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

learn more… | top users | synonyms (2)

0
votes
2answers
49 views

Seamlessly connect a sine curve and a parabola

I want to seamlessly connect an unknown parabola to a known sine wave. The equations are: s(x) = a sin(bx + c) p(x) = Ax^2 + Bx + C I want to draw ...
1
vote
0answers
24 views

I need help with a partial derivative

I was given a function and I need to find a partial derivative of it. The result I got is different from the answer, and I don't know why. Here's the function: $$sin(\theta_{a}) = ...
0
votes
0answers
19 views

Applications of the Lebesgue differentiation theorem

The Lebesgue differentiation theorem states that for every locally integrable function real valued $f$ defined on $R$, the set of Lebesgue points of $f$ has full measure. I understand quite well what ...
3
votes
3answers
90 views

A concave positive function on $[1,\infty)$ is uniformly continuous

Let $f$ be a concave positive function on $[1,\infty)$, then $f$ is uniformly continuous on $[1,\infty)$. This was a true or false problem that I couldn't prove to be true, so I'm thinking that maybe ...
1
vote
0answers
38 views

Poincaré Lemma problems and computing contractions in an economical way

Let $x=(A,B,C,D)$ be coordinates on $\mathbb{R}^4$. $\displaystyle \beta = \frac{(AdB-BdA)\wedge(dC \wedge dD)+(dA \wedge dB)\wedge(CdD-DdC)}{(A^2+B^2+C^2+D^2)^2}$ I would like to compute ...
6
votes
4answers
137 views

Why is the derivative of $x^2$ not $2x+1$?

If the derivative is the change of the function at each step, it could be expressed as: $$f(x)+f'(x)=f(x+1)$$ Therefore if $f(x)=c$ $$c+f'(x)=c \implies f'(x)=0$$ This is also correct for ...
1
vote
2answers
50 views

Evaluating $\frac{d}{dx}\int_1^{x^2}\sqrt{y^2+3}dy $

How to Evaluate $$\frac{d}{dx}\int_1^{x^2}\sqrt{y^2+3}dy $$On doing differentiation of integration we obtain the same function but here limit of integration is also to be applied .How can we do it ...
1
vote
2answers
23 views

Evaluating $\frac{d}{dx} \int_{1}^{3x} \left(5\sin (t)-7e^{4t}+1\right)\,\mathrm dt$

$$\dfrac{d}{dx} \int_{1}^{3x} \left(5\sin (t)-7e^{4t}+1\right)\,\mathrm dt$$ The answer I come up with is: $5\sin(3x)(3)-7e^{4(3x)}(3)$, however this was not on the answer choice. What is the ...
3
votes
1answer
63 views

Real roots of $p(x)=x^n+ax+b$

What can we say about the real roots of $p(x)$? My Work: If $n$ is odd I found that $p$ has at most $3$ real roots if $a<0$ and $p$ has at most $1$ real root if $a\geq 0$. How can I classify the ...
2
votes
1answer
43 views

Evaluating derivative of $\int^{3x}_{2x} \sin(t^3 + 1) \,\mathrm dt$

Maybe I'm not very good at my trig rules but I'm having a tough time finding derivative of $$\int^{3x}_{2x} \sin(t^3 + 1) \,\mathrm dt$$ I believe that $u = t^3 + 1$ and $du = 3t^2$, but I'm not ...
-5
votes
2answers
63 views

Differentiability proof of exponential function $\sum_{n=0}^ \infty \frac{x^n}{n!}$

$$f(x)=\sum_{n=0}^ \infty \frac{x^n}{n!}$$ I want to prove that $f$ differentiable on $x$ in $[0,1]$. I am not clear with using the definition of differentiability. I can prove it is ...
1
vote
1answer
49 views

Evaluating $\frac{d}{dx}\int_{7-2x}^3 \frac{u^3}{1+u^2}du$

Evaluate following expression by using The Fundamental Theorem of Calculus $$\frac{d}{dx}\int_{7-2x}^3 \frac{u^3}{1+u^2}du$$
0
votes
2answers
22 views

Simple Integration Question - Integral of the product of a function and its derivative.

Why does $\int y'(x) y(x) dx = y(x)^2/2 + C$? This seems to be true, at least according to Wolfram, but I do not understand how it is derived.
-1
votes
0answers
13 views

Tangent line parallel to a chord in a parabola

In a parabola with equation $f(x) = ax^2+bx+c$, a chord $AB$, $A=(xa,f(xa))$ and $B = (xb,f(xb))$, is parallel to the tangent line at $x = \frac{xa+xb}{2}$. It's easy to verify that using ...
3
votes
3answers
102 views

$dy$ by $dx$ or $dy$ divided by $dx$

I was always taught do not say "$dy$ divided by $dx$", instead "$dy$ by $dx$" because it's not really dividing. I then studied differentiation from first principles, where one takes two points on a ...
3
votes
0answers
99 views
+50

Use of Poincare Lemma in solving $\nabla \times \textbf{A}(\textbf{r})=\frac{\textbf{r}}{r^3}$

Let $U = \mathbb{R}^3 \setminus \{(0,0,z) \}$ (ie $\mathbb{R}^3$ with the $z$-axis removed ) and consider $\beta$ on $U$ given by $\displaystyle \beta = \frac{x dy \wedge dz + y dz ...
0
votes
1answer
33 views

Finding arc length by approximating

Let's say that a have a smooth curve in 2D and I want to find it's lenght. I split the curve in sections like here: http://www.whitman.edu/mathematics/calculus_online/section10.03.html. The only ...
2
votes
1answer
19 views

Recursive functions.

