Differentiability Definition and 197 Threads

Homework Statement I have an upcoming math test, and these are from the sample exam. I'll post my solutions as I go along. I've submitted this post as is and am going to edit in my attempts. A few of these are "verify my proof is rigorous" others are "i've no idea what I'm doing 1 Using simple...

Homework Statement Given each of the functions f below, describe the set of points at which the Fourier series converges to f. b) f(x) = abs(sqrt(x)) for x on [-pi, pi] with f(x+2pi)=f(x) Homework Equations Theorem: If f(x) is absolutely integrable, then its Fourier series converges to f...

We have a corollary that But I wonder can we prove a function is not differentiable by showing that f_{x} or f_{y} are not continuous? i.e. is the converse of this statement true? By the way, are there any books have a proof on this corollary? Most of the Calculus book state the...

2.b) f is continues in [0,1] and differentiable in (0,1) f(0)=0 and for x\in(0,1) |f'(x)|<=|f(x)| and 0<a<1 prove: (i)the set {|f(x)| : 0<=x<=a} has maximum (ii)for every x\in(0,a] this innequality holds \frac{f(x)}{x}\leq max{|f(x)|:0<=x<=a} (iii)f(x)=0 for x\in[0,a] (iii)f(x)=0 for...

I've been thinking about this for a while... sorta. If a function of two or more variables is differentiable at some point, does this imply that all its partial derivatives are continuous at that point?

Homework Statement differentiability is a tough word to spell. F(x,y) = (x^2 + y^3)^{\frac{1}{3}} Find F_y (0,0) The Attempt at a Solutionhttp://www.wolframalpha.com/input/?i=D[%28x^2+%2By^3%29^%281%2F3%29%2Cy] But I get 0/0 I found the answer to be F_y (0,0) = \frac{\mathrm{d}...

in many programming languages there is a function of two variables called BitXor (which is also known as nim-sum, since it is used in solving de nim game) which represents each number as a string of its binary digits and then takes the Xor of each pair of terms, forming a new number. For...

Homework Statement if g:[-1,1] -> Reals is differentiable with g(0) = 0 and g(x) doesn't equal 0 for x not = 0 and f : Reals -> Reals is a continuous function with f(x)/g(x) ->1 as x->0 then f(x) is differentiable at 0. Homework Equations The Attempt at a Solution I took...

Quoted from a book I'm reading: if f is any function defined on a manifold M with values in Banach space, then f is differentiable if and only if it is differentiable as a map of manifolds. what does it mean by 'differentiable as a map of manifolds'?

I have a matrix A, which contains only positive real elements. A is a differentiable function of t. Are the eigenvalues of A differentiable by t?

If all the first partial derivatives of f exist at \vec{x}, and if \lim_{\vec{h}\rightarrow\vec{0}}\frac {f(\vec{x})-(\nabla f(\vec{x}))\cdot\vec{h}}{||\vec{h}||} = 0 Then f is differentiable at \vec{x} Note: Its the magnitude of h on the bottom. First of all, I don't...

Homework Statement Let g:R->R be a twice differentiable function satisfying g(0)=g'(0)=1 and g''(x)=g(x)=0 for all x in R. (i) Prove g has derivatives of all orders. (ii)Let x>0. Show that there exists a constant M>0 such that |g^n(Ax)|<=M for all n in N and A in (0,1). Homework Equations...

For months I have been staring into this expression, and I cannot visualize what the hell omega represents... f(x)-f(x0)=f'(x0)(x-x0)+\omega(x)*(x-x0) Where \omega(x)(=\omega(x;\Deltax)) is a continuous function in point x0 and equals zero in that point or lim, as x approaches x0 of...

Homework Statement If a function satisfies g'(x) = lim(h->0) {[g(x+h)-g(x-h)/2h}, must g be differentiable at x? Provide a proof or counter example Homework Equations From the formal definition of differentiation, I know that g'(x) = lim (h->0) {[g(x+h)-g(x)]/h} The Attempt at a...

Homework Statement Differentiability- Okay, so I understand that a function is not differentiable if there are either: A. A cusp B.A jump C. f(x) DNE D. Vertical tangent E. Pretty much if there isn't a limit there is no derivative which means its not differentiable. How would one find the...

Homework Statement proof at 0,0 g(x,y) is differentiable Homework Equations notes says i have to write in the form fx(0,0)\Deltax + fy(0,0)\Deltay + E1\Deltax + E2\DeltayThe Attempt at a Solution i compute fx(0,0) = 0 and fy(0,0) = 0 but what's the E talking about? what am i trying to do...

