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

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How fast is this dot moving when the angle $θ$ between the beam and the line through the searchlight perpendicular to the wall is $π/6$?

A searchlight rotates at a rate of $4$ revolutions per minute. The beam hits a wall located $11$ miles away and produces a dot of light that moves horizontally along the wall. How fast (in miles per ...
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3answers
30 views

How to convert this particular expression into some desired form?

The parametric equations of a curve are $$x=\cos(t) \cdot e^{-t} $$ $$y=\sin(t) \cdot e^{-t} $$ Show that $dy/dx =tan(t-\pi/4) $. how to solve this? I can get a $dy/dx$ but i cannot convert into the ...
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0answers
10 views

Why is the rotation not zero and the divergence zero in the figures below?

Why is the rotation not zero and the divergence zero in figure 1 and figure 2 below? Figure 1 Figure 2
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1answer
40 views

Prove $f'''(x) \geq 3$ for some $x \in (-2,2)$, if $f$ is cont on $[-2,2]$ and three times differentiable in $(-2,2)$ & $f(2)=-f(-2)=4$ & $f'(0)=0$

Prove that there exists a $x \in (-2,2)$ such that $f'''(x) \geq 3$, if $f$ is cont on $[-2,2]$ and three times differtiable in $(-2,2)$ with values $f(2)=-f(-2)=4$ & $f'(0)=0$. How do I handle ...
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23 views

Increasing/ decreasing functions

We are given a random variable x with a pdf f(x) and F(x) is its distribution function. Let $$r(x) = \frac {xf(x)} {1-F(x)} $$ Then for $x< e^{\mu} $ and $$f(x) = \frac {e^ {1/2(\log x - \mu)^2}} ...
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1answer
27 views

Derivation of Ln root

$$f(x) = \ln \sqrt{4x-3}$$ I'm practicing derivation for the exam and I'm stuck on this task. Could someone help me out in solving this. But when it comes to root, I'm a bit confused. The result is ...
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1answer
25 views

Derivative for numerical models.

I am working in Mechanical engineering and Computer vision, in which I use a matlab code (or codes) to represent a specific system or process. Of course such model has an input , an implimented ...
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1answer
17 views

Let $f(x) = |x − 1| + |x − 2|^3$ for $x\in \mathbb{R}$ . Then examine the differentiability of $f$ at x = 1 and x = 2 and rest of $x\in \mathbb{R}$ .

Let $f(x) = |x − 1| + |x − 2|^3$ for $x\in \mathbb{R}$ . Then examine the differentiability of $f$ at x = 1 and x = 2 and rest of $x\in \mathbb{R}$ . derivative of the function f at a: ...
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1answer
22 views

directional derivative problem

for a point M(4,1) and a function $z = x y^2 - (x^2/y^3)$ I was tasked with finding a directional derivative in the direction which creates a 30 degree angle with the $x$ axis....I find it a little ...
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25 views

How to find second and third derivative of Probit likelihood function

I am using Probit likelihood function for a classification problem. I need second and third derivatives of the log likelihood function with respect to $f$. The likelihood function is $log\Phi (y\cdot ...
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2answers
18 views

Jacobi's Derivative of the Determinant

I've been given the following theorem for the derivative of the determinant of a matrix: "Let $A\in \mathbb{R}^{n\times n}$ be a square matrix. Then the Fréchet derivative of det$: ...
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1answer
44 views

Diffeomorphism from disk to plane

I want to show that the disk $D = \{(x,y) \in \mathbb R^2 : x^2+y^2 < 1\}$, the open square $K = (-1, 1)^2$ and the whole plane $\mathbb R^2$ are all diffeomorphic to each other. Therefore I want ...
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5answers
135 views

Estimate $\int^1_0 e^{-x^2}\, dx$

Estimate $\int^1_0 e^{-x^2}\, dx$ This is in a section on Taylor series so I would assume that is how it should be solved. I started by using the Taylor series formula for $e^x$ replacing $x$ with ...
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1answer
24 views

Quotient rule for derivatives..am I making this to complicated

This is a straight forward question.. When I have something like 10/x (i.e basically whenever the numerator is just a number with no variables) and I need to take the derivative I go through the ...
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1answer
25 views

Using Taylor series find derivatives of arctan(x)

Using Taylor series for $arctan(x)$, find $f^{(5)}(0)$ and $f^{(6)}(0)$ for $f(x)=arctan(x)$ I figure for this problem I compare the general Taylor series formula to the Taylor series for $arctan(x)$ ...
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0answers
22 views

Show that this function is differentiable at all points [on hold]

n-sphotos-h-a.akamaihd.net/hphotos-ak-xta1/v/t34.0-12/11116109_10206718706905332_835173146_n.jpg?oh=baf1ad15e0f70703e5eb93818b61c9d1&oe=55401033&gda=1430237746_5bfaae4f8271730ad293b579ab0e93ab ...
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0answers
44 views

On a problem about Rolle's theorem

Let $f:[1,3]\to\mathbb R$ be a continuous function such that $\int_1^2 f(x)dx=2$, and $\int_1^3 f(x)dx=3$, then there exists a real number $c\in(2,3)$ such that $$ \int_1^c f(x)dx=cf(c) $$ Note. I ...
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1answer
27 views

If f is continuous and strictly increasing, then the function $f^{-1}:f(I)\rightarrow I$ is continuous and strictly increasing.

