Differentiability Definition and 191 Threads

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...

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...

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?

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...

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}...

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...

given a function F:S-->R such that for every element belonging to "S" has both left hand derivative and right hand derivative and are equal to the derivative at that point. Can we say that the function is differentiable..?

Homework Statement Can someone tell me why and why not following functions are Continous and Differentiable. I am also providing the answer but can some help me understand.. thanks 1) f(x) = x^(2/3) -1 on [-8,8] answer: function is continuous but not differentiable on -8. Is that...

Here's a question I've thought about on several occasions: How many levels of derivatives (rates of change) typically occur for objects in nature? For instance, a car has a position, velocity (1st derivative), and acceleration (2nd derivative), but it can also be said to have a rate of...

[b]1. Suppose that f is differentiable on R and \lim_{x \rightarrow \infty} f'(x) = M. Show that \lim_{x \rightarrow \infty} (f(x+1)-f(x)) also exists, and compute it. [b]3. I am pretty sure the limit will be equal to m. Here is my attempt. \lim_{x \rightarrow \infty} f'(x) = \lim_{x...

This isn't homework per se... It's a question from a book I'm self-studying from. If f is continuous on [a,b] and differentiable at a point c \in [a,b], show that, for some pair m,n \in \mathbb{N}, \left | \frac{f(x)-f(c)}{x-c}\right | \leq n whenever 0 \leq |x-c| \leq \frac{1}{m}...

Ok, so I have f(x,y)=(p(x)+q(y))/(x^2+y^2) where (x,y)NOT=0 and f(0,0)=0. the basic idea of the function is that the numerator contains 2 polynomials>2nd order. and the denominator has a Xsquared+ysquared. I have to prove that if f(x,y) is differentiable at (0,0) then its partial derivatives...