Differentiable Definition and 280 Threads

Homework Statement Let g be a twice differentiable function with g'(x)>0 and g''(x)>0 for all real numbers x, such that g(4)=12 and g(5)=18. Of the following, which is a possible value for g(6)? a. 15 b. 18 c. 21 d. 24 e. 27 Answer: e. 27 Homework Equations The Attempt...

Homework Statement Hi, I have this function: f(x ) = 0 (x is irrational) or f(x) = 1/q for rational p/q in lowest terms. show that this function is not differentiable anywhere The Attempt at a Solution This is the answer from the solutions book: consider [(f(a+h) - f(a ) ) /...

Good day! Well, I've just started this year with Complex Analysis (we're using "Complex Variables and Applications" 8th ed by Brown and Churchill) and as I'm going through some assignment questions, I noticed that one of the questions states: "Is g analytic at any point of C (as opposed to...

Homework Statement For k = 1,2,\ldots define f_k : \mathbb{R} \to \mathbb{R} by f_k(x) = \sqrt{k} x^k (1 - x). Does \{ f_k \} converge? In what sense? Is the limit integrable? Differentiable? Homework Equations The Attempt at a Solution I don't know how to approach this...

Homework Statement f(x,y) is differentiable in (0,0) and f(0,0)=0 q(t) devirate t=0 and q'(0)=1 q(0)=0 let it be g(x,y) = q(f(x,y)) prove that g differentiable in (0,0) and that f_{x}(0,0) = g_{x}(0,0)Homework Equations all calculus The Attempt at a Solution Well my idea is like this First I...

Homework Statement I apologize for not knowing how to use Latex, so I will type the problem as it is read... Prove that for all x greater than or equal to 0, we have the integral from 0 to x for [g(x)]^3 dx which is less than or equal to (the integral from 0 to x for g(x) dx)^2...

Homework Statement Is the composition of two differentiable functions always differentiable? E.x. h(x) = sin(x) k(x) = 1/x for x not equal 0 Does this automatically mean h(k(x)) is differentiable? Thank you, M

Homework Statement Let f: M \rightarrow N , g:N \rightarrow K , and h = g \circ f : M \rightarrow K . Show that h_{*} = g_{*} \circ f_{*} . Proof: Let M, N and K be manifolds and f and g be C^\infinity functions. Let p \in M. For any u \in F^{\infinity}(g(f((p))) and any...

Defining differentiability for multivariable functions we want not only for the partial derivatives to exist but also local linearity. Because my question is the same also for the single variable case, I'll pose it with a single variable function. In one variable we have that local...

Many "theoretical mechanicians" seem to awesome that motion is a {C^\infty } function(at least that is how I learned it). However, it seems like the postulates of Newtonian/Lagrangian/Hamiltonian/Vakonomic mechanics seem to "work" in the general case where only the motion is a {C^2}(ie the...

Homework Statement Let V be a finite dimensional normed vector space and let U= L(V)*, the set of invertible elements in L(V). Show, f:U-->U defined by f(T)= T-1 is differentiable at each T in U and moreover, Df(T)H = -T-1HT-1 where Df(T)= f'(T). Homework Equations Apparently...

Homework Statement Consider a function g : (a, b)-->R. Assume that g is differentiable at some point c in (a,b) and that g'(c) is not = 0. Show that there is a delta > 0 so that g(x) is unequal to g(c) for all x in V_delta(c)\{c}intersect(a,b) Homework Equations The Attempt at a...

I'm looking for a simple example of a differentiable manifold that doesn't have an associated riemann metric. thanks

I am trying to find a Hausdorff topological space that is not second-countable but otherwise a DIFFERENTIABLE n-manifold. I can't figure it out. Does it exist? :smile: I read about the classical example of L=\omega_1\times[0,1) with lexicographical order and the order topology. It's Hausdorff...

I look for a function u(x) with u(0)=0 and u(1)=1, which is single-valued and differentiable on the entire interval x= [0,1] and allows one to choose the derivatives u'(0) and u'(1) through two free parameters. Seems simple enough, right?

For what values of c and d will x be differentiable for all values f(x)=\left\{\begin{array}{cc}cx+d,&\mbox{ if } x\leq -1\\cx^3+x+2d, & \mbox{ if } x>-1\end{array}\right. i took the derivative on each side f'(x) = c f'(x) = 3cx²+1 c = 3cx²+1 so i get c as -(1/2) but...

So, a certain discussion occurred in class today... If f is differentiable, is f ' continuous? At first sight, there seems no reason to think so. However, we couldn't think any counterexample. It also seems logical that f' is continuous since otherwise f wouldn't be differentiable. For...

What's the difference between "analytic" and "continuously differentiable?" I'm reading Gamelin's Complex Analysis book, and he talks about f(z) being analytic if it is continuously differentiable and satisfies the Cauchy-Riemann equations. But if f(z) is continuously differentiable, doesn't...

Is a bijective continuous function:[a,b]->[f(a),f(b)] differentiable? I think it has to be. continuity between two distinct values of f(a) and f(b): it got to take all the values between f(a) and f(b) at x in [a,b], by the intermediate value theorem. if f is bijective, at [a,b], f(x) can't go...

How do you determine this? F(x) x^2 +1 if x<1 F(x) 2x if x >= 1 at x=1 Are there designated steps? I understand that it is the derivative, but I don't understand the differentiable at the indicated point part..

