1
vote
2answers
72 views

What do we mean by derivative of a function? What does it tell? [duplicate]

Taking the derivative of any kind of function is easy but I don't know why we take the derivative? Like $f(x)=x^2$ has the derivative $2x$, so what does it mean? I don't know how to define ...
2
votes
0answers
10 views

Proof that maximal interval of existence exist and bounded

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
-5
votes
0answers
20 views

Change order between integral and differential calculation

Are those right? And I want to ask, in general case, when we can change the order of diff and integral: diff(integrate(L(x,y))) integrate(diff(L(x,y)))
1
vote
0answers
29 views

Proving that maximal interval of existence exists and that solution is unque

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
0
votes
1answer
62 views

Differential and Integral calculus.

Can anyone here explain me, why do we take the Centre of mass of a conical shell using slant height and $dl$ whereas the centre of mass of a solid cone is calculated using the vertical height and ...
1
vote
2answers
29 views

How to find the derivative of improper integral with variable upper limit?

I have the integral from $-\infty$ to $y^2$ of the function $(e^{-|x|})$ and I need to find the derivative of this. That is, $$\frac{d}{dy} \int_{-\infty}^{y^2} e^{-|x|}\,dx$$ Usually derivative ...
1
vote
1answer
76 views

Does a word problem provide all information?

A while ago I asked a similar question about word problems and assumptions. Is it a definition or an accepted-fact that word problems provide all information about the relevant existence/situation in ...
0
votes
0answers
13 views

Differential and a notation problem

Let $df(x) = f'(x)\,dx \:\:\:\:\: (1)$ Now we want to integrate both sides and we get: $f(x) = f(x)$ But now we want to differentiate again and we get $f'(x)=f'(x)$ I just don get it. If we say ...
3
votes
1answer
61 views

Finding $\int_{0}^{1} \frac{\log(1+x)}{1+x^2} {\rm d}x$ by differentiating under the integral sign.

I've tried to find this integral by the method already outlined in the title. I decided to let $$ \displaystyle I(\alpha) = \int_{0}^{1} \dfrac{\log(1+\alpha x)}{1+x^2} \text{ d}x. $$ From this ...
0
votes
0answers
25 views

Integration with matrices

I have written two equations in matrix format as follows $m(t)={\begin{pmatrix} 200\\ 300\\ 400\\ 500 \end{pmatrix}}^T \begin{pmatrix} ...
-1
votes
0answers
19 views

Prove - Integrate/Differentiation of Normal Distribution [closed]

Can someone prove this? $\dfrac{\partial}{\partial x}\left(\displaystyle\int_{-\infty}^{2x}\dfrac{1}{\sqrt{2\pi}}e^{-\frac{(2x)^2}{2}}dx \right)=\dfrac{1}{\sqrt{2\pi}}e^{-\frac{(2x)^2}{2}}\cdot ...
2
votes
2answers
74 views

Is there any geometric explanation of relationship between Integral and derivative?

It is said integral is anti-derivative, derivative is tangent of graph function in each point on the function and integral is the area of the region in the xy-plane bounded by the graph. I can not ...
4
votes
6answers
80 views

Given $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$

If $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$. Here is what I have done so far. I have took $f'(x)=(1+x^2)^{1/2}$ and I have found $1/f'(0)$ which should equal $1$. I don't think this ...
1
vote
1answer
44 views

First and second derivatives of the function $f(x)=x\int_0^x e^{t^2}dt$

I haven't done calculus for a while so I need your help with these two exercises. I am not sure whether my solutions are correct so I'd really appreciate someone's feedback. $$ f(x)=x\int_0^x ...
1
vote
2answers
59 views

Partial Derivative v/s Total Derivative

I am bit confused regarding the geometrical/logical meaning of partial and total derivative. I have given my confusion with examples as follows Question Suppose we have a function $f(x,y)$ , then ...
0
votes
0answers
20 views

Error of Riemann sum is $a/n + o(1/n)$ [duplicate]

A problem from an old qual: For $f$ of class $C^2$, find $a$ such that $$\int_0^1 f(t)dt-\frac1n\sum_{k=1}^{n-1}f\left( \frac {k}{n}\right)=\frac{a}{n}+o\left(\frac1n \right).$$ If we divide ...
1
vote
2answers
57 views

How to prove that a derivative of a formula equals to another formula.

