Differentiable Definition and 280 Threads

let f(x) = 2-x if x<= 1 x^2 - 2x + 2 if x > 1 Is f diff at x = 1? At first I would say yes because f(x) is continuous at x = 1. But when I graph f '(x) it is obvious that the function is not differentiable at x = 1. My questions is... is there another way to determine if...

I'm curious about the conditions for when a function f(x) is not differentiable

Homework Statement K and M are constants. If h is differentiable at x=2, what are the values of k and m. h(x)= kx^2 + 1, 0<x<2 mx - 3, 2<x<5 All of the "<" signs are "less than or equal to" Homework Equations Not sure The Attempt at a Solution I tried setting the...

Hey guys, I'm just wondering if I got this question right: Discuss where the following in R^{2} is differentiable: f(x,y)=(x^{3}+y^{3})^{2/3} So I take the partial derivative: f_{x}(x,y)=\frac{2x^{2}}{(x^{3}+y^{3})^{1/3}} and see that f(x,y) might not be differentiable at (0,0), so...

f(x)= mod x is this function differentiable at all points other than 0.

Homework Statement See photo, part b and c Homework Equations The Attempt at a Solution For part b It seems it is trival, in part a we have proved that f_{x} and f_{y} exist. Obviously, they are differentiable for x and y\neq0 For part c. It seems there are 2 method to do it...

Homework Statement Hi I'm just looking for a link to the proof of this theorem: if the partial derivatives of function f exist and are continuous at a point then the function is differentiable there Or even the name would be helpful Its not a homework assignment per say, just something...

I know that it isn't. I just want to know how I could, step by step, prove that this function is not differentiable at x=0.

[b]1. For two differentiable vector functions E and H, prove that (Delta (dot) (e X h) = H (dot) (delta X e) - e (dot) (Delta X h) [b]2. Cross product and dot product. The Attempt at a Solution First I took did the left side of the equation, I took the cross product of vectors e and...

Homework Statement PROVE: If A(t) is nxn with elements which are differentiable functions of t Then: \frac{d}{dt}(det(A))=\sumdet(Ai(t)) where Ai(t) is found by differentiating the ith row only. Homework Equations I know I should prove this by induction on n The Attempt at a...

Hi, I just started learning differential geometry. Got some questions. Thanks in advance to anyone who can help! Consider the one-dimensional manifold represented by the line y = x for x<0 and y = 2x for x>= 0. Now if I consider the altas with two charts p(x, y)=x for x<-1 and q(x,y)=y for...

Homework Statement Suppose that f is a differentiable function of a single variable and F(x,y) is defined by F(x,y) = f(x^2 - y) Problem: Given that F(0,y) = sin y for all y, find a formula for F(x,y) Homework Equations The Attempt at a Solution This is what the tutor had put...

Hello! I got the following function: \int_{0}^{x} \left(-1 \right)^{\lfloor 2^{i} \cdot t \rfloor} \ dt, \quad x \in \left[0,1 \right] I want to show it is not differentiable at x= 2^{-i} k where k is a natural number greater equal 0. I already calculated the right derivate by...

Can a function f: (a,b) in R be bounded and diffferentiablle, but have an unbounded derivative. I believe it can, but can not think of any examples where this is true. Anyone have any ideas?

f(x)= x=1/x-1 if x ≤ 0 x^2-2x +1 if 0 < x < 1 ln x if x ≥ 1 Here's what I have so far lim f(x)= lim x+1/x-1= -1, LHL= -1 x →0- x→0- lim f(x)= lim x^2-2x+1= (1/2)^2 - [(2)1/2] +1= 1/4, RHL= 1/4 x→0+ x→0+...

Homework Statement 1. if a function is twice continuously differentiable with f''(x) >= 0 for all real values of x then (f(-x) + f(x))/2 >= f(0) ? 2. if a function is twice continuously differentiable with f''(x) >= 0 for all real values of x then tf(x) + (1-t)f(y) >=...

I'm private studying a section in Linear Algebra first dealing with complex numbers. Now, I am ok with most of the answers, but I need help particularly with where this function is differentiable. Homework Statement Let f(z) = ln|z| + i arg z, 0<=arg z<2pi. What is the domain? What is...

I innocently gave my students a problem: Which differentiable functions f: R \rightarrow R are bijective? "Innocently", I say, because I'm finding it hard to come up with any simple set of conditions that are both necessary and sufficient. Here's what I can say so far: (1) If f'(x) \neq 0...

