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

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3answers
175 views

Second derivative of $\arctan(x^2)$

Given that $y=\arctan(x^2)$ find $\ \dfrac{d^2y}{dx^2}$. I got $$\frac{dy}{dx}=\frac{2x}{1+x^4}.$$ Using low d high minus high d low over low squared, I got $$\frac{d^2y}{dx^2}=\frac{(1+x)^4 ...
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0answers
16 views

unimodality and continuous

i would like to ask question about unimodality of probability function ,from wikipedia http://en.wikipedia.org/wiki/Unimodal it says that In mathematics, unimodality means possessing a unique mode. ...
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1answer
80 views

How to show that this function is differentiable?

Let $$\phi: \mathbb{R} \rightarrow \mathbb{\mathbb{C}}, s \mapsto \int_2^{\infty} \frac{e^{isx}}{x^2\ln(x)}dx$$, I want to show that this function is differentiable everywhere. Unfortunately, it ...
0
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1answer
61 views

properties of third order derivative

I have a question in my assignment (about interpolation) which has the condition that the third-order derivative is continuous at $x_2$ and $x_{N-1}$. That is, $S'''(x_2)=S'''(x_{N-2})$. The question ...
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2answers
45 views

Calculate the energy in a circuit containing a resistor

A voltage peak in a circuit is caused by a current through a resistor. The energy E which is dissipated by the resistor is: Calculate E if Can anyone please give me some suggestions where to ...
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1answer
38 views

Fréchet derivative and local maximum

I'm pretty confused with the idea of local maximum in function spaces. Normally having a null Fréchet derivative is a necessary but not sufficient condition for being a local maximum. Computing the ...
3
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0answers
37 views

Prove the following expression is true.

Let $x_1,...,x_{n+1}$ be arbitrary points in $[a,b]$ and let $$Q(x)= \prod\limits_{i=1}^{n+1} (x-x_i)$$Now suppose $f$ is an n times differentiable function and tha P is a polynomial function of ...
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1answer
41 views

Definition of Limit of a Function

I'm a little confused by the definition of the limit of a function-on one hand I feel the definition suggests that your limiting variable is shrinking into a little delta ball- on the other hand when ...
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2answers
222 views

What is the mistake in this proof of product rule of differentiation?

I was trying to derive the product rule of differentiation which states: If $y=u\cdot v$, then, $y'=u'\cdot v+v'\cdot u$. So I assumed it like this: $y=u+u+u+\cdots$ ($v$ number of terms of $u$) ...
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1answer
14 views

Functions with symmetrical behaviour with respect to an axis or a plane

Suppose we have two functions with a symmetrical behaviour with respect to an axis. For the sake of simplicity, let $f(x)$ and $g(x)$ have a symmetrical behaviour with respect to the $y$ axis. A ...
2
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1answer
54 views

supremum of derivatives

Let $f$ twice continuously differentiable on $(a, \infty)$. Let $M_{0} = \sup f$, $M_{1} = \sup f'$, $M_{2} = \sup f''$. Show that $ (M_{1})^{2} \leq M_{0}M_{2}$. Also, How can this be modified ...
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2answers
59 views

Calculus - Derivatives of Polynomials

$f(x)=-3x^2-14x-5$ and $g(x)=\frac{1}{2}x^2-\frac{15}{2}x+29$ are parabolas on the same grid. The tangent to $y=f(x)$ at $P_1(x_1,y_1)$ and the tangent to $y=g(x)$ at $P_2(x_2,y_2)$, intersect at ...
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1answer
71 views

Tangent of a sine - can it disprove the marginal value theorem?

The marginal value theorem is partly explained by this text and graph: As animals forage in patchy systems, they balance resource intake, traveling time, and foraging time. Resource intake within ...
0
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1answer
60 views

How to solve the gradient of a matrix function

Let: $$A=\hat{r}+e^{j\theta}$$ Where $\hat{r}, e^{j\theta} \in \mathbb{C}^n$, and j = (-1)^(.5). Set $L \in \mathbb{R}^{n \ x \ n}$, define $S$: $$S=A^TLA$$ The gradient of $S$ is given by ...
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1answer
42 views

Derivative of $f(x)=\|Ax\|_2^2$

I'm trying to find the derivative of $f(x)=\|Ax\|_2^2$ where $A$ is some matrix and $\|u\|_2$ is the euclidean norm of $u$, $\|u\|_2 = \sqrt{u_1^2+u_2^2+\cdots+u_n^2}$ I know how to do this by ...
0
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1answer
22 views

Finding coefficients of a third degree polynomial

The third degree polynomial $$-x^3 + ax^2+bx+c$$ has an maximum at $(2,10)$ and an inflation point at $(0,-6)$. Find the coefficients $a$ $b$ and $c$. Am I supposed to differentiate the polynomial ...
3
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0answers
137 views

The second derivative as a limit

It is well-known that if $f$ is twice differentiable at $a$, then $$ f''(a) = \lim_{h\to 0} \frac{f(a+2h)-2f(a+h) + f(a)}{h^2}. $$ See e.g. this question or this question. On the other hand, the ...
3
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1answer
119 views

How to prove l'Hospital's rule for $\infty/\infty$

I'm having trouble with this l'Hospital's rule wiki page(the proof of l'Hospital's rule): http://en.wikipedia.org/wiki/LHospital%27s_rule Well, in the case where the limit looks like $0/0$, it's ...
4
votes
2answers
65 views

Are $C^{\infty}$ completely defined by their derivatives?

