Tagged Questions
3
votes
3answers
50 views
How to determine if a function is decreasing, constant or increasing (in a given interval) if its derivative function has no zeroes?
Let us suppose you have a certain function $f(x)$ and you want to find out in which intervals this function is decreasing, constant or increasing. I know you need to follow these steps:
Find out ...
-2
votes
0answers
39 views
Given a certain function, which is the procedure to determine in which intervals that function is decreasing, constant or increasing?
I would like an example if possible.
1
vote
2answers
31 views
Confusion related to smoothness of a function
I just found this thing that $\operatorname{trace}(AB)$ where $A$ and $B$ are two matrices, it is a smooth function. I didn't understand how it is a smooth function. Any suggestions?
8
votes
4answers
233 views
Is it known or where does this lead to?
I am eleventh class student, recently I started learning calculus. I was experimenting on various things, and found a new thing. It is as follows. Let us consider a function $f(x)$which is ...
0
votes
0answers
28 views
Mathematical Microeconomics
I don't know if this is the proper place to ask, but given that it is a question that basically involve mathematical properties of functions, I think its okay.
So the question defines the following ...
11
votes
2answers
213 views
Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?
Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?
Similarly, let $f :\mathbb{C}→ \mathbb{C}$ be a function such that $f^2$ and $f^3$ ...
1
vote
3answers
64 views
How can you show that $\delta′=f(0)\delta′−f′(0)\delta$ for a function f that is infinitely differentiable?
Assume that $f$ is infinitely differentiable. Let $\delta$ be the (Dirac) delta functional.
I know that $f\delta = f(0)\delta$, but I'm not sure how to derive the equation ...
5
votes
2answers
63 views
Inverse and derivative of a function [duplicate]
Find an example of an inverse function f(x) such that its derivative is the same as its inverse.
I tried many different functions but non of them worked.
4
votes
2answers
137 views
Is $y=|x^3|$ a smooth function?
Is this a smooth function? $y=|x^3|$
The graph of this function has no sharp cuts or corners, so I think it is a smooth function but someone told me that it's not.
1
vote
2answers
79 views
Partial derivatives.
Suppose
$$f(x+y, x^2 +xy + z^2) = 0.$$
Show that
$$x + y = 2z\left(\frac{\partial z}{\partial y}-\frac{\partial z}{\partial x}\right).$$
Please help I don't know where to start!
0
votes
2answers
69 views
How to find when function is increasing?
When is the function defined by $f(x)=x^2+e^{-2x}$ increasing? I know you have to take the derivative and use certain values of $x$, but I am confused on how to do this particular problem, and I ...
0
votes
3answers
59 views
How to derive a multi-dimensional function?
In a regular function it possible to derive it by - $$f'(x_0)=\frac{f(x_0+h)-f(x_0-h)}{2h}$$
when $h$ is "little enough" .
How could I derive a multi-dimensional function with the above approch ?
...
1
vote
1answer
74 views
Comparing rates of change: which function increases faster?
I am comparing two functions for $x \ge 1$:
$$f(x) = \ln(\lfloor\frac{x}{9}\rfloor!) - \ln(\lfloor\frac{x}{10}\rfloor!) - \ln(\lfloor\frac{x}{90}\rfloor!)$$
$$g(x) = (2.07766)\sqrt{\frac{x}{9}} + ...
2
votes
1answer
67 views
If $f '(2) = 0$ and $f ''(2) = 4$, what can you say about $f$?
I was doing very well in Calculus up until this point. I realize that concavity and $f'$ and $f''$ require one to really visualize what is happening with a function, but can someone please help me to ...
2
votes
1answer
27 views
Differentiability of first derivative of a function
If a function $f$ is differentiable on domain $D$ and $f'$ is increasing on $D$, is $f'$ necessarily continuous on $D$? Is $f'$ necessarily differentiable on $D$? Counterexamples?
From Darboux ...
4
votes
4answers
188 views
Show that function is strictly monotone increasing
I want to show that $$ f(x)=\dfrac{x-\sin(x)}{1-\cos(x)} $$ is strictly increasing in $(0,2 \pi) $. Unforunately, this is not that easy for me , as the derivative is not very manageable and ...
3
votes
2answers
59 views
Matrix Derivative of $ABC$ with respect to $B$
I have looked throughout the matrix cookbook and other sources, but am a bit confused by this problem. If I have a function $F = ABC$, what is the partial derivative of $F$ with respect to $B$? When ...
3
votes
1answer
53 views
Non-differentiability in $\mathbb R\setminus\mathbb Q$ of the modification of the Thomae's function
Here is the problem I'm struggling with:
Where is the following function continuous, differentiable, continuously differentiable?
$$f(x) =
\begin{cases}
q^{-2} & \text{if $x=\frac{p}{q}$ ...
