1
vote
0answers
69 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
52 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
18 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
18 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
73 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
42 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 $\frac{dy}{dx} = .75 -.005y$ ...
7
votes
2answers
334 views
+100

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
41 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
45 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
62 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
69 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
44 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 ...
8
votes
4answers
130 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
67 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
32 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
67 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
40 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
42 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
22 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
98 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
35 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
57 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 ...
0
votes
1answer
25 views

Left & Right Area Approximation Using Y-Axis - Method Alternatives

Is there a simpler way of solving this then calculating x1(h)+x2(h)+x3(h)+x4(h) by using the given y values (in this case h, the height is one, because the length of each rectangle is one) ...
0
votes
1answer
11 views

Related Rates of Change - Cylinder Question

A cylindrical tank with radius 5 cm is being filled with water at rate of 3 cm^3 per min. how fast is the height of the water increasing? I dont want this question solved, but please help me correct ...
0
votes
0answers
26 views

Regarding methods of finding a derivative.

I read in the American Mathematical Monthly Descartes found away to calculate the slope of a tangent to a curve at a point specified. Called the Double tangent point method ( I think). This method ...
1
vote
0answers
40 views

Taking derivative under the integral sign

Reading a textbook and stuck on this one detail... would like to confirm my understanding. The book defines a function $\eta \in C^1(\mathbb{R})$ satisfying $0 \leq \eta \leq 1$, $0 \leq \eta^\prime ...
2
votes
3answers
65 views

How to find derivative of an integral of this type

$$f(x) = \int _x^{e^x}\:\left(\sin t^2\right)\,dt$$ How to find the derivative $f'(x)$ Attempt: $\sin (e^{x^2}) e^x$
1
vote
2answers
43 views

$\dfrac{\partial}{\partial x}\left(\int_{g(x)}^{h(x)}f(y)\, dy \right)= f(h(x))h'(x)-f(g(x))g'(x)$

I'm trying to prove the following, interesting, relation: $\dfrac{d}{dx}\left(\int_{g(x)}^{h(x)}f(y)\, dy \right)= f(h(x))h'(x)-f(g(x))g'(x)$ I tried to integrate by parts the RHS, but i don't ...
5
votes
2answers
101 views

Is $\int^x \cos \frac1t$ differentiable at zero?

From Spivak's Calculus, 4th ed., exc 14-20: Let $$f(x) = \begin{cases} \cos \frac1x, & x\neq 0\\ 0, &x=0. \end{cases}$$ Is the function $\int_0^xf$ differentiable at zero? I'm having ...
0
votes
1answer
56 views

Anti derivative notation [duplicate]

$F$ is an anti derivative of $f$. $$\int f(x) dx = F(x)+C$$ Can you tell me why there is '$dx$' in the LHS?
0
votes
1answer
20 views

How rigorous is multiplying both sides of an eqaution for the differential of a function?

I have to solve this equation: $$ -C_0 f + \frac{1}{2}f^2 +\frac{d^2 f}{d X^2}=A $$ where $C_0$ and $A$ are two real nonzero constant; $f:\mathcal{R}\to \mathcal{R}$ I have seen that the person who ...
1
vote
2answers
94 views

Lyapunov function for non-autonomous non-linear differential equations

I have read some lecture notes about Lyapunov’s Second Method for autonomous system. Now, I want to deal with the stability of a non-autonomous system. Suppose there is a non-autonomous non-linear ...
0
votes
0answers
50 views

Elementary integration and derivatives

Update Consider that the mean, of let's say a variable N is defined as: \begin{equation} N = E(e\,l) = \int\int e\, l(a) H(a,e) \end{equation} Where $E$ denotes the expected value (the random ...
0
votes
1answer
24 views

Radius of curvature and continuous functions

Let $\kappa (x)$ be radius of curvature function for a continuous function $f(x)$. Is it necessary that $\kappa(x)$ will have extrema when $f(x)$ does. And the nature of extrema is opposite to that ...
0
votes
1answer
39 views

Calculus Review - Differentiating an Integral

I'm trying to review some calculus over the summer and I just wanted to double-check my answer to a simple problem I came up with myself. Thanks. What is: $\frac{d}{dx} \int_a^{g(x)} f(t)\;dt\;$? ...
1
vote
2answers
43 views

On integration when solving differential equations (specifically separable equations)

So here is the differential equation and inititial conditions: $$x \frac{\mathrm{d}y}{\mathrm{d}x}=y(3−y) $$ and $$y(2) = 2$$ We have to find the equation $y$ in terms of $x ~~[y(x)]$ that is a ...
0
votes
2answers
40 views

Why would I want to find the rate at which things were changing? Marginal cost, marginal revenue and profit

I'm learning calc and after learning about how to differentiate using product rule and chain rule etc. I came across marginal cost and marginal revenue. I'm pretty familiar with cost, profit and ...
4
votes
1answer
66 views

Definition of integration

The derivative of a function is defined by $$ f^{\prime}(x)=\lim_{\Delta x \to 0}{\frac{f(x+\Delta x)-f(x)}{\Delta x}} $$ provided the limit exists. For example for $f(x)=\sin(x)$ we can prove that ...
1
vote
1answer
51 views

Find lowest and highest value of function $f(x)=\int_0^x{\frac{2t-2}{t^2-2t+2}}dt$

Find highest and lowest value of function: $$f(x)=\int_0^x{\frac{2t-2}{t^2-2t+2}}dt$$ We need to use first derivative test to find critical points. $$f'(x) = \frac{2x-2}{x^2-2x+2}(x)' - ...
7
votes
3answers
75 views

Show that it is possible that the limit $\displaystyle{\lim_{x \rightarrow +\infty} f'(x)} $ does not exist.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ a differentiable function with continuous derivative and the limit $\displaystyle{\lim_{x \rightarrow +\infty} f(x) }$ exists. Show with an example that it ...
0
votes
2answers
86 views

Find $G'\left( x\right)$.

Let $$G\left( x\right)=\int_{x}^{2x}{f\left( t\right)dt}$$ Find $G'\left( x\right).$ I tried to divide the integration interval but the subintervals are expressed in terms of $x$.
1
vote
0answers
93 views

Will antiderivative always be differentiable?

Suppose f(x) is continuous on [0,1]. Obviously, such a function will be integrable. Will antiderivative be always differentiable on (0,1)? The answer is "Yes" by the Fundamental Theorem of Calculus. ...