If $u= \ln(\tan x+\tan y+\tan z)$ prove $$\sin 2x \dfrac{du}{dx} + \sin 2y \dfrac{du}{dy} + \sin 2z \dfrac{du}{dz}=2 $$ My answwer was like this: $$u' =\dfrac{ 1}{\tan x+\tan y+\tan z} \cdot( ...
1
vote
2answers
41 views

Integrable combinations - I can't seem to arrive at the given answer

I need help! I can't seem to arrive at the answer given in our textbook. I'm new here, so I really need help. The instruction says that I need to solve this D.E by recognizing integrable ...
3
votes
1answer
40 views

Calc II - Definite integral of sqrt(t^2 + t) from 2x to 1?

How do I find $$\int_1^{2x}\sqrt{t^2 + t}$$ with only knowledge from a Calculus I course? I've tried plugging this puppy into Wolfram Alpha and other integral solvers, which report it as solvable ...
2
votes
3answers
140 views

Limit of a Riemann Sum and Integral

I've been trying to solve this problem, but I haven't been able to calculate the exact limit, I've just been able to find some boundaries. I hope you guys can help me with it. Let $f:[0,1] \to ...
1
vote
0answers
91 views

Problem with trigonometric substitution proof

I'm sad, I can't get it. I know perfectly how to integrate using the mechanical process described in the books, but I want to understand the proof of it. My book (Stewart) says: In general we can ...
1
vote
1answer
63 views

Derive the formula for the sum of the first $n$ squares using derivatives and integrals

I wanted to prove the formula for sum of squares without using induction and thought using derivatives would be the easiest approach ...
0
votes
0answers
25 views

Derivative with respect to a function

We have a function ${f(s,{\psi(s)}_{3\times 1})}_{3\times1}\tag1$ Given Data $f,\psi$ are matrices and their dimensions are already given in the question s is not a matrix, it is a scalar ...
1
vote
1answer
30 views

Differentiation under the integral sign (one complex variable)

Let $u(z), u'(z)$ be complex-analytic functions on an open neighborhood $\Omega \subseteq \mathbb{C}$ of the origin. Also, let $f(X)$ be a complex-analytic function. For $s \in [0,1],$ define $$g(s,z) ...
0
votes
1answer
23 views

Derivative of an integral with variable in upper bound and a term of the integrand

So I want to take the first and second derivatives of a function g(Z) which is made up of several terms, one of which is where Z and H are our variables. Taking the derivative of this, it seems ...
3
votes
3answers
74 views

Find the limit and derivative of integral function.

$\psi_m(x)$ is defined as $$\int_0^{\ln|x|}e^{mt}\sin(t)^m\mathop{dt}$$ with $m$ a natural number greater then zero. Now the question is, does $\lim\limits_{x\to 0}\psi_m(x)$ exist. I've tried using ...
1
vote
1answer
48 views

Two methods of solving the differential equation $y' = .75 -.005y$

I am working on a differential equation problem and I am stumped since two different methods seem to give me two different answers Method 1 Given $\frac{dy}{dx} = .75 -.005y$ ...
8
votes
2answers
385 views

Is there an easier way to find the “natural” integration constant?

Suppose we take consequtive derivatives of a function at a point and then interpolate them with Newton series (Newton interpolation formula) so to obtain a smooth curve. ...
2
votes
2answers
43 views

Find volume of cask

I was given the following question: A wine cask has a radius at the top of $30 cm$ and a radius at the middle of $40 cm$. The height of the cask is $1m$. What is the volume of the cask in litres, ...
3
votes
2answers
51 views

Find arc length of curve on the given interval

I was asked to find the arc length of the curve of the following curve: $24xy = x^4 + 48$ from $x = 2$ to $x = 4$ This has turned out to be a very difficult problem, I get stuck using the arc length ...
2
votes
0answers
67 views

Definite Integral involving matrices

We have a definite integral of the form given below $ f(t) = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1-\alpha) X(t)}\,d\alpha \tag 1$ Given Data in the question $X(t)$ is a ...
0
votes
0answers
37 views

Use of the anti-derivative

Given a velocity function $dx/dt$, I am asked to find when a certain particles changes direction. This would then be when $dx/dt=0$. Let's say that $dx/dt$ has roots at $t= -1$ and $ t = 3$. I am ...
2
votes
1answer
71 views

ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.
1
vote
0answers
46 views

Differentiation with respect to the index of the summation notion?

$$\sum_{t=1}^k \binom{N-1}{t-1} \int[1-F(s)]^{N-1}[F(s)]^{t-1}g(s)\,ds$$ $k\in\mathbb Z ^+$ If I want to find out the effects of changing $k$ (comparative statics), what can I do? Differentiation ...
6
votes
4answers
131 views

How to find the derivative of a function defined by an integral? Namely, $f(y)=\int_0^{y^2} e^{-x^2y^2}dx$