Homework Statement Suppose that the function f: R^n --> R is continuously differentiable. Let x be a point in R^n. For p a nonzero point in R^n and alpha a nonzero real number, show that (df/d(alphap))(x)=alpha(df/d(p))(x)Homework Equations A function f: I --> R, defined on an open...

Homework Statement Let f denote the function defined by f(z)= _z^2 /z if z is not 0 0 if z=0 show that f satisfies the Cauchy-Riemann equations at z=0 but that f is not differentiable there Homework Equations The Attempt at a Solution it is easily to show the...

Homework Statement For any (x,y) other than (0,0). f(x,y)=\tan(x·y)\sin\left(\frac{1}{x^2+y^2}\right) For (x,y) = (0,0) f(x,y) = 0 Is f totally differentiable? Homework Equations The Attempt at a Solution If the function is not continuous, it can't be differentiable...

If a function f is differentiable at a point x = c of its domain, then must it also be differentiable in some neighborhood of x = c?

Homework Statement Given: w=9(cos(\theta)+sin(\theta)) and u=r*w. 1) Is u continuously differentiable? 2)Is it possible to get u continuously differentiable with a different w? Homework Equations The Attempt at a Solution 1) u is continuously differentiable since w is in terms...

f(x) and g(x) are differentiable on 0 f(0)=g(0)=0 \lim _{x->0}\frac{cos(f(x))-cos(g(x))}{x^2}=\lim _{x->0}\frac{-2sin(\frac{f(x)+g(x)}{2})sin(\frac{f(x)-g(x)}{2})}{x^2}=-2 because i can use (sin x)/x=1 here is it ok??

Homework Statement F f(x)={(2x-1)/Absolute value(2x-1) x cannot equal (1/2) { 0 x = (1/2) a) is f continuous at X = (1/2) explain b) is f differentiable at x = (1/2) explain Homework Equations I have made the graph and x is a point...

This came up in a recent discussion about l'Hopital's Rule. Suppose f(x) has a derivative at x=0, that is f'(0) exists. Is it necessarily true that f(x) is differentiable in some open interval containing x=0? Others--who know calculus better than I--say no, f(x) is not necessarily...

r twice continuously differentiable function proof... Homework Statement Help :frown: if f:[a,b] \rightarrow R is twice continuously differentiable, and f(x)\geq 0 for all x in [a.b] and f ''(x) \leq 0 for all x in [a,b] prove that 1/2 (f(a) + f(b)) (b-a) \leq \int f(x)dx \leq(b-a)...

prove that this function differentiable endles times on x=0 ?? http://img502.imageshack.us/img502/6778/83126617mm0.th.gif i was told "once we can express the function as a power series around zero and it is differentiable at zero, we know it is infinitely differentiable" differentiable...

Homework Statement Suppose g is a differentiable on [a,b] and f = g', then does there exist a function f which is not integrable? Homework Equations The Attempt at a Solution I've tried to look at pathological functions such as irrational, rational piecewise functions. but the...

Homework Statement Show that f(z) = zRez is differentiable only at z=0, find f'(0) The Attempt at a Solution This should be easy. I find the limit as z_0 approaches 0 of [f(z+z_0) - f(z)]/(z_0) for this function...expand it out, simplify, and find what the limit is when z_0 is...

I'm having trouble with this inequality: let f be (real valued) continuously differentiable on [0,1] with f(0)=0, prove that sup_{x\in[0,1]} \left|f(x)\right| \leq \int^{1}_{0}\left|f\acute{}(x)\right| dx Thanks for any help.

Homework Statement Define h(x)=x^3sin(1/x) for x\neq0. and h(0)=0. Show h is differentiable everywhere and that h is cont everywhere, but fails to have a derivative at one point. Homework Equations The Attempt at a Solution [h(x)-h(0)]/[x-0]=x^2sin(1/x) h is diff everywhere...

Homework Statement how could i prove that cos x= sum (n=1 to 00) [((-1)^n) * x^(2n)/((2n)!)] is continuous and differentiable at each x in R Homework Equations the Taylor Expansion of cosine is the given equation The Attempt at a Solution basically i need to prove that the...

Let B(V,V) be the set of bounded linear transformations from V to V. Let U be the set of invertible elements of B(V,V) and define the map ^{-1}: U\rightarrow U by ^{-1}(T)=T^{-1} Show that the map ^{-1} is differentiable at each T \in U.

If M is a topological manifold, a smooth structure A (or maximal atlas) on M is a set of smoothly compatible charts of M that is maximal in the sense that if we consider any chart that is not in A, then there is some chart in A with whom it is not smoothly compatible. Now, it is a fact that...

differentiable and uniformly continuous?? Homework Statement Suppose f:(a,b) -> R is differentiable and | f'(x) | <= M for all x in (a,b). Prove f is uniformly continuous on (a,b). Homework Equations The definition of uniform continuity is: for any e there is a d s.t. | x- Y | < d...

Homework Statement F(x)= {1/2x+1 when x=<2 {squareroot(2x) when x=>/=2 is it differentiable at x=2. Homework Equations (f(x)-f(2))/(x-2) The Attempt at a Solution So i know i ahve to take the limit from both the negative and positive of 2, and determine if they are equal. But...

Homework Statement Let f be a function differentiable function with f(2) = 3 and f'(2) = -5, and let g be the function defined by g(x) = xf(x). Which of the following is an equation of the line tangent to the graph of g at point where x=2? a. y=3x b y-3 = -5(x-2) c y-6 = -5(x-2) d y-3 =...

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 -...