1
vote
1answer
88 views
How to find $\frac{d^2x}{dy^2}$
How to find $\dfrac{d^2x}{dy^2}$ in terms of $\dfrac{d^2y}{dx^2}$ and $\dfrac{dy}{dx}$ for any implicit function which is twice differentiable w.r.t. both x and y.
I tried but we can't write ...
2
votes
1answer
44 views
Derivative of a function equals the reciprocal of that function
I need to solve for a function that satisfies the following condition: $$f'(x)f(x)=1.\tag{1}$$ The solution suggests that we guess and verify. The guess is $$f(x)=Cx^k,$$ which implies that ...
1
vote
0answers
64 views
Different Definitions on the Differentiability of Functions on a closed set.
I have encountered three different definitions on the differentiablity of functions on a closed set.
In the following, suppose that $\Omega\subset M$ is a (open) domain, where $M$ is a manifold. The ...
0
votes
1answer
46 views
Power and automatic differentiation
I'm programming a math application where users can define functions and the application evaluates it at a given point. I'm trying to develop the automatic differentiation so I have developed a type ...
3
votes
1answer
65 views
If $f'(t) = g'(t)$ then $f(t) = g(t) + k$ for all $t\in\mathbb R$
Let $f,g: I \to\mathbb R$ for some interval $I\subset\mathbb R$. Then $f'(t) = g'(t)$ for all $t\in I$ if and only if there exists $k\in\mathbb R$ such that $f(t) = g(t)+k$.
Necessary condition ...
1
vote
2answers
102 views
Differentiating piecewise functions.
Say we have the piecewise function $f(x) = x^2$ on the interval $0 \le x < 4$; and it equals $x+1$ on the interval $ x \ge 4$. Why is it that, when I take the derivative, the intervals loose their ...
3
votes
1answer
82 views
Determining the bigger of two numbers : $\left(\frac12\right)^e$ or $\left(\frac1e\right)^2$
The question says - use the function $f(x)=sin(x)^{sin(x)}$, where $0<x<\pi$, to determine the bigger of two numbers: $\left(\frac12\right)^e$ or $\left(\frac{1}{e}\right)^2$. Any tips on how to ...
2
votes
2answers
310 views
How to find critical points of an absolute values function
I am asked to find How many critical points does the function $g(x) = |x^2 − 4|$ have?
I know that the result is $3$ but I can only find $2$. What I do, is to equal the equation to $0$, so $x^2-4=0$ ...
1
vote
1answer
28 views
Is the function is differentiable at $x$ or $D$?
I know that a) and b) is differentiable at the given points, would you maybe explain how I should show that ?
a) $f:\mathbb{R}\rightarrow \mathbb{R},\quad x\rightarrow 0,{ \quad x }_{ 0 }=0$
b) ...
0
votes
1answer
58 views
Continuity of a function, Differentiable function
The following function is given:
$$f:\mathbb{R}\rightarrow \mathbb{R}, \ x\rightarrow \begin{cases} x^2\cos{\left(\frac{1}{x}\right)} & \text{for } x \neq 0\\ 0& \text{for } x =0\end{cases}$$
...
1
vote
1answer
138 views
Statements about a twice differentiable function
Can you help me to prove or disprove?
We have a function $f:(0,+\infty)\rightarrow\mathbb{R}$ twice differentiable, such that as $x\rightarrow+\infty$
(a) $xf(x) \rightarrow+\infty$
(b) $xf''(x) ...
4
votes
4answers
190 views
Question on functions and derivatives
I can't seem to get this subject very well.
Let $f(x)$ be twice differentiable on $[0,1]$, and that there is a constant $A$ so that $|f''(x)|\le A$. Show that if $f(0)=f(1)=0$, then $|f'(x)|\le ...
1
vote
2answers
63 views
Derivative for log
I have the following problem:
$$
\log \bigg( \frac{x+3}{4-x} \bigg)
$$
I need to graph the following function so I will need a starting point, roots, zeros, stationary points, inflection points ...
3
votes
1answer
132 views
Functions in calculus - notation
I don't have an extensive formal training in calculus, but I'm doing quite a lot of differential calculus work at the moment and there's something which bothers me.
Say I have the differential
...
0
votes
1answer
51 views
subadditive function example
Can somebody give me an example of a continuous function $F:\mathbb{R}^2\rightarrow \mathbb{R}$. Such that
$$\frac{\partial F(x_1,x_2)}{\partial x_i} \geq 0 \;\; \forall x_i, \;i=1,2$$
but,
$$
...