Find at each point of its domain the derivative of the function $f: \mathbb{R} \rightarrow \mathbb{R}$ $$f(y)=\int_0^{y^2} e^{-x^2y^2}dx$$ $$$$ Is the domain of the function $\mathbb{R}$ because of ...
3
votes
4answers
85 views

Differential equation which has following solution $y=\frac{1}{1+\exp(ax)}$

Is there any linear differential equation which has following solution $$y=\frac{1}{1+\exp(ax)}$$ $a$ is constant. something like: $$ y'' + by' +cy + \alpha = 0$$ where $b$, $\alpha$ and $c$ are ...
2
votes
1answer
35 views

Solve 2 connected ODEs describing a domain

This problem confused me for a long time. I have 2 ODEs which describe part of our domain. They are connected at middle: $$ \frac{d^2}{dx^2} u = -a, x<x_0 $$ $$ \frac{d^2}{dx^2} u - \frac{u}{b^2}= ...
1
vote
4answers
68 views

Differential equation with the solution of $(1+ax/2)\exp(-ax)$

Is there any linear differential equation which has following solution $$y=(1+ax/2)\exp(-ax)$$ $a$ is constant.
1
vote
0answers
33 views

how to differentiate an integral

the integral is of the form below $$ \frac {d {\int y(x, t)h(x) dx}}{dy(x, t)} $$ what does the differentiation give? $h(x)$ and what about $$ \frac {d {\int y(x, t)h(y(x,t)) dx}}{dy(x, t)} $$ ...
0
votes
1answer
38 views

Calculus formula doubt

I am having a confusion in some of the formulas of differential and integral calculus. If $y=\ln x$, then $dy/dx=1/x$ and integral of $\tan x$ is $\log|\sec{x}|$ and also similarly of $\cot x$ and ...
0
votes
1answer
38 views

Absolute continuity and derivatives of integrals

I am preparing for a comprehensive at the end of the month, so I would appreciate any input I could get on this solution. I am pretty confident if the first part, but I think the second answer could ...
0
votes
1answer
69 views

Two methods of finding a function $f$ such that $Mdx+Ndy=0$ on the curves $f(x,y)=c$

this problem is from my class,i did one way and got one answer,professor did it in another way and got another answer.question is:Find $f(x,y)=constant$ where differential equation is ...
0
votes
1answer
42 views

Calculating the value of an integrals derivative given then value of the integral

I am given the following informations about a function: $$f\in C^1(\mathbb{R}),\quad f(3)=7,f(7)=13,\quad \int_{3}^{13}f'(x)\,dx=12$$ and i need to find the value of $$\int_{7}^{13}f'(x)\,dx.$$ A ...
4
votes
1answer
44 views

How to show that $\int\limits_{-\infty}^{+\infty}(n-1)\Phi(x)^{n-2}\phi(x)^2dx$? decreases in $n$?

I was working on a research project that involves taking the integral of $$(n-1)\int\limits_{-\infty}^{+\infty} \Phi\left(x\right)^{n-2}\phi\left(x\right)^2dx,$$ where $\Phi(.)$ is the CDF for ...
4
votes
0answers
23 views

Distributions and primitives

I was wondering: if distributions are seen as a generalization of functions that "removes obstructions" to the operation of derivation, is there a generalization of functions that would remove any ...
0
votes
3answers
108 views

Assumptions in Word Problems.

My dilemma has been that I am confused on how we make mathematical assumptions in WORD problems. Suppose you are given a related-rates word problem. (Q#) Air is being pumped into a spherical balloon ...
0
votes
0answers
42 views

ODE with multiple simple conditions $f'(x)=f(x)(Ax+D ) $

I have an ODE to solve . The main issue is,in addition to solving it I have to keep some conditions too in the solution of f(x).. I am bit confused regarding how to deal with it. Equation is given ...
1
vote
1answer
36 views

Question about the Fundamental Theorem of Calculus

So I have studied the FOTC, but not really sure of what I read so this question is just to help me learn the FOTC and understand how to do problems like it. $$ if $$ $$F(x)=\int_0^x\sqrt{sin^3(t)}dt$$ ...
4
votes
2answers
58 views

How can you explain implicit differentiation?

So I am taking calculus 1 online from a local college (bad idea, but the only thing that fit my schedule). The professor used the notation $f'(x) =$ for EVERY function up until two weeks ago. All of ...
-1
votes
1answer
94 views

How to find the derivative of the function $ \int_{x}^{x^2}\frac{t}{\ln(t)}dt$? [closed]

The problem is to find $\displaystyle\frac{d}{dx}\int_{x}^{x^2}\frac{t}{\ln(t)}\,dt$ I could do this if I had the first clue on how to integrate $\dfrac{t}{\ln(t)}$ but even wolframalpha is giving ...