If you have a recursive function $$g(x) = f(f(x))$$ and you know that $$f(0) = 0, f'(0) = 1, f''(0) = 2$$ Will then $$g(0) = 0, g'(0) = 1, g''(0) = 2$$ ?
0
votes
1answer
20 views

Use MVT to prove $(k+1) \cos \left ( {\frac{\pi}{k+1}} \right ) - k \cos \left( \frac{\pi}{k} \right) > 1 \qquad \forall k \geq 2 $

I want to prove $$(k+1) \cos \bigg ( {\frac{\pi}{k+1}} \bigg ) - k \cos \bigg ( \frac{\pi}{k} \bigg ) > 1 \qquad \forall k \geq 2 $$ using Mean Value Theorem. My thought was to apply MVT to $f(x) = ...
1
vote
0answers
38 views

When the function f is constant?

Let $ f: \mathbb R^n \to \mathbb R^n$ be a differential function. Let $Df(x)$ be the derivative of $f$ at $x$ in $\mathbb R^n$. Then which of the following is/ or correct? $Df(0)(u) = 0$ for all $u$ ...
0
votes
1answer
61 views

Identity concerning Lie derivative of $k$-form $\omega$

Let $X$ and $Y$ be vector fields on $\mathbb{R}^n$. Show that for $\omega$, a $k$-form on $\mathbb{R}^n$, $(L_XL_Y-L_YL_X)\omega=L_{[X,Y]}\omega $. I try using Cartan's magic formula and get that ...
0
votes
0answers
51 views
+50

Identity of the pushforward of a vector field using a Jacobi bracket.

Let $Z(u,v)$ be the vector field $Z(u,v)=(u^2+u,v^2+v)$, let $\Gamma_t$ denote its flow. I have shown that $[X,Z]=Z-X$. Show that $(\Gamma_t)_*X=e^{-t}X-(e^{-t}-1)Z$. Could someone please show me ...
0
votes
0answers
23 views

Derivative with constraint

Consider function $F=F(q(t),p(t))$ with constraint $p(t)=q'(t)$, where $'$ denotes time derivative. Let $\displaystyle G=F'=\frac{\partial F}{\partial q}q'+\frac{\partial F}{\partial p}p'$, I want to ...
1
vote
1answer
34 views

is there a way to simplify $x^{2} ( v' (x^{n})' )$?

so I have what is probably an algebra question, if I have $x^{2} ( v' (x^{n})' )$ where the ' denotes a derivative, is there a way to simplify this expression?
2
votes
1answer
50 views

use fundamental theorem of calculus to find a function $f(x)$ and a number $a$

I thought I understood the fundamental theorem of calculus but I'm confused on the following problem.. Use the Fundamental Theorem of Calculus to find a function $f(x)$ and a number $a$ so that ...
4
votes
2answers
67 views

There is a function which is continuous but not differentiable

I have a function which is a convergent series: $$f(x) = \sin(x) + \frac{1}{10}\sin(10x) + \frac{1}{100}\sin(100x) + \cdots \frac{1}{10^n}\sin(10^nx)$$ This function is convergent because for any E ...
0
votes
1answer
29 views

How can we find another path

Can you help me about this question please, Thank you..
2
votes
4answers
36 views

How can I differentiate this equation?

I need to differentiate this: $$ y = b(e^{ax}-e^{-ax}) $$ I've got the solution from a book, but I don't found the process to differentiate it. The solution is: $$ y = ab(e^{ax}+e^{-ax}) $$ Here ...
1
vote
1answer
32 views

If $f(z)$ is an entire function, prove that it has a zero at $z_0$ of order $k\ge 1$ iff $z_0$ is a simple pole of $\frac{f'(z)}{f(z)}$

Let $f(z)$ be an entire function. Prove that $f(z)$ has a zero at $z_0$ of order $k\ge 1$ iff $z_0$ is a simple pole of $\frac{f'(z)}{f(z)}$ and the residue of $\frac{f'(z)}{f(z)}$ at $z_0$ is $k$. ...
1
vote
1answer
29 views

Finding the Lie derivative of a 2-form exercise

Let $\beta=-x dx \wedge dy + ydy \wedge dz$. The vector field is $X=(y,0,z)$.Find the Lie derivative. I try that $\begin{align}L_X \beta &=L_X (-x dx \wedge dy + ydy \wedge dz) = -x L_X(dx ...
1
vote
3answers
71 views

Derivative=0 implies double root

Let a be the root of a polynomial f(x) and let f'(a)=0. Then a is a double root of f(x). What is the name of this theorem and does someone know a simple (high school level) proof?
0
votes
1answer
18 views

Identity about composition of the push forward of diffeomorphisms

I am able to do part a) and I believe it should be used in solving part b). I think that for part b) we should that $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$, $G: \mathbb{R}^n \rightarrow ...
0
votes
3answers
54 views

Find the 27th derivative

Let $f^{\left(n\right)}\left(a\right)$ denote the n-th derivative of $f$ at $a$. If $f\left(x\right)=\left(1+x^2\right)^{15}$, then $f^{\left(27\right)}\left(1\right)\:=$? Hints appreciated!
1
vote
3answers
39 views

What is the value of the constant C?

