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

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4
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1answer
48 views

doubt regarding Definition of differentiability .

The definition of a differentiable function is as follows: A function $f:A\to Y$ is said to be differentiable at a $\in A$ if there is a linear map $T\in L(X,Y), $ such that : ...
1
vote
1answer
152 views

Related Rates - Implicitly Deriving Geometric Formulas for Time?

So the step where you implicit derive the geometric formulas for time seem to be giving me the most trouble. How would this be done for volume of a cylinder $$(v=pi^2h)$$ and area of a triangle ...
0
votes
1answer
29 views

How to take the integral of a derivative to obtain desired result?

I am aiming for the form of derivative below computed over time that causes its differentiated variable V to go from an initial -.001 and increase to reach 10. I will explain my current calcs below ...
3
votes
4answers
52 views

Prove that $\frac{d}{dx}(\tan^{-1}(x))=\frac{1}{1+x^2}dx$

Prove that $$\frac{d}{dx}(\tan^{-1}(x))=\frac{1}{1+x^2}$$
1
vote
2answers
76 views

What is the difference between $\delta x$ and $dx$

Sometimes I see the $\delta x$ and $dx$ but I don't know exactly what is the difference between them.
1
vote
1answer
29 views

Finding the directional derivative of $f(x,y)$.

I have this problem: Let $f(x,y,z)=xyz$. Find the directional derivative in the direction of the velocity vector of the curve $\gamma(t)=(cos(3t),sin(3t),3t)$ on $t=\frac{\pi}{3}$. Is that the max. ...
0
votes
1answer
36 views

On which points of $xy=(1-x-y)^2$ is the tangent parallel to the $x$-axis?

On which points of $xy=(1-x-y)^2$ is the tangent parallel to the $x$-axis? All I get is the derivative of the function, as far I know, I set the derivative equals to zero.
4
votes
1answer
51 views

Prove that $f(c)=\frac12(c-a)(c-b)f''(\xi)$

A function $f:[a,b] \rightarrow \mathbb R$ is continuous on $[a,b]$ and $f''(x)$ exists $\forall x\in (a,b)$. If $a<c<b$ and $f(a)=f(b)=0$, prove that there exists a point $\xi$ in $(a,b)$ such ...
2
votes
1answer
58 views

Partial derivative of convolution

I have a convolution: $$g(x,\alpha) = \int_D \phi(t)f(x-t,\alpha)dt,$$ where $D$ is compact. I need to calculate $\frac{\partial}{\partial \alpha}g(x,\alpha)$. Under what conditions: ...
1
vote
1answer
38 views

Finding range of a function with derivatives

we have a function such that $$f(x):[0,1] \mapsto R$$ ,f(x) is twice differentiable and also $ f(0) =0=f(1) $ and also satisfies $$f''(x)-2f'(x) +f(x) \ge e^x $$where x belongs in [0,1] so what is ...
2
votes
1answer
24 views

Verify Lagrange's Mean Value Theorem for $f(x)=4-(6-x)^{\frac23}$ on $[5,7]$

$f(x)=4-(6-x)^{\frac23}$ Well, $f(5)=3, f(7)=3$, so, $\frac{f(7)-f(5)}{7-5}=0$ And $f'(x)= \frac23 (6-x)^{-\frac13}$ So, by MVT, $\exists c\in (5,7)$ such that $\frac{f(7)-f(5)}{7-5}=f'(c)$, i.e. ...
0
votes
1answer
19 views

Chain rule for multiple variables

If a $C^1$ function $f(u,v)$ has the following partial derivatives $$\frac{\partial f}{\partial u} (u,v) = 2uv \\ \frac{\partial f}{\partial v} (u,v) = u^2$$ and we have $g(t) =f(2t,t^2)$, what is ...
1
vote
2answers
50 views

Maple differentiation syntax?

