For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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0
votes
2answers
18 views

differential equation FP2

I am stuck on an FP2 edexcel differential equation. Q5. $(1-x^2)\frac{dy}{dx} + xy = 5x$ The answer is $y=5 + c(1-x^2)^{1/2}$ Question from fp2 edexcel book, pg 85
-2
votes
1answer
38 views

evalute $\int \frac{1}{x^6+x^4+7x^3+7x}\mathrm{ d}x$

I want to evalute this integral $$\int \frac{1}{x^6+x^4+7x^3+7x}\mathrm{ d}x$$
0
votes
0answers
26 views

How can show that the following function is non-negative?

I was working on a problem and reduced it to show the following inequality: ‎‎ $$\sum_{\substack{i,j=1\\i<j}}^{n}x_i^{\alpha}x_j^{\alpha}(\ln x_i-\ln x_j)^2+A_1\Big[\sum_{i=1}^{n}x_i^\alpha (\ln ...
0
votes
1answer
24 views

Find $f(x,y)$ integrable such that $f_x(y)$ isn't integrable

Find $f(x,y)$ integrable such that $f_x(y)$ isn't integrable, where $f_x(y)$ is in fact $f(x,y)$ while $x$ is a parameter. I thought of using $\log$ in some variation, but I think it is problematic ...
0
votes
0answers
7 views

Approximation of logistic function

Given the function: \begin{eqnarray} y=\frac{1}{1+\exp(-\sum_j w_j x_j-b)}. \end{eqnarray} How could I work out this approximation: \begin{eqnarray} \Delta \mbox{y} \approx \sum_j ...
-3
votes
1answer
49 views

Book recommendation (not a maths story book) [duplicate]

I am new here, so I don't know the rules, but can you please suggest me a very very good book in calculus, and other very important books that will also help me doing physics, like helping me write ...
1
vote
1answer
38 views

Evaluation of $\int_{0}^{1}\frac{\arctan x}{1+x}dx$

Evaluation of $$\int_{0}^{1}\frac{\tan^{-1}(x)}{1+x}dx = \int_{0}^{1}\frac{\arctan x}{1+x}dx$$ $\bf{My\; Try::}$ Let $$I = \int_{0}^{1}\frac{\tan^{-1}(ax)}{1+x}dx$$ Then $$\frac{dI}{da} = ...
4
votes
2answers
63 views

If $\tan x$ is not a differentiable function then why does its differentiation $\sec^2(x)$ exists?

$\tan x$ is not differentiable at $(2n + 1)90$ points, which means function itself is not differentiable. So, why does its differentiation $\sec^2(x)$ exists?
1
vote
2answers
38 views

Find the equation of tangent at origin to the curve $y^2=x^2(1+x+x^2)$

How do I find the equation of tangent at $(0,0)$ to the curve $y^2=x^2(1+x+x^2)$ ? Differentiating and putting the value of $x$ and $y$ gives an indeterminate form. Can we trace the curve and ...
2
votes
1answer
36 views

Integration involving greatest integer function : $\int_0^{\pi} [cot(x)]dx$

What the integral of $$\int_0^{\pi} [\cot(x)]dx$$ where $[\cdot]$ represents greatest integer function. I know integral of $\cot$ is $|\log(\sin(x))|$ but $\log$ is not defined for $0$ or is there ...
0
votes
1answer
15 views

$g(x) \ge 0$ Riemann integrable on $[a,b]$ then for each subinterval $\int^b_a g(x)dx \ge \int^d_c g(x)dx$

Let $g(x) \ge 0$ Riemann integrable on $[a,b]$. Show using Riemann sums that for each subinterval $[c,d] \subset [a,b]$: $$\int^b_a g(x)dx \ge \int^d_c g(x)dx$$ I thought that we should ...
0
votes
1answer
37 views

Definite integral with cube roots of trig functions [on hold]

Find a closed form for the following integral: $$\int _{\pi/6} ^{\pi /3} \frac {\sqrt[3]{\sin x}}{\sqrt [3]{\sin x} + \sqrt[3]{\cos x}}dx$$ I think the answer is found out using some properties. So ...
1
vote
1answer
43 views

Power series of $\frac{1+x}{(1-x)^2}$

This question is continuing from the previous question here: Power Series representation of $\frac{1+x}{(1-x)^2}$ I am trying to calculate the power series representation of the equation: $$ ...
0
votes
1answer
23 views

Calculus: Sketching a graph that satisfies the following conditions

The following question is from our reviewer for our upcoming exams: $\ast$ 3. Sketch the graph of a function $f$ satisfying the following conditions: (i). $-5,-3,-2,-1,0$ and $2$ are the only ...
2
votes
1answer
10 views

Proof that distinct numerator polynomials are equal for all x when over the same denominator polynomial

