In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp.
More generally, for x0 as an interior point in the domain of a function f, then f is said to be differentiable at x0 if and only if the derivative f ′(x0) exists. In other words, the graph of f has a non-vertical tangent line at the point (x0, f(x0)). The function f is also called locally linear at x0 as it is well approximated by a linear function near this point.
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'?
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
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...
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...
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...
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..?