral.
(a) Let Ω = [4, 101 and let A = 16, 6], [8, 10]} 2. (i) Find F(A) (ii) Let X : 2->R be defined by X = 2-1[4,5]-3 . 1 (6,8) Is X, F(A)-measurable? Justify your answer. (b) Let (2, F) be a measurable space, and let X :2-R. Suppose that X+ is F-measurable. Does this imply that X is F-measurable? Either prove it or give a counterexample. (a) Let Ω = [4, 101 and let A = 16,...
This is a MATLAB question so please answer them with MATLAB steps. Let f(z) = V3z sin(#) and P(z) =r-x-1. 1. Find f(e) 2. Find the real solution(s) to Px) 0. Hint: use the roots command. 3. Find the global minimum for f(x). Hint: plot f over [0,2] 4. Solve f()P. Hint: plot f and P over [0,21. 5. Find lim,→0+ f(x). Hint: make a vector hi make a table [x + h; f(x + h)]". 6. Find '(In 2)....