The curve $y=Cx^{\frac{1}{5}}$ (where C is constant) is tangent to the line $y=\frac{x}{20}+\frac{32}{5}\:$ somewhere. What is the value of constant C?
1
vote
3answers
49 views

Complex Mean Value Theorem: Counterexamples [closed]

This thread is just to collect some examples... Given an open domain $\Omega\subseteq\mathbb{C}$. Consider a holomorphic function $f:\Omega\to\mathbb{C}$. What would be a counterexample to: ...
1
vote
7answers
43 views

What is the smallest possible value of their sum?

The product of two positive numbers is 36. What is the smallest possible value of their sum? so far I got $$xy=36$$ $$y=\frac{36}{x}$$
0
votes
1answer
66 views

What's wrong when computing $\frac{d\dot f}{d\dot x}$

Let $x(t)=t^3$ and $f(x)=e^x$. By direct calculation, I obtain $\dot x(t)=3t^2$, $f'(x)=e^x$ and $\dot f(t)=f'(x)\vert_{x=x(t)} \dot x(t)=3t^2e^{t^3}$. Now I want to compute $\displaystyle{\frac{d\dot ...
0
votes
0answers
15 views

Calculating total derivatives

I have been asked to calculate the total derivative of this linear transformation and this function: $ M = \left[ \begin{matrix} A & B & C \\ D & E & F \\ G &H &I\end{matrix} ...
0
votes
1answer
33 views

derivative of a function from $\mathbb R^n$ to $\mathbb R$

Let $ L:\mathbb R^n\to\Bbb R$ be the function $L(x)=\langle x,y\rangle$ , where $\langle,\rangle$ denotes some inner product on $\Bbb R^n$ and y is a fixed vector in $\Bbb R^n$. Further denote by DL, ...
0
votes
1answer
25 views

Partial derivatives as a basis?

Back when I was studying multivariable calculus, I had a sort of "intuition" for the formula for taking directional derivatives (the scalar product of the gradient and the vector that gives the ...
0
votes
4answers
61 views

How to differentiate $2\left | x \right |$?

According to the answer of my calculus textbook, the derivative of $2\left | x \right |$ is $2x\left | x \right |^{-1}$, but why? Here is the answer:
0
votes
1answer
26 views

Parametrization and computing distance traveled

I have no idea how to get started on this problem, and would greatly appreciate being pointed in the right direction: A car travels at a constant speed of $1 miles per hour$ from point $A(1,1)$ to ...
0
votes
2answers
40 views

How to differentiate $\ln(a^x)$?

Can someone give me the process to differentiate this (with respect to $x$)? $$ \ln(a^x) $$
0
votes
1answer
29 views

Is the gradient vector of a function the derivative of the function

Is the gradient vector the derivative of any function? I was wondering this because this works with a function of one variable.
0
votes
3answers
57 views

Proving $\frac{\mathrm d}{\mathrm dx}(\tan x)=\sec ^2x$

I can find the derivative of $x^2$ by using $$\lim_{h\rightarrow 0}\dfrac{(x+h)^2-x^2}{h}$$ By the same way, if I want to find the derivative of $\tan x$ $$\lim_{h\rightarrow 0}\frac{\tan (x+h)-\tan ...
-1
votes
0answers
29 views

Absolutely continuous function and differentiability

let $m(E)=0$ in $[a,b]$ Can we define a absolutely continuous function on $[a,b]$ which is not differentiable only at $E$.
1
vote
1answer
24 views

Taking the Derivative: Power Rule with Respect to Vector

I'm trying to take the derivative of \begin{equation} \phi\left(\mathbf{x}\mathbf{\theta}\right)\mathbf{x}^{\top} ...
1
vote
1answer
49 views

How many critical values does $f$ have on $(0,10)$, given $f '(x)=\frac{\cos^2 x}{x} -\frac{1}{5}$? [closed]

How many critical values does f have on open interval $(0,10)$ given $$f'(x)=\frac{\cos^2x}{x}-\frac{1}{5}$$ I'm in calculus AB and this is a review question. I think the next step is to make it ...
0
votes
2answers
51 views

Does this piecewise function contradict the fact that all differentiable functions are continuous?

I learned that all differentiable functions are continuous. Why does the following equation not violate this rule: $$f(x)=\begin{cases}x^2+3 \quad &\text{when } x>1 \\ x^2 \quad ...
0
votes
2answers
42 views

Using Differentials Problem: Can't Separate x and y

I've been asked to estimate a y coordinate by using differentials. This normally isn't overly difficult, however, I'm not sure what to do in a case like this when y cannot be separated and used as a ...