Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

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1answer
27 views

Finite difference formula to approximate second derivative

I have one question which asks to derive a finite difference formula to approximate $f''(x)$ in the form of $$f''(x)\approx Af(x+2h)+Bf(x+h)+Cf(x)$$ with the method of undetermined coefficients. ...
0
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1answer
52 views

Can Lebesgue Dominated Convergence always be used?

Suppose I want to find the derivative $$\frac{d}{dx}\int f(x,y) dy.$$ I want to know under what condition it would be equal to $$\int \frac{d}{dx}f(x,y) dy.$$ Of course, if I can find a suitable ...
0
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1answer
56 views

Interchanging Expectation and Derivative

Suppose I have a random function, $f(x)(\omega)$. And that for fixed $\omega$, we have the derivative $g(x)(\omega)=\frac{d}{dx}f(x)(\omega)$. For a fixed $x$, I can find the expectation $E(f(x))$. ...
0
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0answers
40 views

Substitution in differential equation

I have a differential equation in the following form, where a, b and R are constants and $\delta (x,t)$ Dirac function. I know that substitution holds $z=x-c y$, where $c$ is a constant also $$a \...
0
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1answer
30 views

Proving that a plane reaches a certain velocity at least two times during a flight

I am asked to prove the following: A plane initiates its departure at 2pm. The distance it will travel is $2500~\text{mi}$. The plane arrives at its destiny at 7:30pm. Prove that, at least two times ...
2
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1answer
59 views

Writing higher order derivatives using the limit definition of derivative?

Studying Taylor series, I wanted to get a sense for what higher derivatives really express in precise terms using the limit definition of the derivative. Is this correct? $$\frac{d^2y}{dx^2} = \...
2
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1answer
102 views

Differentiability of a integral

Define $$L(x)=\int_0^x \left(\sqrt{1+\cos^2(1/u)}\right)du$$, then is L right-differentiable at 0? As far as I'm aware, l'hopitals rule doesn't apply here, but I still think it should be ...
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0answers
16 views

Is $w(x,y)=\frac{x^2+3y^2+2xy}{3 (x^2+y^2)^\frac{4}{3}} \ dx - \frac{3x^2+y^2+2xy}{3 (x^2+y^2)^\frac{4}{3}} \ dy$ an exact differential form?

$$a_1=\frac{x^2+3y^2+2xy}{3 (x^2+y^2)^\frac{4}{3}}$$ $$a_2=- \frac{3x^2+y^2+2xy}{3 (x^2+y^2)^\frac{4}{3}}$$ I verify if $w$ is a Closed differential form: $$\frac{d \ a_1}{d \ y}=\frac{d \ a_2}{d \ ...
2
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2answers
79 views

Is $(x^2+y^2+z^2) \ \sin \frac{1}{\sqrt{x^2+y^2+z^2}}$ Differentiable in $(0,0,0)$?

$$f(x,y,z)=\begin{cases} (x^2+y^2+z^2) \ \sin \frac{1}{\sqrt{x^2+y^2+z^2}} \qquad (x,y,z) \ne (0,0,0) \\ \\ 0 \qquad (x,y,z)=(0,0,0) \end{cases} $$ At first, I study the continuity in the origin. I ...
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0answers
29 views

Explaination of the proof of directional derivative formula

Hi guys :) i read that proof of the formula of the directional derivative and i didn't understand the sceond step and what is h(o) and where does it come from? Where does the dot product between the ...
-3
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2answers
32 views

Whst is the maximum viewing angle? [closed]

While working on a math workbook(it doesn't have solutions), I got stuck on this problem: Two vehicles, A and B, start at point P and travel east at rates of 10km/h and 30km/h, respectively. An ...
2
votes
1answer
84 views

Jacobian of a matrix-valued function

I am fairly sure the very reason my problem arises in the first place is that I am going about it wrong, or have gotten some basic concept wrong, so any answer to point out my misunderstanding would ...
0
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1answer
41 views

Unconventional Differentiation Rules

We all know the stock-standard and conventional differentiation rules, such as the Sum and Difference Rule, Product Rule, Chain Rule etc. But are there other more advanced rules that are not treated ...
5
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2answers
88 views

