Differentiable Definition and 290 Threads

This is a problem many of the grad students have probably encountered, it's in Chapter 0 of Riemannian Geometry by Do Carmo. Do Carmo proved that the tangent bundle of a differentiable manifold is itself a differentiable manifold by constructing a differentiable structure on TM, where M is a...

Hi all. Having a little trouble on this week's problem set. Perhaps one of you might be able to provide some insight. Homework Statement f:[a,b] \rightarrow \mathbb{R} is continuous and twice differentiable on (a,b). If f(a)=f(b)=0 and f(c) > 0 for some c \in (a,b) then \exists...

Suppose the real valued g is defined on \mathbb{R} and g'(x) < 0 for every real x. Prove there's no differentiable f: R \rightarrow R such that f \circ f = g.

Hi, I've been thinking about a problem in Spivak's Calculus on Manifolds and noticed that it can be proven quite cleanly if the following is true: Let g:R^n->R^n be a differentiable 1-1 function. Then we can find a point s.t. det g'(x) != 0. Geometrically this means that the best linear...

Hello, I was trying to understand Green's function and I stumbled across the following statements which is confusing to me. I was referring to the following site http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node79.html Here the author says the following "What if $ u$ is...

Homework Statement Show through direct application of definition that the function f(x,y) = xy is differentiable at (1,1) The Attempt at a Solution I know that all functions of the class C1 are differentiable and that a function is of the class C1 if its partial derivatives...

f is differentiable on (a,\infty) and \lim_{x\to\infty}\frac{f(x)}{x}=A I am trying to prove that there exists a sequence \{x_n\}, x_n\rightarrow \infty, such that f'(x_n)\rightarrow A. Any help would be appreciated.

I am trying to understand differentiable manifolds and have some questions about this topic: We can think of a circle as a 1-dim manifold and make it into a differentiable manifold by defining a suitable atlas. For example two open sets and stereographic projection etc. would be the...

suppose f is real, continuously differentiable function on [a,b], f(a)=f(b)=0 and integral f^2dx=1 show [integral(xf(x)f'(x)dx)= -1/2 over [a,b]

I have to prove x^3 is differentiable at x=4 using the definition of what it means for something to be differentiable. So I was wondering if I just have to show that f'(4) = \lim_{x \to 4} \frac {f(x) - f(4)}{x - 4} exists, where f(x) = x^3. So... f'(4) = \lim_{x \to 4} \frac {x^3 -...

[SOLVED] Is |x|^3 differentiable? Homework Statement Is |x|^3 differentiable? Homework Equations Def: \ Let \ f \ be \ defined \ (and \ real-valued) \ on [a,b]. \ \ For \ any \ x \in [a,b], \ form \ the \ quotient \phi(t)=\frac{f(t)-f(x)}{t-x} \ \ \ \ (a<t<b, \ t\neqx), \\...

I can prove derivative of 1/f(x) = -f'(x)/(f^2). But how can this be differentiable at x=0 (if f(x) is not equal to 0) as my class notes claim Thanks

Homework Statement If f is differentiable at x=0 and g(x) = [f(x)]^2, f(0) = f'(0) = -1, then g'(0) = Homework Equations MC Answers: (A) -2 (B) -1 (C) 1 (D) 4 (E) 2 The Attempt at a Solution The only thing I could think of was that if g(x) = (f(x))^2 then g'(0) = (f'(0))^2...

[SOLVED] Real analysis - show convex functions are left &amp; right differentiable Homework Statement Let f:R-->R be convex. Show f admits in every point a left derivative and a right derivative. Homework Equations A function f:R-->R is convex if x1 < x < x2 implies f(x)\leq...

prove f(x) is differentiable at x=1: f(x)=2x^2 x(less than or equal to)1 4x-1 x>1

I have to find the values of a and b in terms of c so that this function is differentiable. Attached is the problem and my work, but I think that there's an error somewhere in my attempt. Any advice?

Homework Statement Determine where the function f has a derivative, as a function of a complex variable: f(x +iy) = 1/(x+i3y) The Attempt at a Solution I know the cauchy-riemann is not satisfied, so does that simply mean the function is not differentiable anywhere?

Sometimes I've encountered functions f:\mathbb{R}^n\to\mathbb{R}^m being called N-times differentiable. What does it mean, precisely? I know that for a function to be differentiable, if is not enough that the partial derivatives \partial_i f_j exist, but instead the derivative matrix Df must...

please help me to find out the solution of this question its very simple but i am confused.the question is "CAN WE CHECK A FUNCTION WHICH IS NOT DIFFERENTABLE EXCATLY AT TWO POINT>IF YES THEN HOW WE CHECK IT" can i use sin or cos function,or is the graph is straight or not

Hello need help with this one. f:[0,1] --> [0,1] f( .x1 x2 x3 x4 x5 ...) = .x1 x3 x5 x7 ( decimal expansion) prove that f is nowhere diffrentiable but continuous. i tried by just picking a point a in [0,1] and the basic definiton of differentiability about that point...doesnt seem...

f(x)= x^4 x less than or = 2 mx+b x is greater than 2 Find the values of m and b that make f differentiable everywhere. so what i was trying to do was to find where the graph of x^4=mx where x=2 so that the mx+b function would start at where ever x^4 left off at x=2 i...