I study Calculus by myself, and I tried to solve the following question. I got the answer, but is my solution consistent? Thank you in advance. 1. The problem statement Let f be a function such that |f(x)| ≤ x² for every x. Show that f is differentiable in 0 and that f'(0) = 0. 2...

I'm doing a little self study on complex analysis, and am having some trouble with a concept. From Wikipedia: "In mathematics, the Cauchy–Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential...

Homework Statement Argue that if y=f(x) is a solution to the DE: y'' + p(x) y' + q(x) y = g(x) on the interval (a,b), where p, q, and g are each twice-differentiable, the the fourth derivative of f(x) exists on (a,b). Homework Equations The Attempt at a Solution Its a general...

Let F(x,y,z) be a function which is defined in the point M_0(x_0,y_0,z_0) and around it and the following conditions are satisfied: 1. F(x_0,y_0,z_0)=0 2. F has continuous partial derivatives in M_0 and around it 3. F'_z(x_0,y_0,z_0)=0 4. gradF at (x_0,y_0,z_0) != 0 5. It is known that...

Homework Statement Hi I'm currently trying to revise for a Calculus exam, and have very little idea of how to do the following: Let f be defined by f(x,y) = (y+e^x, sin(x+y)) Let g be of class C2 (twice differentiable with continuous second derivatives) with grad(g)(1,0) = (1,-1) and Hg(1,0)...

Hi all, I am having a little trouble understanding one of the concepts presented in my calculus class. I do not understand how the endpoints of an open interval can be differentiable. My teacher says that the endpoints of a closed interval can not be differentiable because the limit can not...

Homework Statement "A real valued function, f, has the following property: \left|f\right| is differentiable at x=0 Prove that if we specify that f is continuous at 0, then f is also differentiable at 0." Homework Equations Since \left|f \right| is differentiable we know the...

Homework Statement [PLAIN]http://img261.imageshack.us/img261/1228/vectorcalc.png Homework Equations The Attempt at a Solution f({\bf a+h})-f(\bf{a})={\bf c\times h} + \|{\bf a+h} \| ^2 {\bf c} - \|{\bf a}\| ^2 {\bf c} f({\bf a+h})-f({\bf a}) = {\bf c} \times {\bf h} + {\bf...

Homework Statement Suppose that f is differentiable in some interval containing "a", but that f' is discontinuous at a. a.) The one-sided limits lim f'(x) as x\rightarrow a+ and lim f'(x) as x\rightarrowa- cannot both exist b.)These one-sided limits cannot both exist even in the sense of...

Homework Statement [PLAIN]http://img261.imageshack.us/img261/1228/vectorcalc.png Homework Equations The Attempt at a Solution I have the definition but what do I do with f({\bf a+h})-f(\bf{a})={\bf c\times h} + \|{\bf a+h} \| ^2 {\bf c} - \|{\bf a}\| ^2 {\bf c} ?

Homework Statement So I am to prove that cosine is continuous on R and differentiable on R. I already proved it for sine which was simple by using the identity of sin(x +- y)=sin(x)cos(y)+-cos(x)sin(y) Now I need to prove it for cosine and also we cannot use the identity of...

Homework Statement Let f be the function defined as ƒ(x)={ lx-1l + 2, for x<1, and ax^2 + bx, for x (greater or equal to) 1, where a and b are constants. Homework Equations A) If a=2 and b=3, is f continuous for all of x? B) Describe all the values of a and b for which f is a...

Homework Statement Show that f(x,y) = |xy| is differentiable at 0.Homework Equations The Attempt at a Solution I thought absolute value functions are not differentiable at 0?

Homework Statement I have some doubts about the demonstration of the differentiability. If I'm asked to proof that an average function is differentiable on all of it domain, let's suppose its a continuous function on all of its domain, but it has not continuous partial derivatives. How should I...

For a function ƒ defined on an open set U having the point X:(x1,x2,...,xn) and the point ||H|| such that the point X + H lies in the set we try to define the meaning of the derivative. \frac{f(X \ + \ H) \ - \ f(X)}{H} is an undefined quantity, what does it mean to divide by a vector...

Consider the real function f(x,y)=xy(x2+y2)-N,in the respective cases N = 2,1, and 1/2. Show that in each case the function is differentiable (C\omega) with respect to x, for any fixed y-value. whats the strategy for proving C\omega-differentiability here? i have to show with induction that f...

Given: f(x+y)=f(x)f(y). f'(0) exists. Show that f is differentiable on R. At first, I tried to somehow apply the Mean Value Theorem where f(b)-f(a)=f'(c)(b-a). I ended up lost... Then I tried showing f(0)=1, because f(x-0)=f(x)f(0) and f(x) isn't equal to 0. However, with that...