3
votes
1answer
71 views
Prove the Borel Lemma
I'm trying to prove the Borel Lemma, which is:
For every series $a_0,a_1,a_2,\dots$ in $\mathbb{C}$ exists $f \in C^{\infty}(\mathbb{R})$ such as $$ f^{(k)}(0) = a_k $$ for every $k \in ...
4
votes
2answers
129 views
Existence of an infinitely differentiable function $ f $ with $ {f^{(n)}}(0) = 0 $ for all $ n \in \mathbb{N} $.
How can one show that there exists an infinitely differentiable function $ f: \mathbb{R} \to \mathbb{R} $ such that $ {f^{(n)}}(0) = 0 $ but $ f^{(n)} \not\equiv 0 $ for all $ n \in \mathbb{N} $?
4
votes
1answer
79 views
Meaning of different Orders of Derivative
I have been trying to analyse the meaning of higher order derivatives and their geometrical significance.
Given a function $f(x)$ what are the unique geometric interpretation of its higher orders?
...
2
votes
2answers
75 views
Finding functions extremes, convexity and up/downards intervals.
I know title sounds weird but i had to translate it and that was best i could put. Anyhow i have the following function:
$$
f(x) = x\cdot e^{-x^{2}}
$$
I have to find the following:
1. Intervals ...
1
vote
1answer
64 views
A differentiable injective function with Lipschitzian Inverse
I'm having difficulty with the following question which was given to me following studying the inverse mapping theorem.
Let $U\subseteq\mathbb{R}^{n}$
be an open set and let $f:U\to\mathbb{R}^{n}$
...
0
votes
1answer
42 views
Derivative functions count vs Integral functions count
Let's consider set of all functions that can be expressed in quadratures (not sure if 'quadratures' is right word in english. I mean functions that are built from standard functions like + - * / log, ...
-1
votes
1answer
191 views
Prove $\sup \left| f'\left( x\right) \right| ^{2}\leqslant 4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) \right| $
Let $f\left( x\right)$ be a $C^{2}$ function on $\mathbb{R}$. Show that $$\sup \left| f'\left( x\right) \right| ^{2}\leqslant4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) ...
0
votes
3answers
26 views
Derivatives of Functions
Suppose $ F(x)=f(g(x)) $
$g(1)=3$
$g'(1)=4$
$f'(1)=6$
$f'(3)=5$
What is $F'(1)$ ?
1
vote
2answers
114 views
second derivative of the inverse function
I know that the derivative of the inverse function of $f(x)$ is $g'(y) = \frac{1}{f'(x)}$
But how to derive the formula for the second derivative of g(y) knowing that $[\frac{1}{f(x)}]' = ...
0
votes
2answers
87 views
Minimizing The Cost
I have this exercise that I would like anyone to suggest the required steps in order to solve it
A cylindrical can is to be made to hold $250 \pi\; cm^3$. Find the dimensions of the can that will ...
0
votes
3answers
225 views
How to prove this partial derivative?
Consider $u:\mathbf{R}\times\omega\rightarrow\mathbf{R}$, where $\omega\subset\mathbf{R}^{n-1}$ is a bounded domain. For each $y\in\omega$ and each $\lambda>0$, consider ...
0
votes
2answers
53 views
Anyone have an idea for a function, that its derivative equals 0 along y=x, y=-x (|y|=|x|)
I am looking for a differentiable function, thats its partial derivatives equals 0 along
$$|y|=|x|$$
I have thought about $$u(x,y)=\frac{(x-y)^2(x+y)^2}{2}$$ but I am looking for another one
if ...
1
vote
2answers
81 views
Calculus (First Year) - Derivatives Question
This is should be a straightforward question for me but I'm blacking out right now.
Let $g$ be a differentiable function that satisfies $g(x) + x^3
\sin(g(x)) = x^4 + 4x$ around $x=1$.
If ...
5
votes
2answers
377 views
Why use the derivative and not the symmetric derivative?
The symmetric derivative is always equal to the regular derivative when it exists, and still isn't defined for jump discontinuities. From what I can tell the only differences are that a symmetric ...
3
votes
4answers
227 views
Finding the derivative of a function using the Product Rule
I'm home teaching myself calculus because I'm 16 and therefore too young to take an actual class with a teacher, so I apologise if this seems simple.
I understand the definition of the Product Rule ...
0
votes
2answers
108 views
Piece-Wise Discontinuity & Continuity
When is the following function continuous? How would i go about listing the removable
discontinuities and then redefine the function so that it is now continuous
in those places?
$$f(x)= \begin ...
1
vote
0answers
209 views
Using Rouche's theorem
Let $p>1$.
Consider $\phi(p)=\int_0^{\infty}\left|\frac{\sin t}{t}\right|^pdt$.
Function $\phi(p)$ is analytic on its domain.
It's derivative, $\phi'(p)=\int_0^{\infty}\left|\frac{\sin ...