I am just curious about this part of the proof. The question is this: Suppose that F, G, and Q are polynomials and $\frac{F(x)}{Q(x)}=\frac{G(x)}{Q(x)}$ for all x except $Q(x)$ = 0 Prove that ...
-1
votes
1answer
32 views

differentiate by a differential [on hold]

This is a difficult question to phrase so I will show it mathematically. Let $f(\theta) = \sin(\theta)$ Is it possible to do: $\frac{d f(\theta)}{d \dot{\theta}}$ I have tried to do $ t = ...
0
votes
0answers
14 views

Construction of the Area Function

I am following calculus by Tom M Apostol in which he has given the Axiomatic definition of the Area Function We assume there exists a class M of measurable sets in the plane and a set function a, ...
0
votes
1answer
9 views

Monotonic optimal value function

Are there any theorems/sufficient conditions about when the optimal value function of a parametrized optimization problem is monotonic in the parameter? Specifically, are there simple conditions ...
1
vote
3answers
24 views

Book recommendation for differential and integral calculus of one and of several variables

I want to re-study differential and integral calculus of one and of several variables and be able to solve all kinds of low to high level problems related to them. Iit has been 3-4 years I solved ...
0
votes
1answer
11 views

Finding a curve given only its basic form and its tangent line

The basic form is $f(x)=k\sqrt{x}$ and the tangent line is $4x+36$. I've spent over half and hour with wolfram alpha and my notes on this problem: I've tried $(k\sqrt{x})'=4x+36$ and got ...
1
vote
2answers
17 views

Average $y$ from a range of $x$ in a parabola

Given a parabolic/quadratic formula such as $ax^2 + bx + c =y$, how do I get the average value of $y$ given a range of $x$ ($x_{min}$ to $x_{max}$). Real world example: if my formula represents the ...
2
votes
2answers
66 views

How is the second derivitive derived? [on hold]

As everyone knows that the derivitive of a function is notated as $\frac{dy}{dx}$ The question is: How is the second derivitive $\left(\frac{d^2y}{dx^2}\right)$ notation derived?
0
votes
1answer
44 views

How to find the center of mass in this problem

How can I find the centre of mass of the surface of the sphere $x^2+y^2+z^2=a^2$ that is contained in the cone $z\tan(\gamma)=\sqrt{x^2+y^2}$, $0 \lt \gamma \lt$ $\pi/2$ a constant, where the density ...
0
votes
1answer
33 views

If $f(x)=\lim_{n\to\infty}n^2(e^{\frac{k}{n}\ln\sin x}-e^{\frac{k}{n+1}\ln\sin x})$ where $0<x<\pi$, $n\in\mathbb{N}$

If $f(x)=\lim_{n\to\infty}n^2(e^{\frac{k}{n}\ln\sin x}-e^{\frac{k}{n+1}\ln\sin x})$ where $0<x<\pi$, $n\in\mathbb{N}$ and $\int_0^{\frac{\pi}{2}}f(x)dx=-\frac{\pi}{k}\ln4$, then the value of ...
0
votes
3answers
25 views

Intersection of two exponential functions (of different bases)

What is the point of intersection (x,y) of the two functions $y=x^{1/2}$ and $y=e^{-3x}$? (This is for a volume of a solid of revolution problem.)
2
votes
1answer
30 views

Complex Analysis - Argument - Need Explanations [on hold]

Can anyone explain this solution? How did we get $ - i(\pi/2 + 2n\pi) $?
1
vote
1answer
40 views

Integrating inverse trig function with radicals

$$\dfrac{x + 5}{\sqrt{9-(x-3)^2}}$$ It's a inverse trig integration problem. I tried to separate the numerators but made my problem worse. Any advice?
2
votes
0answers
25 views

Volume of rotation of $y=x^2-1$ and $y=3-x^2$

Find the volume generated by the rotation about the $Y$ axis of the area between the curves in the first and fourth quadrant $y=x^2-1$ and $y=3-x^2$ (using cylindrical shells). The problem with this ...
1
vote
0answers
18 views

Second order total derivative

Suppose we have a function $g: \mathbb R^2 \to \mathbb R$ and $$\nabla g(u,v)=(5v^4-2u\exp(v-u^2), \exp(v-u^2)+20uv^3), (u,v)\in\mathbb R^2$$ Can the function $g$ be twice differentiable, i.e. does ...
-1
votes
1answer
36 views

Help with vector triple integral problem

Prove that $$\iiint_{D}(\vec a \cdot \vec R)(\vec b \cdot\vec R)(\vec c \cdot\vec R) \,dx\,dy\,dz=\frac{(\alpha\beta\gamma)^2}{8r}$$ Where the $\vec a , \vec b,\vec c$ are constant vectors, $\vec ...
-1
votes
0answers
16 views

What is the saddle-point approximation? [on hold]

I want to take advice which books are useful to understand saddle point approximation. Can you give suggestion about that ? Also, if you explain what is the saddle point approximation, I will be so ...
0
votes
0answers
19 views

Volume generation by rotation of $y=x^2-1$ and $y=3-x^2$

Find the volume generated by the rotation about the $Y-axis$ of the area between the curves in the first and fourth quadrant $y=x^2-1$ and $y=3-x^2$ using method of cylindrical shells. Now height of ...
2
votes
2answers
75 views

Does $\frac{\sin(x\ln x)}{x\ln x}$ as $x$ approaches $0$ from the right side have a limit?