This question has been on my mind for some time. Here's my process. Firstly, is it possible to construct a function such that it's defined with a different expression on different intervals, but that ...
2
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0answers
49 views

Is little-o preserved under integration and derivation of another variable?

Given an integrable function $g:\mathbb{R}\longrightarrow\mathbb{R}$, and a function $f:\mathbb{R}^2\longrightarrow\mathbb{R}$ such that $f(x,y)=o(x^{-1})$ when $x\rightarrow\infty$, i.e. ...
0
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1answer
39 views

Find the volume of a cone whose length of its side is $R$

How can i compute the volume of a cone whose length of its side is $R$ and the vertex of the cone forms an angle $2θ$ . The top cone is a cap of a sphere of radius $R$. I tried to solve first in 2 ...
2
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2answers
70 views

Derivative $ \frac{ d^2}{d^2 x} \frac{\Gamma(x+1) }{\Gamma(x+3)}$

I know this may seem like a really low level/silly question, I apologize in advance. I do not know how to differentiate the gamma,beta,digamma function, for ex: $$ \frac{ d^2}{d^2 x} ...
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1answer
25 views

How do I convert the limit definition of differentiability to different variables?

I want to convert this: $$\lim_{h \to 0} \frac{f(x_0 + h) - f(x_0) - J(h)}{\|h\|} = 0$$ Into the version of the limit where it has $\lim\limits_{(x,y) \to (0,0)}$ instead of $h$. How do I do this?
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1answer
73 views

Describe the graph of f if the graph of its integral its given

Describe the graph of $f$ if the graph of its integral $g(t) = \int_{0}^{t} f(s) ds $ is: graphic of g graphic of f I analyze the derivative and the sign of the derivative and try to find ...
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2answers
34 views

Help finding a function given its derivative

I have shown in a problem that $ f'(x) = -f (x) $ . I can't seem to find any function that has this property. Does anyone have any ideas?
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2answers
62 views

fence a circular land and a square land.

With a wire mesh of 1000 mts divided into two parts , we want to fence a circular land and a square land. a)Calculate the lengths of each of the parties such that the total area enclosed is ...
0
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1answer
38 views

Find f with A plane curve whose equation is $y - f (x) = 0$ passes through the origin.

A plane curve whose equation is $y - f (x) = 0$ passes through the origin.Consider the rectangle $R_x$ formed by the coordinate axes and lines parallel to the axis passing through the point $(x, f ...
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1answer
16 views

Proof about First order derivative

Show that if $f'(c)>0$ then there exists $\delta>0$ such that $x \in (c,c+\delta) \ \ \implies \ \ f(x)>f(c)$ $x \in (c-\delta,c) \ \ \implies \ \ f(x)<f(c)$ My Attempt Now ...
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2answers
43 views

For what value(s) of $a,b$ does this limit hold?

For which value(s) of $a$ and $b$ is $$ \lim_{x \to 0}(x^{-3}\sin(3x)+ax^{-2}+b)=0$$ My Attempt Firstly I rewrote it into this form; $$ \lim_{x \to ...
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1answer
32 views

$f$ is continuous and satisfies the equality given for all $0 \leq x $

How can i compute $f (2)$ if $f$ is continuous and satisfies the equality given for all $0 \leq x $ : $\int_{0}^{f(x)} t^2 dt = x^2(1-x)$ and $\int_{0}^{x^2(1-x)} f(t) dt = x$ Some help for this ...
3
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2answers
50 views

Length of Curve $6xy=3+x^4$

Question : Find the length of the curve $6xy=3+x^4$ between $x=1$ and $x=2$. Answer = 17/12 I have tried this: I obtain a different answer. Where did I do wrongly? Thank you for your ...
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1answer
121 views

Every closed subset $E\subseteq \mathbb{R}^n$ is the zero point set of a smooth function

In Walter Rudin's Principles of mathematical analysis Exercise 5.21, it is proved that for any closed subset $E\subseteq \mathbb{R}$, there exists a smooth function $f$ on $\mathbb{R}$ such that ...
2
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1answer
50 views

Let $z=\ln \tan\frac xy.$ What is $z_x$ and what is $z_y$?

Let $$z=\ln \tan\frac xy.$$ What is $z_x$ and what is $z_y$? Thanks ahead:) What I have tried: $$z_x=\frac{1}{\tan \frac xy} \frac{1}{1+(\frac xy)^2} \frac 1y=\frac {y}{\tan \frac xy (x^2+y^2)}$$ ...
2
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4answers
96 views

Differentiability at a point $(0,0)$

How would i show $$\frac{xy(x^2-y^2)}{(x^2+y^2)^{3/2}}$$ is not differentiable at $(0,0)$
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1answer
22 views

Derive the “Marginal Product” of x and z by taking the partial derivatives of the production function.