Homework Statement is x/(1+e1/x) differentiable at x =0 Homework Equations The Attempt at a Solution i calculated the right and left hand derivative and got them as 0 and 1 respectively , so it should not be differentiable, is it right my book says that it is differentiable ?

Homework Statement if f and g are two continuous functions and f+g is differentiable...are f and g differentiable? if not give a counter example! Homework Equations The Attempt at a Solution

Homework Statement If f is differentiable at x then f is continues at x Any help would be great. Homework Equations MUST USE epsilon delta definition to prove The Attempt at a Solution

Hi all, I was just wondering if a function that is continuous and differentiable for all xεR, but where the domain is restricted to closed interval [a,b], does the derivative exist at x=a or x=b?

Homework Statement three functions: y=\begin{cases}\arctan \frac{1}{x}\ x\neq0\\ 0\ x=0\end{cases} y=\frac{1}{x}, y=|x^2-1| and what about inflection point? The Attempt at a Solution first function is concave on left of 0, convex on right, so from definition it should be inflection point...

Homework Statement Find the continuous points P and the differentiable points Q of the function f in {R}^3, defined as f(0,0,0) = 0 and f(x,y,z) = \frac{xy(1-\cos{z})-z^3}{x^2+y^2+z^2}, (x,y,z) \ne (0,0,0). Homework Equations The Attempt at a Solution If you want to look at the limit I'm...

Homework Statement Show that f(z) = x^2 + i(y^2) is diff at all points on y=x. Then show that is not analytic anywhere.Homework Equations Cauchy Riemann equations: fy = ifx <=> function is differentiable (I'm still unclear about the implications of CR-equations. It says in my book that if f is...

1. Prove f is differentiable at x=xo implies f is continuous at x=xo using epsilon and delta notation. 2. I have gotten this far: absolute value(f(x)-f(xo)) <= absolute value(x-xo)*(epsilon + absolute value(f '(xo))) <= means less than or equal to. 3. I need to get here: absolute...

Homework Statement Is f(y)=\sum_{m=1}^\infty \frac{1}{y^2+m^2} differentiable? Homework Equations The Attempt at a Solution From the graph, it is obvious that f is not differentiable at y=0, but I don't know how to prove that. I proved that \sum_{m=1}^n f_m=\sum_{m=1}^n\frac{1}{y^2+m^2}...

Let g:[0,1]-->R be twice differentiable(both g and g' are differentiable functions) with g''(x)>0 for all x in [0,1]. If g(0)>0 and g(1)=1, show that g(d)=d for some point d in (0,1) iff g'(1)>1. I thought I might use the MVT. g'(c)=g(1)-g(0)/1=1-g(0) g'(c)<0 then

Homework Statement Let f: R-->R be continuous. For δ>0, define g: R-->R by: g(x) = (1/2δ) ∫ (from x-δ to x+δ) f Show: a) g is continuously differentiable b) If f is uniformly continuous, then, for every ε>0, there exists a δ1>0 such that sup{∣f(x) - g(x)∣; x∈R} < ε for 0<δ≤δ1The Attempt at...

Homework Statement Let I be an interval, and f: I --> R be a continuous function that is known to be differentiable on I except at c. Assume that f ' : I \ {c} --> R admits a continuous continuation to c (lim x -> c f ' exists). Show that f is in fact also differentiable at x and f ' (c) =...

Homework Statement In which points the surface \{\left(x,y,z\right)\in\Re^{3}|x^{3}-y^{3}+xyz-xy=0\right\} is a differentiable manifold (subvariedad diferenciable in spanish). Calculate its tangent space in the point (1,1,1). Homework Equations NA The Attempt at a Solution I've...

Homework Statement Let r>0, and let f be a function from B_{r}(\textbf{0}) \rightarrow \textbf{R} , and suppose that there exists an \alpha > 1 such that |f(\textbf{x})| \leq ||\textbf{x}||^{\alpha} for all \textbf{x} \in B_{r}(\textbf{0}). Prove that f is differentiable at 0. What happens...

Homework Statement give an example of a function f: R --> R that is differentiable n times at 0, and discontinous everywhere else. Homework Equations ---The Attempt at a Solution i got one, and i proved everything, i just want to make sure what i did is correct: f:x n+1 when x is rational...