If you have assigned a function to a label, how can you take it's derivative? For example I defined fun2 as $x^3$ but doing ...
1
vote
1answer
44 views

Implicit differentiation: Describing where a graph is increasing or decreasing

Considering $s$ is implicit to function of $p$, given by $s^6 - p^4 = 1$. For what $s$ is it increasing and decreasing? Well, I answered first like following: Calculating the first derivative using ...
0
votes
1answer
77 views

Homogeneoused ODE: $x' + 2x + 2x'' + x' = 0$

I have $$x'+2x+2x'' + x' =0,$$ which goes to $$x'' + x' + x = 0$$ and I take the characteristic equation... $$a^2 + a + 1 = 0$$ and this comes out to be $a = \frac{-1 \pm \sqrt 3i}{2}$. Is ...
0
votes
1answer
203 views

A cubic function is a polynomial of degree 3; that is, it has the form f􏰪(x) = ax^3+􏱎bx2 􏱎+ cx􏱎 + d,where a not=0

A cubic function is a polynomial of degree 3; that is, it has the form f􏰪(x) = ax^3+􏱎bx2 􏱎+ cx􏱎 + d,where a not=0 (a) Show that a cubic function can have two, one, or no critical number(s). Give ...
2
votes
0answers
194 views

Need help finding the volume of the solid of revolution

I need finding the volume of the solid of revolution (a cup), convert the volume to ounces and having the calculation be a close approximate to the measured ounces. I'm using the equation of the line ...
0
votes
2answers
46 views

If $f$ is monotone on $[a,b]$, is $f'$ bounded a.e. on $[a,b]$?

All the counterexamples I can develop for $f'$ being unbounded when $f$ is monotone only fail at one point. So I am wondering if it can only happen at a few points so that $f'$ is still bounded almost ...
1
vote
2answers
45 views

How can I differentiate correctly in this problem so that the units work out correctly?

I have this homework problem: Two sides of a triangle are 4 m and 5 m in length and the angle between them is increasing at a rate of 0.06 rad/s. Find the rate at which the area of the triangle is ...
1
vote
2answers
26 views

Solving Rate of Change when variable isn't given

I'm completely stuck in solving questions with rates of change when no variable is given in the formula. Example: The volume, $V\text{ cm}^3$, of water in a container is given by $V=(1/3)\pi h^3$ ...
0
votes
1answer
43 views

Non-differentiable in a null set

This is a problem from Stein's real analysis book that I have been working on. Show that exists a non-negative integrable f in $\mathbb{R}^{d}$ so that $\liminf_{m\left(B\right)\rightarrow0,x\in ...
2
votes
0answers
46 views

Is this a legitimate math-move?

I have assumed $P=f(x)$, where $f'(x)<0$, $f''(x)>0$. I want to find $\frac{\partial x}{\partial P}$. Since I have $P = f(x)$, I can find $\frac{\partial P}{\partial x} = f'(x)$. From that ...
1
vote
4answers
37 views

Derivative of limit using L'Hôpital's rule

Original question is: $$\lim_{x \to 0}\frac {1}{x} \cdot \ln (e^x+x)$$ Which is $$\frac 00$$ So I use L'Hôpital's rule and use the derivative but I'm not sure how to do that. I tried and got: ...
0
votes
2answers
39 views

quadratic matrix derivation using chain rule

Giving $$f(x) = x^T A x$$ with $x \in \mathbb{R}^{n} $ and $A \in \mathbb{R}^{n \times n}$, than $$ \frac{\partial f(x)}{\partial x} = (A + A^T) x$$ I tried to prof this with the chain rule. With ...
0
votes
1answer
7 views

Help with scond derivative test $\cos x-x$ interval $[0, 4\pi]$

So I am confused on how the interval can go to four pi, because when you find the critical numbers, you get $\frac\pi2, \frac{3\pi}{2}, \frac{5\pi}{2},$ and $\frac{7\pi}{2}$. However, the last two ...
0
votes
1answer
28 views

