0
votes
0answers
24 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
39 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
64 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
40 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
99 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
54 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
19 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
87 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
38 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
37 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
61 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
49 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)' - ...
8
votes
3answers
71 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
84 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
92 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. ...
0
votes
1answer
66 views

Differentiation of multivariable function proof

I'm looking for the differentiation of multivariable function integral $$\frac{\mathrm{d} }{\mathrm{d} x} \int_{v(x)}^{u(x)}f(t,x)dt=u'(x)f(u(x),x)-v'(x)f(v(x),x)+\int_{v(x)}^{u(x)}\frac{\partial ...
1
vote
1answer
37 views

Why is $(\sec x)' = \tan x\sec x$ and not $\tan x$?

As far as I understood, the Fundamental Theorem of Calculus states that the integral of a function is its anti-derivative. And yet, although the integral of $\tan x$ is $\sec x$, the derivative of ...
0
votes
4answers
42 views

derivative $\frac{df(t)}{dt}$ of $f(t) = \int_0^t\ln{(s^2+t^2)} ds$

Let $f(t) = \int_0^t \ln{(s^2+t^2)} ds$, how can I find the derivative $\frac{df(t)}{dt}$? The function $\,\int_0^t \ln{(s^2+t^2)} ds$ is defined to be continuous in $s^2+t^2 > 0$ and $ s^2+t^2 ...
1
vote
1answer
109 views

showing $\int _a^b\left(f'\left(x\right)\right)dx\:=\:f\left(b\right)-f\left(a\right)$

Let $f(x):[a,b]\to \mathbb R$, be differentiable on $[a,b]$ (and continuous) so that $f'(x)$ is integrable on $[a,b]$. I need to show that: $$\int _a^b\left(f'\left(x\right)\right)\mathrm dx = ...
0
votes
1answer
20 views

Solving an ODE using variations of parameters and Wronskian theorem.

So I am attempting to solve this differential equation by trying to follow an example that my professor did in class. I am just not too sure about my answer seeing as WolframAlpha gives me this: ...
0
votes
1answer
55 views

$F(x) = \int_0^{x} t^2 e^{t^2}dt$

Let $y_0 = f''(2) + f'(1) + f(0)$ if $f$ is a real function defined by $f(x) = \int_0^{x} t^2 e^{t^2}dt$. How can I calculate the value of the expression $y_0$. I tried use the fundamental theorem ...
6
votes
2answers
164 views

simple way to show $|| \partial_x \int_{B(x,\epsilon)} \frac{x-y}{|x-y|^3} f(y) dy||_{\infty} = O(||f||_{\infty})$ in $\mathbb{R}^3$

We are set in $\mathbb{R}^3$. Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ be a $C^1_0$ function, i.e. continuously differentiable with compact support. Let $\epsilon > 0$ be small. I need to show ...
0
votes
0answers
25 views

Calculating the peak of a gaussian curve from the area

I run an online service, and I'm trying to calculate the maximum number of users that we'll have online at any given time, given the following assumptions: The total number of active users is 1000 ...
0
votes
1answer
45 views

Why is this integral not returning to the original equation when derived?

A projectile is fired into fluid at a rate of $60$ (nevermind the units on this one.) It decelerates such that $a=(-.4)v^3$. This is all fine and dandy. The book provides this solution. ...
2
votes
3answers
42 views

Question on integral, notation and Nikodym derivative

I have a very general question for those measure theoric, real analysis guy out there . I am very confused by the concept of Nikodym derivative. If $v << \mu$, we can find a non negative ...
-2
votes
2answers
54 views

Solve the Integral Example [closed]

$$\int_{-1}^{-2}\dfrac{6x}{\sqrt{2-\frac{x}{2}}}\mathrm dx.$$ I cant solve this problem please someone help me :)
-3
votes
1answer
70 views

I am trying to find derivative $f$

I want to find the derivative of $f: [1, \infty] \to \mathbb{R}$ defined by formula $$f(x) = \int_0^{x^4} e^{t^2} dt$$ Here is what I have done: $F(b) = \int_0^b e^{t^2} dt$ and knowing ...
1
vote
2answers
67 views

Integrable function on $[0,2]$ and its antiderivative

I got this question: Let $f$ be the integrable function defined on the interval $[0,2]$ by the rule: $f(x)= \begin{cases} 4x^3 & \text{if $0 \leq x \leq 1$} \\ x^2+2 & \text{if $1<x \leq ...
0
votes
1answer
41 views

Differentiation to Integration

Suppose we have the following relation: $$\frac{dx}{dt}=v$$ Then how does this imply the following: $$\int_{t_0}^{t} v\,dt=\int_{x_0}^xdx$$ The differentials cannot be treated as numbers since they ...
0
votes
2answers
63 views

How to find the derivative of $g(x)$?