What's the condition for f(x,y) to be differentiable in its domain? I googled for it but couln't find... Thanks in advance.

Hi, I don't understand why mathematicians would need to define the mathematical concepts of diffferentiabilty and conitnuity. To be honest, I don't even understand why "f(x) tends to f(a) as x tends to a" describes continuity. Also, I am wondering why f(x) = mod x is not differentiable at...

(PROBLEM SOLVED) I am trying to think of a complex function that is nowhere differentiable except at the origin and on the circle of radius 1, centered at the origin. I have tried using the Cauchy-Riemann equations (where f(x+iy)=u(x,y)+iv(x,y)) \frac{\partial u}{\partial x}=\frac{\partial...

I want to show x->abs(x) is not differentiable at 0 Some techniques in analysis are required... how should i do?

Homework Statement show f(x)=\left\{e^{\frac{-1}{x}} \\\ x>0 f(x)=\left\{0 \\\ x\leq 0 is differentiable everywhere, and show its derivative is continuous Homework Equations Product Rule and Chain Rule for derivatives. Definition of a derivative f^{'}(a)=\frac{f(x)-f(a)}{x-a}...

Let I be an open interval in R and let f : I → R be a differentiable function. Let g : T → R be the function defined by g(x, y) =(f (x)−f (y))/(x-y) 1.Prove that g(T ) ⊂ f (I) ⊂ g(T ) (The last one should be the closure of g(T), but I can't type it here) 2. Show that f ′ (I) is an interval...

Homework Statement Suppose a>0 is some constant and f:R->R is given by f(x) = |x|^a x sin(1/x) if x is not 0 f(x) = 0 if x=0 for which values of a is f differentiable at x=0? Use calculus to determine f'(x) for x is not equal to 0. For what values of a is f' a continuous function defined...

Homework Statement Hello, I can't find any way to prove if this funtion is or isn't differentiable in if (x,y)=(0,0) : {f(x,y)=\displaystyle\frac{x^{3}}{x^{2}+y^{2}}} if (x,y) \neq(0,0) f(x,y)=0 if (x,y)=(0,0) The Attempt at a Solution Partial derivatives don't exist...

Homework Statement Hi all I wish to show differentiability of g(x)=x on the interval [-pi, pi]. This is what I have done: g'(a) = \mathop {\lim }\limits_{h \to 0} \frac{{g\left( {a + h} \right) - g\left( {a} \right)}}{h} \\ = \mathop {\lim }\limits_{h \to 0} \frac{h}{h} \\ = 1...

Homework Statement Hi all I have f(x)=|x|. This I write as f(x) = -x for x<0 f(x) = x for x>0 f(x) = 0 for x=0 If I want to show that f(x) is not differentiable at x=0, then is it enough to show that f'(x) = -1 for x<0 f'(x) = 1 for x>0 and from this conclude that it is...

Hi all, I'm looking at the following problem: Suppose that f:\mathbb{R}^2\to\mathbb{R} is such that \frac{\partial{f}}{\partial{x}} is continuous in some open ball around (a,b) and \frac{\partial{f}}{\partial{y}} exists at (a,b): show f is differentiable at (a,b). Now I know that if both...

Homework Statement f(x) = {x, x rational, 0, x irrational g(x) = {x^2, x rational, 0, x irrational Show that f(x) is not differentiable at 0. Show that g(x) is differentiable at 0 Homework Equations f'(x) = lim(h->0) f(x+h) - f(x)/h I suppose The Attempt at a Solution Just...

Homework Statement Determine that, if f(x) = {xsin(1/x) if x =/= 0 {0 if x = 0 that f'(0) exists and f'(x) is continuous on the reals. (Sorry I can't type the function better, it's piecewise) Homework Equations The Attempt at a Solution For f'(0) existing, For x ≠ 0...

Homework Statement Prove that (ax^n)' = nax^n-1 using induction. I am very weak with induction proof, and I haven't had much trouble proving the basis step, but I can't seem to finish it... Homework Equations The Attempt at a Solution 1. Prove (ax)' = a (a(x+h) - a(x))/h =...

Homework Statement Dear all, How can I show that the function f(x,y)=xy is differentiable? Thanks Dimitris Homework Equations The Attempt at a Solution

Is there a convenient sufficient condition for knowing whether a function of two variables is differentiable? Isn't it something like if both the partial derivatives exist and are continuous, you know the derivative \mathbf{D}f exists?

Hallo. If we consider Rolle's Theorem: "If f is continuous on [a, b], differentiable in (a,b), and f (a) = f (b), then there exists a point c in (a, b) where f'(c) = 0." Why do we need to state continuity of f in interval and differentiability of f in open segment? Why can't we say f...