Finding a general solution(s) to a differential equation in first order form

I cannot understand why we need $\alpha > 0$ ensure the continuity of $\ddot{q}$ at $q=0$. Nor do I understand why to have $\dot{q}(0)=0$ we must have that $b >1$. Furthermore I cannot ...
1
vote
2answers
46 views

Value of k so that $ke^x -x$ has one solution

We are given equation as $ke^x -x$ and we need to get value of k so that it has at least one root. We can rewrite equation as $$ke^x =x$$ for us to get solution graph of $y=x$ and $y=ke^x$ must ...
0
votes
2answers
55 views

Showing that a particular f'(x) is not continuous at $x=0$

I really dont understand what the highlighted section of this proof is trying to show. I feel I am missing some theorem of which it is trying to apply.
1
vote
1answer
30 views

Smoothness of a particular function in two variables

I cannot understand why this doesnt work for $v \neq 0$ and $u=0$? I think it may be to do with my lack of understanding of why $v=0$ and $u \neq 0$ works which I believe has come from the fact that ...
0
votes
0answers
59 views

Non-commuting flows and obtaining a new expression about the pullback of a function

Let $U \subset \mathbb{R}^n$ and be an open set. If $\mathbb{X}, \mathbb{Y} \in \mathcal{X}(U)$, the set of smooth, complete vector fields on $U$. Let $\Phi_t,\Psi_s$ are their respective flows and ...
0
votes
3answers
49 views

Critical point of $x - 5x^{\frac{1}{5}}$

$f(x) = x - 5x^{\frac{1}{5}}$ Find the critical point This is what I did. \begin{align*} f(x) & = x - 5x^{\frac{1}{5}}\\ f'(x) & = 1 - x^{-\frac{4}{5}}\\ f'(x) & = \frac{x^{\frac{4}{5}} ...
7
votes
7answers
385 views

Differentiate $f(x)=\int_x^{10}e^{-xy^2}dy$ with respect to $x$

I am trying to find $f'(x)$ when $0\leq x\leq 10$. I know I could use the formula given on this wikipedia page: http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign but I have been ...
1
vote
2answers
53 views

Is the derivative of a monotone function non-negative a.e.?

I know that a monotone increasing function $f$ is differentiable a.e. I want to immediately say that $f' \ge 0$ where it exists but I do not actually have a proof. I only think it is true from ...
1
vote
0answers
38 views

Integral of a matrix exponent

What is the analytic closed form expression of $\int e^{A_1+A_2s} \ ds \tag 1$ where A and B are constant skew symmetric matrices NB $A_1=\left( \begin{array}{ccc} 0 & -c_0 & b_0 ...
6
votes
3answers
252 views

If $f$ is bounded and twice differentiable in $\mathbb{R}$, show that there exists $\xi\in\mathbb{R}$, s.t. $f''(\xi)=0$.

My idea: If $f$ has maximum and minimum, then $f'=0$ at these two points, and the conclusion is further derived using Mean Value Theorem. But what if $f$ has no maximum/minimum, like ...
1
vote
1answer
31 views

Prove that $\lVert L(x)\rVert \leq C\cdot \lVert x \rVert$.

I have to prove the next statement: Let $\ L:\ \mathbb{R}^n \to \mathbb{R}^m$ lineal. Then, there's a constant $C$ such that $\lVert L(x)\rVert \leq C\cdot \lVert x \rVert$, with $x\in ...
0
votes
1answer
39 views

problem with a functional derivative

I've the following problem with a functional derivative (I'm not a specialist). Let's start with something I know (hope!): given a functional $\mathcal{F}[\psi]$, say $$ \mathcal{F}[\psi] = ...
0
votes
2answers
47 views

Is a total derivative the same thing as a directional derivative?