Let $I \subseteq \Bbb R$ be a non-degenerate open interval, and let $f:I\rightarrow \Bbb R$ be a function. Suppose that f is strictly monotone. If f is continuous and strictly increasing (or ...
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1answer
18 views

Initial value problem through origin

$\frac{dz}{dt}=8t*e^z$, Through the origin I have never done an initial value problem before, but I took it to mean that it gave me the initial value of the differential equation (0, 0) and that I ...
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2answers
26 views

Find solution to the differential equation

$\frac{dB}{dx}+2B=50$ $B(1) = 50$ I tried separating the variables but that didn't work, and without separating the variable I'm not sure what to do.
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2answers
24 views

Differentiating inverse hyperbolic function

I am trying to differentiate $\tanh^{−1}\left(x/(1 + x^2)\right)$, but am finding it difficult understanding what to do. I think you have to place the differential of the angle of the hyperbolic ...
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3answers
24 views

Differentiating inverse trig function

When differentiating $\sin^{-1}(x/2)$, I got $\frac{1}{2}(4-x^2)^{-1/2}$ but the answer I'm given does not include being multiplied by half. Can anyone explain if the answer I'm given is right and ...
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2answers
15 views

Differentiation of a constant function from first principles

How do you differentiate a constant $K$ from first principles to show that it equals zero? $f(x) = K$ but what does $f(x+h)$ equal to where $h$ is the change in $x$?
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2answers
31 views

Stationary points of $ \ln(x + 1)$?

I'm trying to find the stationary points of $f(x) =\ln(x + 1)$. When I differentiate, I get $f'(x) =\dfrac{1}{ (x + 1)}.$ I then set that to zero and end up getting $1 = 0$? I'm not sure what this ...
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2answers
49 views

Find the solution to the differential equation

Assume $x\gt 0$ $$x(x+1)\frac{du}{dx} = u^2$$ $$u(1) = 4$$ I started off by doing some algebra to get: $$\frac{1}{u^2}du = \frac{1}{x^2+x}dx$$ I then took the partial fraction of the right side of ...
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1answer
14 views

Is this function differentiable w.r.t. a variable in an indicator?

I have $$y^i = x^i - \alpha \sum_{j \epsilon N} (x^j - x^i) I_{x^i \lt x^j} - \beta \sum_{j \epsilon N} (x^i - x^j) I_{x^i \ge x^j}$$ where N = {1, 2, ..., n}, and $I_{x^i \lt x^j}$ is 1 when $x^i ...
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1answer
23 views

Let $f\colon [a,b]\to\mathbb R$ is continuous and $G(x,t)=t(x-1)$ when $t\leq x$ and $x(t-1)$ when $t\geq x$.

Let $f\colon[a,b]\to \mathbb R$ is continuous and $$G(x,t)=\begin{cases}t(x-1)&\text{when $t\leq x$,}\\x(t-1)&\text{when $t\geq x$.}\end{cases}$$ Let $$g(x)=\int_0^1f(t)G(x,t)\,\mathrm dt.$$ ...
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3answers
33 views

A problem of Schwarz derivative [on hold]

I need help with the following problem analysis: Suppose $f$ is defined on an interval around $x$. The limit $$\lim_{h\to0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2},$$ if it exists, is called the Schwarz ...
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0answers
13 views

Consider the Plane Curve?

Consider the plane curve $$\gamma(t) = \left( \cosh(t) \cos(t), \cosh(t) \sin(t) \right), \;\; t \in \mathbb R.$$ Is $\gamma$ regular? If $\gamma$ is not regular, can you restrict the parameter ...
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2answers
10 views

Time dependence of velocity from position dependece of velocity

I know dependence of velocity on position $v(x)$ and I wan't to know dependence of velocity on time $v(t)$ I was thinking that using some chain rules or derivative of inverse it would be possible to ...
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0answers
20 views

Find global minimum of the function

I need to find the global minimum of the function $$f ( x) = \langle Ax,x \rangle + 2\langle b ,x\rangle+c$$ where $c \in \mathbb{R}$ is constant, $b \in \mathbb {R}^n$, and $A$ is a positive ...
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1answer
17 views

Problem with real differentiable function involving both Mean Value Theorem and Intermediate Value Theorem

Problem: Let $a,b \in \Bbb R$, $a<b$, and let $f$ be a differentiable real-valued function on an open subset of $\Bbb R$ that contains $[a,b]$. Show that if $\gamma$ is any real number between ...
3
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3answers
156 views

Finding the derivative of the integral using the Fundamental Theorem of Calculus.