Determine $\frac{f''(\frac{1}{2})}{f'(\frac{1}{2})}$ if $f(x) = \sum_{k=0}^{1000} \ {2015 \choose k}\ x^k(1-x)^{2015-k}$

Problem : Determine $\frac{f''(\frac{1}{2})}{f'(\frac{1}{2})}$ if $f(x) = \sum_{k=0}^{1000} \ {2015 \choose k}\ x^k(1-x)^{2015-k}$ Trying to simply brute force the problem, yields the following ...
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2answers
52 views

Fifth differentiation of a function

Let a function y= $x/(x^2-1)$ And we have to find $y^{\left(\mathtt{V}\right)}(0) $ I wrote $y= {1\over 2}\left[{1\over (x-1)} + {1\over (x+1)}\right]$ But I am now stuck please help me to proceed ...
1
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2answers
45 views

Non-differentiable at how many points?

There are two functions whose domain is $[-1/2,2]$ and whose co-domain is the real numbers $\mathbb{R}$. They are $f(x) = \lfloor x^2-3\rfloor$, where $\lfloor x \rfloor$ denotes the greatest ...
7
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0answers
85 views

Let $f$ be a twice differential function on the open interval $(-1,1)$ such that $f(0)=1$. Show that $f'(0) ≥ -\sqrt2$ [duplicate]

Q. Let $f$ be a twice differential function on the open interval $(-1,1)$ such that $f(0) = 1$. Suppose $f$ also satisfies $f(x) ≥ 0$, $f'(x) ≤ 0$ and $f″(x) ≤ f(x)$, for all $x ≥ 0$. Show that $f'(0) ...
2
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1answer
32 views

Why the norm in the definition of differentiability?

A function $f: \Bbb R^m \to \Bbb R^n$ is differentiable at $x_0$ iff $$\lim_{h \to 0} \frac{\|f(x_0+h)-f(x_0)-J(h)\|_{\Bbb R^n}}{\|h\|_{\Bbb R^m}}=0$$ Is there any particular reason we use the norm ...
1
vote
1answer
83 views

Proof of equivalence of two ways of calculating directional derivative

I am seeking the connection between two formulas that I saw to compute the directional derivative of function $f$ in the direction of a vector $\vec v$. One of them is : $$\nabla_{\vec v} f(\vec x_0)...
0
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1answer
28 views

Check of differentiability

We want to check differentiability of $x^\frac{1}{3}$ at $x=0$. For that I tried to find the left derivative and the right derivative and I discovered they are not equal. But at $x=0$ the curve is ...
0
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2answers
32 views

Differentiation of a double integral

I have to compute the following: \begin{equation} \frac{d}{db}\int_x^b \left (\int_s^b f(y)dy \right)g(s)ds \end{equation}
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0answers
20 views

derivative of convolution integral

I'm confused for a derivation related to the derivative of convolution. Given that $$ C_{im}(x,t)=\omega e^{-\omega t}*C_m(x,t)+C_{im}(x,0)e^{-\omega t} $$ By taking derivative of the above equation ...
0
votes
2answers
29 views

Find inflection points and concavity intervals of $f(x)=(x^2-1)^{(2/3)}$

I have a problem with this exercise. Look this image. I have answered two questions about it but, I could not find the way to establish the next: Is the function concave upward in $(1,\infty)$? Is $-...
0
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0answers
44 views

ODE inside ODE question

Given the equations of motion: $$ x'' = \frac{F - .375(\theta'' \cos\theta - (\theta')^2 \sin\theta)}2$$ and $$\theta'' = \frac{2g\sin\theta - \cos\theta (F+.375(\theta')^2 \sin\theta)}{1.5 - .375\...
0
votes
1answer
25 views

$f:[0,\infty] \rightarrow \Re$, $f(0)=a= \lim_{x \rightarrow \infty} f(x)$, f is infinitely differentiable, Prove that all differentials have roots.

$$ f:[0,\infty] \rightarrow \Re $$ $$ f(0)=a= \lim_{x \rightarrow \infty} f(x) $$ f is infinitely differentiable Prove that all differentials have roots. My take: If $f$ is constant, then it's ...
2
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0answers
25 views

Related Rates: rate of change of the speed of sound with respect to temperature.

