Let f(x) be a differentiable function with inverse of f(x) such
that f(0)=0 and f'(0) is not 0. Prove lim(x->0) f(x)/f −1(x)
=f'(0)^2
f-1(x) is f inverse of x
Let f(x) be a differentiable function with inverse of f(x) such that f(0)=0 and f'(0) is not 0. Prove lim(x->0) f...
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 f be defined on an open interval I containing a point a (1) Prove that if f is differentiable on I and f"(a) exists, then lim h-+0 (a 2 h2 (2) Prove that if f is continuous at a and there exist constants α and β such that the limit L := lim h2 exists, then f(a)-α and f'(a)-β. Does f"(a) exist and equal to 2L? Let f be defined on an open interval I containing a point a...
Q3 (Prove that P∞ k=1 1/kr < ∞ if r > 1) . Let f : (0,∞) → R be a twice differentiable function with f ''(x) ≥ 0 for all x ∈ (0,∞). (a) Show that f '(k) ≤ f(k + 1) − f(k) ≤ f '(k + 1) for all k ∈ N. (b) Use (a), show that Xn−1 k=1 f '(k) ≤ f(n) − f(1) ≤ Xn k=2 f '(k). (c) Let r > 1. By finding...
Let f(x)={user user = { x 8. Prove the following 10 a. Prove lim f(x) = 0 b. Prove lim f(x)=1 c. Prove lim f(x) does not exist. 1-2
Part 1: Part 2: 1 2/2 Calculate lim +0 ( + ) Let f(x) be a differentiable function such that lim f(x) = 00. 200 In(f(x)) (a) Compute lim 10 f(x) e-2f(x) (b) Compute lim 2400 arctan( f(x)) –
differentiable function and there exists 0 <A < 1 (6) Suppose that f : R" -> R" is a such that |f'(x)|< A, for all x E R". Prove that the function F(x)= x - f(x) maps R" one-to-one and onto R". (Suggestion: Use the Contraction Mapping Principle Why not use the Inverse Function Theorem?) differentiable function and there exists 0
Suppose that f is twice differentiable function where f(0)=f(1)=0. Prove that strategy Suppose that f is a twice differentiable function where f(0) = f(1) = 0. 1 Prove that f f"(x)f (x) dx a. Using part a, show that if f"(x) = wf (x) for some constant w, then w 0. Can you think of a function that satisfies these conditions for some nonzero w? b. strategy Suppose that f is a twice differentiable function where f(0) = f(1) =...
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...
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
4. Let f be a differentiable function defined on (0, 1) whose derivative is f'(c) = 1 - cos (+) [Note that we can confidently say such an f exists by the FTC.) Prove that f is strictly increasing on (0,1). 5. Let f be defined on [0, 1] by the following formula: 1 x = 1/n (n € N) 0, otherwise (a) Prove that f has an infinite number of discontinuities in [0,1]. (b) Prove that f is nonetheless...