2. Show that the set of all infinitely differentiable functions f:R → R is an R-module...
Show that the relation R on the set of differentiable functions from R to R consisting of all pairs of functions f, g such that f(x) -() is an equivalence relation. What functions are in the equivalence class of f(x) ?? Show that the relation R on the set of differentiable functions from R to R consisting of all pairs of functions f, g such that f(x) -() is an equivalence relation. What functions are in the equivalence class of...
vectors pure and applied Exercise 11.3.1 Let Co(R) be the space of infinitely differentiable functions f R R. Show that CoCIR) is a vector space over R under pointwise addition and scalar multiplication. Show that the following definitions give linear functionals for C(R). Here a E R. (i)8af f (a). minus sign is introduced for consistency with more advanced work on the topic of 'distributions'.) f(x) dx. (iii) J f- Exercise 11.3.1 Let Co(R) be the space of infinitely differentiable...
1) Let f:R-->R be defined by f(x) = |x+2|. Prove or Disprove: f is differentiable at -2 f is differentiable at 1 2) Prove the product rule. Hint: Use f(x)g(x)− f(c)g(c) = f(x)g(x)−g(c))+f(x)− f(c))g(c). 3) Prove the quotient rule. Hint: You can do this directly, but it may be easier to find the derivative of 1/x and then use the chain rule and the product rule. 4) For n∈Z, prove that xn is differentiable and find the derivative, unless, of course, n...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ] -» R, equipped with the inner product f(t)g(t)d (f,g) = (a) Let UC V be the subspace spanned by B = (sinr, cos x, 1) (you may assume without proof that B is linearly independent, and hence a basis for U). Find the B-matrix [D]93 of the "derivative linear transformation" D : U -> U given by D(f) = f'. (b) Let WC V...
help me. 5. consider set F(R):ff: f:R-R), but set all function with set real number in domain and codomain. Show "addition" in any two function it.eCE(R) to produce new function such as given: ttgR2R which is every xER such as given:(tg)lx)-fx)+g(x), and any real number k ER, multiply it with any element f EF(R) to produce new function as given: kfRR in every value xER such as given:(k:0(x):-kfx)(observe it with multiply dua real number) (a) Show. FIR) ith addition and...
If 3.80 fig: [a,b] → R 2 Alonspiciens differentiable functions and we suppose Fca) = f(b) =. The wronskien of these a functions is the function TW Cf. g): [a, b] R defined by wCfg) () = det (FX) 906) -F68) g'(x)=9(x)}f'(X) (f'(x) g(x)) If W (f, g) (x) #0 for all x E [a,b], show that it exist a c E Ca,b) such that g (c) = 0.
show all steps please 5. Functions f:R R: 9: R R are both one to one on the set of real numbers. Is the function f +9 also one to one? [Hint: Be creative! Make up two one to one functions.
Let V = Cº(R) be the vector space of infinitely differentiable real valued functions on the real line. Let D: V → V be the differentiation operator, i.e. D(f(x)) = f'(x). Let Eq:V → V be the operator defined by Ea(f(x)) = eax f(x), where a is a real number. a) Show that E, is invertible with inverse E-a: b) Show that (D – a)E, = E,D and deduce that for n a positive integer, (D – a)" = E,D"...
please show all work, even trivial steps. Here are definitions if needed. do not write in script thank you! 4. Letf: R2 → R2, by f(x,y) = (x-ey,xy). a. Find Df (2,0). b. Find DF-1(f (2,0)) Inverse Function Theorem: Suppose that f:R" → R" is continuously differentiable in an open set containing a and det(Df(a)) = 0, then there is an open set, V, containing a and an open set, W, containing f(a) such that f:V W has a continuous...
2. Let S be the set of all functions from R to R. For f.g es, we define the binary operation on S by (fog)(x) = f(x) + g(x) + 3x*, VX E R. (1) Find the additive identity in S under the operation . (ii) Find the additive inverse of the function w es defined by w(x) = 5x - 8, VXER [4] under the operation .