, then the domain of (f.g)(x)is: 7) Let f(z) = VE + 1 , 0,2) g(1)--IT-6 D) I-1,0o)/12) E) None of the above.
(8) Let E C R" and G C R" be open. Suppose that f E G and g G R', so that h = go f : E → R. Prove that if f is differentiable at a point x E E, and if g is differentiable at f (x) E G, then the partial derivatives Dihj(x) exist, for all and j - ...., and 7m に! (The subscripts hi. g. fk denote the coordinates of the functions h, g....
(8) Let E c R" and G C Rm be open. Suppose that f E -G and g:GR', so that h -gof:E R'. Prove that if f is differentiable at a point x E E and if g is differentiable at f(x) є G, then the partial derivatives Dh,(x) exist, for all , SO , . . . , n, and and J-: に1 The subscripts hi, 9i, k denote the coordinates of the functions h, g, f relative to...
9. If f(x) = x2 - 6x + 2 and g(x) = -2x, find a. f.g(2) b. g•f(x)
4. Let F be a field. Prove that for all polynonials f(x), g(x), h (z) є FI2], if f(x) divides g(x) and f(z) divides h(r), then for all polynomials s(r),t() E Fr, f() divides s()g(r) +t(x)h(r).
4. Let F be a field. Prove that for all polynonials f(x), g(x), h (z) є FI2], if f(x) divides g(x) and f(z) divides h(r), then for all polynomials s(r),t() E Fr, f() divides s()g(r) +t(x)h(r).
S f(r)da= g(x)dz. Prove a,bsuch that (8) Suppose f and g are continuous functions on that there is ro e (a, b) such that f(ro) = g(xo). (9) Prove that if the function f is continuous on a, b, then there is c E [a, b such that f(x)dax a Ja f(e)
S f(r)da= g(x)dz. Prove a,bsuch that (8) Suppose f and g are continuous functions on that there is ro e (a, b) such that f(ro) = g(xo). (9)...
Please write legibly and show all work!
The goal is to prove the product rule for polynomials over a field F. Let f(x),g(x) E Fx. Prove that d )g))g) This will be done in three steps. (a) Show it is true when fx)s) are monomials f(x)-a,stx) (b) Show it is true when f(x) -as any polynomial but g(x) bx is a i-0 monomial Use your result from (a) and the proat (x)g) 1n (c) Show it is true in the...
Let f.g,h: R + R be functions. Prove that the followings is true or not. If not show an example. a) (g+h) • f = (gºf) + (hof) b) f(g+h) = (fºg) + (f • h)
Prove the following Green's identity for function.....
4. (a) Prove the following Green's identity for functions f.g E Co(2) where2C R'" where the notation : ▽ Vf n, where n is the outward pointing unit normal vector. You may use the divergence theorem, as well as the identity (b) Let G(x.xo) denote the Green's function for the Laplacian on Ω with Dirichlet boundary con- ditions, that is, 4,G(x, xo) = δ(x-xo), for x 62 (x,x;)= 0 for x Eon By...
·J (I) < 0 for all such y. (Hint: let g(x)--f(x) and use part (a)) 3. In this problem, we prove the Intermedinte Value Theorem. Let Intermediate Value Theorem. Let f : [a → R be continuous, and suppose f(a) < 0 and f(b) >0. Define S = {t E [a, b] : f(z) < 0 for allェE [a,t)) (a) Prove that s is nonempty and bounded above. Deduce that c= sup S exists, and that astst (b) Use Problem...