Homework Statement Find the points where the function given by is not differentiable. The Attempt at a Solution I got the doubtful points as +-1, 2 How do I check the differentiability now? The mod. function is confusing me a bit.

Homework Statement Find the points at which the function is not differentiable. Homework Equations It is not asked to check differentiability at a particular point. How do I find the points which are not differentiable? The function is not continuous at x=0

Homework Statement Give an example of a function f that is differentiable on [0,1] but its derivative is not bounded on [0,1] Homework Equations The Attempt at a SolutionOk, I know that the derivative f' cannot be continuous, because then it would be bounded on [0,1]. I also know that it...

I don't this this is an overly complicated proof but it is one I have never seen or done before. Let f be a polynomial with atleast two non-zero terms having different degrees. Prove that the set {f(x),xf'(x)} is linearly independent in P Proof: With out loss of generality we can...

Use implicit differentiation to find dy/dx. y is a differentiable function of x 2y^2+4xsiny = cos4x Here is what I did: 4y*dy/dx + 4siny+ 1/cosy*dy/dx = -sin4x + 4 4y*dy/dx + 1/cosy*dy/dx = -sin4x - 4siny + 4 dy/dx(4y + 1/cosy) = -sin4x - 4siny + 4 dy/dx = (-sin4x - 4siny + 4) /...

Okey, I have problem with the foundation of lie algebra. This is my understanding: We have a lie group which is a differentiable manifold. This lie group can for example be SO(2), etc. Then we have the Lie algebra which is a vectorspace with the lie bracket defined on it: [. , .]. This...

Say I have the function f(x) = x^5ln(x) I can differentiate f(x) say 6 times and I'm left with just ln(x). But this is not a proof! How do I go about proving it? Could I use the Taylor expansion?

Homework Statement Hi there. I got next problem, and I must say if it is differentiable in all of its domain. f(x,y)=\begin{Bmatrix} (x+y)^2\sin(\displaystyle\frac{\pi}{x+y}) & \mbox{ if }& y\neq{-x}\\0 & \mbox{if}& y=-x\end{matrix} So, I thought trying with the partial derivatives...

Homework Statement Let f: R\rightarrow R be a differentiable even function. Prove that f' is an odd function. Also, prove that if f is a differentiable odd function, then f' is an even function. Homework Equations The Attempt at a Solution I tried to use definition, so I should...

For a game I am thinking about making I would need to know how to project points from a differentiable bounded 3-manifold to a Euclidean plane (the computer screen). The manifold would be made from a 3-dimensional space with two balls cut out of it and a hypercylinder glued onto it at the holes...

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Homework Statement Show that the function f(x) = { x/2 if x is rational { x if x irrational is not differentiable at 0 Homework Equations If f is differentiable at 0 then for every e > 0 there exists some d > 0 such that when |x| < d, |(f(x)-f(0))/x - L | < e...

Homework Statement If f is differentiable at x = a, evaluate lim[h->0] (f(a+2h)-f(a+3h))/h Homework Equations We know that f'(a) = lim[h->0] (f(a+h)-f(a))/h The Attempt at a Solution I have done the following, and I am not sure if it is correct, though the result makes sense...

Homework Statement Suppose f is differentiable on an open interval I and let x* \in I. Show that there exists a sequence {x_n}\subset I such that lim[n->inf]{x_n}=x* and lim[n->inf]{f'(x_n)}=f'(x*). Homework Equations We know that a function g is continuous iff for any sequence...

Consider the manifold of the real-line R with a differentiable structure generated by the map x^3 : M \rightarrow \mathbb{R} . This example is given in a textbook I'm looking at, but I don't really understand how this can work. The inverse map is clearly not smooth at x=0. Do they mean that...

Given http://www.mathhelpforum.com/math-help/vlatex/pics/105_fde5ac6b051b4fac473487c7b4afa9e5.png Is f(x) differentiable at x=1? I know that we have to prove http://www.mathhelpforum.com/math-help/vlatex/pics/65_6fae3c52eaa96aaafdf2c225a900ea48.png exist/does not exist at x=1. But...

The definition of having multiple differentiable structures is that given two atlases, {(U_i ,\phi_i)} and {(V_j,\psi_j)} (where the open sets are the first entry and the homeomorphisms to an open subset of Rn are the second entry), that the union {(U_i,V_j;\phi_i,\psi_j)} is not necessarily...