The speed of sound, v, in air is a function of the temperature, T, of the air: $v = 331.4 + 0.6(T − 273)$ with v in meters per second and T in kelvins Suppose the rate of change of air temperature ...
0
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2answers
32 views

Infinite differentiability with a removable discontinuity?

I'm still a beginner with calculus. But this puzzled me. Let's say you had $f(x) = \frac{x^2-1}{x+1}$. It's discontinuous at one point. If you took the derivative infinitely many times, would the ...
2
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1answer
110 views

RESEARCH Q: Finding the n-th derivative of the Quotient Rule

I am a sophomore at a community college so if my writing sounds a bit gibberish please ask for clarification. My goal is to find a sequence/series that can summarize the nth derivative of a $u/v $ ...
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2answers
43 views

When this formula working “smooth”, Where from coming(with which logic)?Under what conditions working right?

$$f(x_0)=y_0 \quad \Longleftrightarrow \quad f^{-1}(y_0)=x_0$$ $$\dfrac{d}{dx}(f^{-1}(y_0))=\dfrac{1}{f'(x_0)}$$
1
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0answers
21 views

Formula for $2$nd order - difference quotient - properties of limits

I have found several threads discussing and proving: $ f''(x) = \lim\limits_{h\rightarrow0}\dfrac{f(x+h) - 2 f(x) +f(x-h) }{h^2} $ like: here or there and I am sure, there are probably more ...
0
votes
1answer
52 views

Maximium and minimum value of area.

Given that the equation of parabola is $y=x^2+1,1\leq x\leq 3$ What is the maximum and minimum value of area formed by x-axis,tangent,normal at any point on parabola. Now I wrote the equation as $x^2=...
2
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2answers
164 views

The way of calculate $\cfrac{\partial}{\partial x} \ \mathsf{A}^x $

Problem: The square matrix $\mathsf{A}\in\mathbf{R}^{n\times n}$ is not function of $x\in \mathbf{N}$. Then, how do to calculate the following matrix derivative? $$ \cfrac{\partial}{\partial x}...
1
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0answers
11 views

How do I calculate the determinant of the functional matrix?

The figure $f: \left [0,\infty \right ),\left \lfloor 0,2\pi \right \rfloor, \left [-\pi/2,\pi/2\right ]$ $\begin{pmatrix}r\\\varphi \\\vartheta \end{pmatrix}\mapsto\begin{pmatrix}rcos\vartheta\cos\...
0
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1answer
56 views

If the $100$-th derivative of $f$ vanishes on $\Bbb R$, then $f$ is a polynomial.

I have the following statement: If $f^{100}(x) = 0$ for every real number $x$, then $f$ is a polynomial. I couldn't find a counter example so I would like to get some help for prove/disprove. ...
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0answers
26 views

Uniform convergence of the function sequence $f_n(x)=n(f(x+\frac{1}{n})-f(x))$

I'm new to uniform converge in sequence function, so I have: Let $f(x)$ be a continuous differentiable function in $R$. $f_n(x)=n(f(x+\frac{1}{n})-f(x))$. I need to find $\lim_{n\rightarrow \infty}...
0
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1answer
15 views

Several variables, differentiablity, continuity and primitive function

Let $F_1 , F_2 : \mathbb{R^2} \rightarrow \mathbb{R}$ be functions defined by $F_1 (x_1,x_2 )=-x_2/(x_1^2+x_2^2)$ and $F_2 (x_1,x_2 )=x_1/(x_1^2+x_2^2)$ Then (i) $∂F_1/∂x_2=∂F_2/∂x_1.$ (ii) ...
0
votes
1answer
27 views

Predicting accelerating object collisions in a friction-less environment

Background I'm building a simple game where two ships are launching missiles at each other in space. Stuff got complicated when the ships started moving in a friction-less environment and the ...
1
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1answer
29 views

Cauchy-Riemann equations not satisfied for $\log z$ when $v(x,y)$ is defined via $\arccos$

If we define, for $z:=x+yi$, where $z,y \in \mathbb{R}$, then $\log z = u(x,y) + iv(x,y)= \ln(\sqrt{x^2+y^2})+i\underbrace{\arccos\left(\frac{x}{\sqrt{x^2+y^2}}\right)}_\Theta$ with some branch, say, $...
1
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2answers
23 views

Is the function an entire function?