I'm still not entirely solid on the concept of the Fundamental Theorem of Calculus, but I believe that the first step of the theorem will give us $$2x-1$$ which is the derivative of F(x). Usually, ...
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2answers
42 views

Find when the population is growing the fastest, under the logistic model

The population $P$ of an island $y$ years after colonization is given by the function: $\displaystyle P = \frac{250}{1 + 4e^{-0.01y}}$. After how many years was the population growing the fastest? ...
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2answers
111 views

Derivative of the power tower

May somebody help me to correctly calculate the dervative of the $n$-th power tower function? $$ \begin{align} f_1(x)&=x\\ f_n(x)&=x^{f_{n-1}(x)}\\ &=x^{x^{x^{...^x}}}\text{ where ...
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3answers
55 views

What does the 2nd degree derivative of a cubic Bezier curve actually represent?

I have a $3D$ Bezier curve. Each co-ordinate along its path is defined by the equation: $$ f(t) = t^3 \bigl(a_2+3(c_1-c_2)-a_1\bigr) + 3t^2 (a_1-2c_1+c_2) + 3t(c_1-a_1) + a_1 $$ where $a_1, a_2$ are ...
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1answer
27 views

Continuity and Directional Derivatives

Does every absolutely continuous function on a compact set possess a left and right hand derivative everywhere on its interior? Although the two need not be equal of course.
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29 views

Why can't we differentiate one variable with respect to another variable?

This sounds a little immature, but why can't we differentiate $y$ with respect to $x$ ? Why does $y$ have to be written in terms of $x$ to differentiation? Why cant $\frac{dy}{dx} = 3a^2$ where ...
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13 views

Representation of Rate of change

My question is from Irrigation Engineering. A Canal Outlet has a property called Flexibility (F). It is defined as the "Ratio of rate of change of discharge of an outlet to the rate of change of the ...
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1answer
31 views

Show that the function is differentiable

I have to prove that the following function is differentiable and to find its derivatives at any point. $$f: \mathbb{R}^2 \rightarrow \mathbb{R}, (x,y) \rightarrow x^2+y^2$$ In my book there is a ...
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3answers
57 views

If $ f'(c) > 0 $, then there is an $ x $ such that $ f(x) > f(c) $.

Here is the homework question that I have: If $ f: [a,b] \to \Bbb{R} $ is differentiable at $ c $, where $ a < c < b $ and $ f^{\prime}(c) > 0 $, prove that there exists an $ x $ such ...
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1answer
37 views

Derivatives - Show equality

Let $y(x)$ be defined implicitly by $G(x,y(x))=0$, where $G$ is a given two-variable function. Show that if $y(x)$ and $G$ are differentiable, then ...
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0answers
14 views

Function intersecting 3 points & deriviate is positive for a range of x values

Thank you for taking the time to help out on this question. I'm looking for a function that intersects 3 points, and a derivative for every value of x between x=0 and x = 365 where dy/dx >= 0. My ...
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10 views

How to get the Riesz representative of the derivative of $L(K):=\text{tr}(\Lambda^* K A)$

$\DeclareMathOperator{\tr}{tr}K,\Lambda, A$ here are appropriate matrices. The question is not completely accurate as I can differentiate it, but I would prefer it to be in the form $⟨DL,h⟩$ for some ...
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2answers
25 views

Let $f:[0,\infty]\to R$ be differentiable on $(0, \infty)$, and $f'(x)\to b$ as $x \to \infty$. Show that $\lim_{x \to \infty}\frac{f(x)}{x}=b$

This is actually part (c) of the original question. Part (a) asks to prove for any $h>0$, we have $\lim_{x\to\infty}\frac{f(x+h)-f(x)}{h}=b$. Part (b) asks to prove if $f(x) \to a$ as $x\to\infty$, ...
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How to show that each tangent vector to $\mathbb{R}^n$ at a point $a$ is of the form $\xi(f) = \sum_{i} c_i \frac{\partial f}{\partial x_i}(a)$? [on hold]

How to show that each tangent vector to $\mathbb{R}^n$ at a point $a = (a_1, \ldots, a_n) $ is of the form $\xi(f) = \sum_{i=1}^n c_i \frac{\partial f}{\partial x_i}(a)$? Thank you very much. Edit: ...
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0answers
13 views

Fixed point of a differentiable function on a closed interval

Given a differentiable function $h:[0,3]\to [0,3]$ such that $h(0)=1 h(1)=2, h(3)=2$. (a) Argue that there exists a point $d ∈ [0,3]$ where $h(d)=d$. (b) Argue that at some point c we have ...
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3answers
41 views

Trig and derivatives: If condition holds for derivative, does it hold for the original equation?

Let's say I have some trigonometric identity such as $\sin(x) + 1 = -\cos(y)$. As we can see, the derivative of this identity gives $\cos(x) = \sin(y)$, which implies that $x + y = \pi/2$. Does that ...
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0answers
26 views

Solution to a path problem with maximum values of derivatives

I want to minimize the travel time from known position A to known position B while the derivatives of path are below their maximum value. I have: ...
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2answers
52 views

Matrix Derivative d(AXA)^(-1)/dX

I am having trouble figuring out the following matrix derivative $\frac{\partial(B X A')(AX A')^{-1}}{\partial X}$, where $X$ is square $n\times n$, A is $m\times n$, with $m<n$. and B is dimension ...