F. Te QB 2. (10 points) Let f and g be differentiable functions of x, and...
2. Let g(x) In f(x) where f(x) is a twice differentiable positive function on (0, o) such that f(x + 1) = x f(x) Then for N 1, 2, 3 find g" N+ 2 2. Let g(x) In f(x) where f(x) is a twice differentiable positive function on (0, o) such that f(x + 1) = x f(x) Then for N 1, 2, 3 find g" N+ 2
2. Problem 2 Let g(z) be a differentiable function defined on is shown below. Also suppose that g(2)-3 realnumbers. The graph of its derivative, g'(z), g'(a) Also define the differentiable, odd function hz) on all real numbers. Some values of h(z) are given below 0 12 3 4 5 h(z 02-42 2 (a) Calculate each of the following quantities or, if there isn't enough information, explain why i. (g'(x) +2) dr i.h() da ii. (h'(z) +2z) dr iv. 8h(x) dr...
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...
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...
21 Let f and g be functions from R3 to R. Suppose fis differentiable and V f(x) - g(x)x. Show that spheres centered at the origin are contained in the level sets for f; that is, f is constant on such spheres.
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...
(10 points) Let f and g be functions such that f(0) = 2, 9(0) = 6, FO)= -12, 5'0) = 1. Find H (0) for the function () = g(*)f(*). HO) =
please explain your solution with details. 7) Let f and g be differentiable functions such that 2< f(x)<4 and 2 s g(x)< 4 for all x. a) Find good upper and lower bounds on the arc O to x 4.(5 pts) length of the graph of f(x) from x= Ax,2 494, ANS L4 . Are Cang th Say tnt 1 + We can b) Can we find a good lower bound on the length of g(x) from x 0 to...
(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...
(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....