Is the following an entire function? (Here $z\in \mathbb{C}$) $$\sum_{n=0}^\infty \frac{2^n}{n!}z^{3n}$$ ($***$) So, here I first note that the function is a sum of powers of $z$. Now if I show that ...
1
vote
5answers
65 views

derivative with square root

I have been trying to figure this equation for some time now, but have come up empty. I have tried multiple ways on solving it. Whether by using the Quotient Rule or some other method, I can't seem to ...
3
votes
2answers
50 views

Finding the second derivative of$f(x)=x^2\sqrt{4-x}$

Find the second derivative of the function following: $$f(x)=x^2\sqrt{4-x}$$ Here I go... $$f(x)= x^2(4-x)^{1\over 2}$$ \begin{align*} f'(x) &= 2x(4-x)^{1\over 2}+{1\over 2}x^2(4-x)^{-{1\over ...
1
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1answer
40 views

Find f'(x) at the given value of x

Find f'(x) at the given value of x $f(x)=3\sqrt{x}$ Find f'(5) For this one in my attempt to find the derivative I ended up with 9/0 which would lead me to believe that the value does not exist at ...
1
vote
5answers
68 views

Find $f'(x)$ at given value of x

Find $f'(x)$ at the given value of x $f(x)=\sqrt{x+2}$ Find $f'(7)$ My question for this one is do I approach this question by trying to find the derivative of the initial equation and then once I ...
0
votes
0answers
15 views

Finding stationary points of a function

Let $f (x) = x^ m(1-x)^n$, where $m$ and $n$ are both integers greater than $1$. Show that $f ′ (x) = (\frac m x − \frac{ n}{ 1 − x} ) f (x)$. Show that the curve $y = f (x)$ has a stationary ...
4
votes
1answer
49 views

Find all functions differentiable and convex

Find all functions $f:[0, \infty) \rightarrow [0, \infty)$, differentiable and convex, so that $f(0)=0 \tag1$ and $ \ f'(x)\cdot f(f(x))=x, \forall x \tag2$ Obviously, $f(x)=x$ is a solution, ...
0
votes
0answers
10 views

How to produce a continuous variation of a discontinuous function?

I have a differential equation that connects the "velocity" of a point in the FOV of a camera with the velocities of a robot's joints, that is $$\dot s=J(s) \dot q$$ where s is a vector with the $x$,$...
2
votes
3answers
43 views

Simplifying derivative result

I am doing the derivative of $$f(x) = \frac{x^2 -4x +3}{x^2-1}$$ So my result is the following $$f'(x) = \frac{4x^2 -8x +4}{(x^2-1)^2}$$ I am sure the answer is correct, but in my solutions book ...
3
votes
1answer
26 views

Real Analysis & Continuity

If $f:\mathbb{R} \to \mathbb{R}$ is a continuous function such that $f(x)=x$ has no real solution, then show that $f(f(x))=x$ has no real solution either. Is the proof trivial as it seems or does it ...
2
votes
0answers
36 views

Derivative solution of $\frac{(x\cos\,\theta-y\sin\,\theta)^2}{a^2}+\frac{(x\sin\,\theta+y\cos\,\theta)^2}{b^2}=1$

The equation $$\frac{(x\cos\,\theta-y\sin\,\theta)^2}{a^2}+\frac{(x\sin\,\theta+y\cos\,\theta)^2}{b^2}=1 \ \ \ \ \ \ \ \ (*)$$ has the following expression for the derivative: $$\frac{\mathrm dy}{\...
0
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0answers
26 views

Twice differentiable function in a point implies differentiability in a neighborhood of the point

I want to prove the following result that I found in the literature: Let $X,Y$ be normed vector spaces and $U$ an open set of X. If $f:U\rightarrow Y$ is a twice differentiable function in $a\in U$, ...