A Firm has the production function Q= 0.95 x + ln(z) + 100 where x and z are a variable inputs. Derive the "marginal product" of x and z by taking the partial derivatives of the production ...
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4answers
55 views

what is the derivative of the function $f(x)=2.5\sqrt{400-x^2}+1.5\left(80-x\right)$?

$$f(x)=2.5\sqrt{400-x^2}+1.5\left(80-x\right)$$ Please explain how you would find the derivative in the scenario, thanks
4
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1answer
140 views

Showing $f^{(n-1)}(\xi) = 0$ for some $\xi$

Let $f$ be an $n$ times differentiable function on the interval $A$. If $x_1 < x_2 < \cdots < x_p$ are points on $A$ and $n_i, 1 \leq i \leq p,$ are natural numbers such that $n_1 + n_2 + ...
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0answers
32 views

A question on self-adjointness of the shape operator

In the book Elementary Differential Geometry of Christian Bar, CUP, on the page 108, of the proof of theorem 3.5.5, the author wrote: Theorem: Let $S ⊂ \mathbb{R}^3$ be an orientable regular surface ...
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0answers
25 views

Polar coordinates: Slope of tangent

Would anyone mind telling me how to solve this problem? It seems strange as my answer is $-1$. Do I have to apply this formula, $(r'sinθ +rcosθ)/(r'cosθ -rsinθ)$ ?
2
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1answer
114 views

Does the existence of partial derivatives imply Frechet differentiability?

Let $f : \mathbb R^n \rightarrow \mathbb R^m$ and $a \in \mathbb R^n$such that $\forall i \in [1,n], \large \frac{\partial f}{\partial x_i}(a)$ exists. Is $f$ Frechet differentiable ? I'd say no, ...
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2answers
64 views

Polynomial interpolation using derivatives at some points

Given $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4), (x_5, y_5)$, we can interpolate a polynomial of degree 4 using Lagrange method. But, when we are given $(x_1, y_1), (x_2, y_2), (x_3, y_3), ...
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0answers
37 views

Generalizing the interpretation of a derivative as a fraction

Sometimes derivatives are (inappropriately) interpreted and handled as if they were fractions. For instance, when applying a chain rule, $df/dx$ would be handled as if $df$ was the numerator and $dx$ ...
3
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1answer
122 views

About equivalent norms

Consider $E$ the space of the functions $f: [0,1] \to \mathbb{R}$ such that $f(0) = 0$ and $f$ satisfies a Lipschitz condition. We define two norms: $$\|f\| = \sup_{x \in [0,1]} |f(x)|$$ and ...
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0answers
25 views

Finding a derivative through the definition

Let $f$ be a function of domain $\mathcal D$ and codomain $\mathcal C$, both subsets of $\mathbb R$, and $\mathcal D_{\mathrm{cl}}$ the set of cluster points of $\mathcal D$. I want to find $f'$ by ...
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4answers
225 views

Differentiating $\ln(4x)$.

Why is the derivative of $\ln(4x)$ equal to: $\frac{1}{x}$ Shouldn't it be? $$\frac{1}{4x}$$ This seems so wrong to me because the derivative of $\ln x$ is $\tfrac{1}{x}$.
2
votes
4answers
184 views

Find the equation of the tangent line

Find the equation of the line tangent to $f(x) = \tan x$ at $x = \frac{\pi}{4}$. I'm trying to incorporate the slope point formula using $f'(x) = \sec^2 x$ but I'm nowhere near!
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1answer
28 views

For what interval does this power series converge and for what interval does it determine a differentiable function?

For what range of values of $x$ does $\sum_{n=1}^{\infty } \dfrac{1}{n}(1+\sin x)^n$ converge? Find with proof an interval on which it determines a differentiable function of $x$ and show that ...
0
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1answer
61 views

Mean value theorem for the second derivative: $f(a+h)-2f(a)+f(a-h) = h^2f''(\xi)$

Attempt; First Part Here I just have to stat the definition of Mean Value Theorem; If $g$ is continuous on $[z-a,z+a]$ and differentiable on $(z-a,z+a)$ the there exists $\xi \in (z-a,z+a)$ such ...
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1answer
48 views

Can someone explain the concept of continuity and differentiability for functions of several variables?

Can someone explain the concept of continuity and differentiability for functions of several variables? Illustrated examples will definitely help, on how to solve problems(or establish proofs) of the ...
0
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1answer
34 views

Implicitly differentiate cone volume

I think I have the derivative with respect to $ h $ right for the equation $V = \frac {1}{3} r^2h $. I got $$\frac {dV}{dh} = \frac{1}{3} \pi r^2 + \frac {2}{3}\pi rh \frac {dr}{dh} $$. Is that ...