Total derivatives are calculated very similar to directional derivatives and I was wondering are they one and the same? If not, what is the difference between them?
2
votes
2answers
44 views

Expansion of function , defined on a open interval containing $0$ , in terms of $\sin$ function

Let $I$ be an open interval in $\mathbb R$ containing $0$ and $f:I \to \mathbb R$ be a twice differentiable function , then is it true that $$\lim_{x \to 0}\dfrac {f(x)-f(0)-f'(0)\sin x - ...
1
vote
0answers
62 views

Switching the order of differentiation

I have a problem with the problem below https://imagizer.imageshack.us/v2/532x710q90/674/BngnD8.jpg (the line with the red arrow) Why is the first equality true, can you change the order of ...
1
vote
0answers
22 views

Negative a.e. right derivative implies monotonicity?

Let $f \colon [0, 1] \to \mathbb{R}$ be a continuous function such that for a.e. $t \in [0, 1]$ there exists the right derivative $f'_+(t)$. Suppose also that for some $\varepsilon > 0$ and for ...
1
vote
2answers
58 views

Trigonometric differential equation

Is it possible to solve the following ordinary differential equation: $\theta'(t)=x(t)\sin(\theta(t))-y(t)\cos(\theta(t)),\ \forall t\in I$, $I-$ interval from $\mathbb{R}$, where ...
0
votes
1answer
45 views

Find all natural $n$ , such that $f^{(n)}$ exists on all of $\Bbb{R}$ , for a defined function.

Exercise: Define $f$ on $\Bbb{R}$ by $$ f(x):= \begin{cases} x^3& x\ge0\\\ 0 &x<0 \end{cases} $$ Find all natural $n$ , such that $f^{(n)}$ exists on all of $\Bbb{R}$. Attempt: ...
3
votes
0answers
64 views

If $\overline f=f-f'(a)$ then how is $\overline {f'(a)}=0$?

Below is the definition of a function being differentiable at a point, given in my notes: A function $f:A \rightarrow Y$ is said to be differentiable at $a \in A$ if there is a linear map $T \in ...
0
votes
3answers
17 views

Find the limit $\quad\lim_{x \to 0}\frac{x^3}{\tan^3(2x)}$

I'm trying to find the limit $$\quad\lim_{x \to 0}\frac{x^3}{\tan^3(2x)}$$ but I'm at a loss. I've tried expanding $\tan^3(2x)$ using $\tan2x = \frac{\sin2x}{\cos2x}$, and then using double-angle ...
3
votes
6answers
110 views

Let $f(z)$ be a holomorphic function on C. Show that $\overline{f(\bar{z})}$ is holomorphic on C

Since $f(z)$ is holomorphic, I used Cauchy-Riemann equations and got $u_x = v_y ,\ u_y = -v_x$ Then I wanted to check if Cauchy-Riemann equations are satisfied for $\overline{f(\bar{z})}$ It does. ...
0
votes
1answer
18 views

Ordinary Differential Equations Form

Okay, I just want to understand the basic concept of this ODEs. Let's get to the form of this ODEs, which forms is : $$F(\ x,\ y,\ y',\ y'',\ \cdots\ ,\ y^{(n)}\ )=0$$It says that the ODE is a ...
2
votes
1answer
25 views

Value of the sum (numerical analysis)

Let $x_0, x_1, \dots, x_n$ are different real numbers and $\omega(x) = (x-x_0)(x-x_1)\dots(x-x_n)$. Then what is the value of the following sum: $$\sum_{k=0}^{n}\frac{\omega''(x_k)}{\omega'(x_k)}$$ ...
2
votes
3answers
56 views

Calculus Notation Question

What is the difference between $\frac{dy}{dx}$, $\frac{\delta y}{\delta x}$ and $\frac{\Delta y}{\Delta x}$? I was reading the derivation of a formula and when I came across this.. as $\Delta x$ ...
3
votes
2answers
28 views

An inequality concerning derivative of functions

Let $f:\mathbb R \to \mathbb R $ be a twice differentiable function such that $|f(x)|\le1, |f''(x)|\le1 , \forall x \in \mathbb R$ , then is it true that $|f'(x)|\le2 , \forall x \in \mathbb R$ ?