I want to find this derivative, but I don't know what to do with the term $(x-t)^2$: Let $f:[0,1]\to\mathbb{R}$ be continuous. Define $g:[0,1] \to \mathbb{R}$ as follows: \begin{equation} ...
0
votes
1answer
42 views

True rigorous meaning of dx symbol in general? [duplicate]

From Multivariate integration of a derivative w.r.t. a single variable, from words_that_end_in_GRY's answer. From elementary calculus, I always thought $\int^{}$ and $dx$ were just a kind of ...
1
vote
1answer
51 views

Computing the derivative of an integral

There are similar questions on the same topic, yet I could not figure out why the following equation (taken from an economics solution manual) holds: $$ \frac{\partial}{\partial C(i,j)} ...
0
votes
1answer
26 views

derivate of indicator function

What is the derivative of the indicator function: \begin{equation} f(x)=\begin{cases} 1 & x^{\min} x\leq x^{\max}\\ -\infty &\mbox{otherwise}? \end{cases} \end{equation} thank you
1
vote
1answer
38 views

How to take the derivative of a function $F(x)$

The function $F(x)=\int_{-1}^{x}\sqrt{1-t^2}dt$. I believe this to be the representation of the area under the curve between $-1$ and $x$, where $\int_{-1}^{x}\sqrt{1-t^2}dt$ is a function of $x$: ...
4
votes
4answers
354 views

Prove that $\int_a^c f(t)dt - \int_c^b f(t)dt = f(c)(a+b-2c) $, for some $c\in(a,b)$

Let $f$ be a continuous on $[a,b]$ then prove that there exist some $c$ that lies in $(a,b)$ such that $$\int_a^cf(t)\,dt - \int_c^b f(t)\,dt = f(c)(a+b-2c) $$ and hence prove that $\int_a^c ...
2
votes
1answer
48 views

The Fundamental Theorem of Calculus and Derivatives

How do I show this in a convincing manner? I know I need to use the Fundamental Theorem of Calculus, but I find it difficult to show any steps in between, as it appears obvious?
1
vote
1answer
16 views

How to find the values of constants when there is one stationary point, no stationary point, and determining the maximum number os stationary points.

b) values of x is when f'(x) = 0 c) how do i solve this without using common sense and knowing that if a=0 there will be no turning points/inflections d)how do i solve this? e) max number of ...
1
vote
1answer
40 views

get the length of a curve with integral

I need to get the length of a curve which equation is : $$y= (4-x^\frac{2}{3})^\frac{3}{2}$$ I need to find the length using the method : $$L=\int_a^b \sqrt{ 1 + \left(\frac{dy}{dx}\right)^2}$$ So ...
6
votes
2answers
153 views

Do you feel comfortable with integral u-substitution? (reverse chain rule)

I've made this post both to see if I'm thinking right and to let others read and understand where the "u-substitution" method for integration comes from. I really hate substitutions, because you lost ...
1
vote
1answer
31 views

Finding the point where a function turns smaller then another

Sorry, couldn't explain better on the title. I mean, if you have a function for the income over time $I(t)$ and another one for costs $C(t)$ and you want to find out the time $t$ for which the profit ...
1
vote
0answers
27 views

Can one obtaining a mean value form of the Taylor series remainder using the integral remainder?

Can we show that $$(\exists \epsilon \in[0,x])\left(\int_{0}^x \frac{(x-s)^n f^{(n+1)}(s)}{n!}ds= \frac{x^{n+1}f^{(n+1)}( \epsilon)}{k!}\right)\text{ ?}$$ Thanks in advance!
0
votes
0answers
14 views

Maximum Principle - Proof

We want to show the maximum principle for a function $f = f(x,t)$ on a n-dimensional hypersurface $M,$ that is, (Corollary) Let $f = f(X,t)$ be a function on M, let $\vec{a}$ be a vector field on ...
0
votes
1answer
31 views

Formula for area under the curve

I don't know that the equation that I am going to explain below is correct or not, and this is why I am asking this question. So, I have found out that area under the curve could be found out by ...
0
votes
1answer
130 views

Approximation of $x!$ - Proof needed

By drawing a graph of the geometric derivative of $x!$, $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}$, i guessed that $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}\sim_{+\infty}(x+1/2)$. ...
0
votes
2answers
39 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 ...
2
votes
0answers
29 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
votes
1answer
29 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 ...