I need to know the definition of a differentiable function at a point in Banach spaces, my notes has a certain ambiguity and I can't find a book with the definition. Thanks.

I need to find an example of such a function. I know that x^2sin(1/x) is differentiable but not C^1, but I'm having trouble extending this to C^2.

Here is a piecewise polynomial function: f(x) = x^2 + 1 if x <= 1 f(x) = 2x if x > 1 I need to prove that this function is differentiable at x = 1? It's a parabola that turns into a line. It doesn't have any gaps or corners. The limit of f(x) as x approaches 1 is 2, and the limit of...

I have a problem regarding the function f (x,y) = {x*y*(x^2-y^2)/(x^2+y^2) if (x,y)!=(0,0) and f(x,y)=0 if (x,y)=(0,0). I am asked if this function is differentiable. Running it through a graphing program it looks differentiable. I know the partial derivatives of it in terms of x and y are...

:smile: For all real numbers x, f is a differentiable function such that f(-x) = f(x). Let f(p) = 1 and f'(p) = 5 for some p>0. a) Find f'(-p). b)FInd f'(0). c)If ß1 and ß2 are lines tangent to the graph of f at (-p,1) and (p,1) respectibely, and if ß1 and ß2 intersect at point Q, find the...

Alright, here is the problem. For all real numbers x, f is differentiable function such that f(x)=f(-x). Let f(p)=1 and f'(-p)=5, for some p>0 a) Find f'(-p) b) f'(0) c) If L1 and L2 are lines tangent to the graph of f at (-p,1) and (p,1) respectively, and if L1 and L2 intersect at...

How do you determine when f is differentiably from a real analysis standpoint (no graphs and calculus)? Would I simply look for a point of discontinuity? We have 4 problems on our homework assignment involving this issue and I don't see one example in my notes or the book adressing it. Here is...

"Suppose f is continuous on [a,b] and c in (a,b). Suppose f is differentiable at all points of (a,b) except possibly at c. Assume further that lim(x->c)f'(x) exists and is equal to k. Prove that f is differentiable at c and f'(c)=k" Since the lim f'(x) as x->c exists, f'(c) either equals k...

infinitely differentiable doesn't care if all the higher derivatives are zeroes (like for polynomials), it only has to be defined...correct?

closed form?? let f:u \rightarrow R^n be a differentiable function with a differentiable inverse f^{-1}: f(u) \rightarrow R^n . if every closed form on u is exact, show that the same is true for f(u). Hint: if dw=0 and f^{\star}w = d\eta, consider (f^{-1})^{\star}\eta. i don't...

Thomae function f(x):(0,1)->R f(x)=p/q x is rational number (p and q are relatively prime natural number) f(x)=0 x in irrational number show that f is not differrentiable. l can show that this function is not differentiable at rational number. But i can't sequence that is example. not...

can the line x=0 be differentiable? the slope would be infinity right? so does that mean it is differentiable?

For what values of x is │x^2-4│ not differentiable? Is there a way to solve it without looking at the graph?

The Harvey Mudd College Math dept presents the Weierstrass' function: f(x)=\sum_{n=0}^{\infty} B^nCos(A^n \pi x) as an example of a continuous function nowhere differentiable if 0<B<1 and AB>1+\frac{3\pi}{2}. Surely it converges to a continuous function if 0<B<1 regardless of the value of...

Hello again, another problem: given: a function f:[0,\infty)\rightarrow\mathbb{R},f\in C^2(\mathbb{R}^+,\mathbb{R})\\ The Derivatives f,f''\\ are bounded. It is to proof that \rvert f'(x)\rvert\le\frac{2}{h}\rvert\rvert f\rvert\rvert_{\infty}+\frac{2}{h}\rvert\lvert...

Hi, I am interested to know whether a theory exists that allows to answer the following sort of question. Does a solution of initial value problem of second order differential equation is infinitely differentiable on the set of positive real numbers? For example, 1) the solution of...

Hi I am a calc student in great need. If any1 can please help me thank u very much. Here it is For the func. f(x) = { 0 x < or/and = 0 2x +1 x > 0 Proove that f(x) is not differentiable at x=0 Also 2. A two piece ladder leaning against a wall is...

I'm only in high school, and I was wondering: Why are some functions not differentiable at certain points?

If f is a function defined by the fomula f(x)=xe^(modx), then show that f is differentiable at every point c, with f'(c)=(mod(c) +1)e^(modx) The hint that is given is 'consider separately the cases cgreater than 0, c less than 0 and c=0 To prove that f is differemtiable at...