I am trying to determine if $$\lim_{x\rightarrow0^{+}}\frac{\sin(x \cdot \ln(x))}{x\cdot \ln(x)}$$ has a limit ? Since $$\ln(x) \rightarrow -\infty\text{ as }x \rightarrow 0^{+}$$ I have tried ...
1
vote
0answers
11 views

Stationary points with matrix

I have an exercise but I do not even know where I should start: Consider the normalised quadratic form $(x^T Ax)/(x^T x)$ where $x∈R^2$, $A$ is a general 2x2 matrix. Find the vectors that make this ...
0
votes
0answers
20 views

Derivative I calculated does not match code (or intuition)?

I want to take the derivative with respect to the $x$ co-ordinate of a Hankel function with the norm of a 2d vector as its argument. Let $\mathbf{x} = (x_1, x_2) \in \mathbb{R}^2$. We have ...
1
vote
1answer
49 views

Evaluation of $\int_{0}^{a}\sin^{-1} \sqrt \frac{x}{a-x} dx$

How can we evaluate $$\int_{0}^{a}\sin^{-1} \sqrt \frac{x}{a-x} dx$$ I tried substitutions like $x=a\cos(t)$ and also tried applying property $$\int_{0}^{a}f(x) dx=\int_{0}^{a}f(a-x) dx$$ which gave ...
0
votes
3answers
77 views

Integrate $\int{\frac{1}{(x+1)(x+2)^2(x+3)^3}dx}$

How to integrate this $$\int{\frac{1}{(x+1)(x+2)^2(x+3)^3}dx}$$ I tried to use that $$\int{\frac{1}{(x+1)(x+2)^2(x+3)^3}dx} = P_{1}(x)/Q_{1}(x) + \int{P_{2}(x)/Q_{2}(x)dx}$$ where ...
2
votes
0answers
38 views

Getting tangent to point on a function in 2 variables

We have a function $h(x, y) = \ln(x^2 - y) + x^2y + 4 \cos(\pi(y - x))$ in two variables. First we want to show that $h(x, y) = 8$ can be solved for $y$ in point $(2, 3)$. This is an implicit ...
1
vote
0answers
16 views

Is truncated normal CDF a decreasing function of $\mu$?

Suppose $F(x;\mu,\sigma^2,0,1)$ is the CDF of a $N(\mu,\sigma^2)$ random variable truncated on the unit interval $(0,1)$. I'd like to show that $\frac{\partial}{\partial \mu} F(x;\mu,\sigma^2,0,1) ...
1
vote
0answers
35 views

Indefinite integration and definite integration calculation [on hold]

Given the functions: $$\begin{cases} A(x)= \dfrac {x\ln(1+\ln(x))}{1+x^{4/3}}\\[2ex] B(x)=A(2x-1)-A(2x) \end{cases}$$ for $b>1$ find out: $$\int_{b}^\infty A(x) dx - \frac12\int_{2b-1}^{2b} B(x) ...
0
votes
1answer
16 views

Trapezoid rule for finding coefficient

If we know that $\int_{a}^b t(x)=h \sum_{k=1}^2 dk * t(a+kh)+O(h^m)$ where $h=\frac{b-a}{3}$, how do we find the coefficient d1, d2 and m in the equation? Answer says that d1=3/2, d2=3/2, m=3 I ...
0
votes
0answers
24 views

Finding equation of tangent at given point

I am trying to find the equation of the tangent line for the given point, given $f(x) = log x$ and point $P(2, log2)$. I know the derivative of $log x$ is $\frac{1}{x ln10}$, and as a result the ...
-5
votes
0answers
33 views

Find the volume of the following room [1] [on hold]

I was working on a project which required me to calculate the volume of the room. The picture of the room is given below: I tried splitting the shape across the diagonals but each time end up ...
-6
votes
1answer
39 views

A question on polunomial [on hold]

Let $m\in (0,1)$ and ${a_n}{x^n} + .... + {a_1}{x^1} - f(m) = 0$ and $x\in \mathbb{C}$ $f(m) $ is continuous decreasing function of $m$. $a_i\ge0$ for all $i$. $k(m)$ is positive zero of first ...
0
votes
0answers
30 views

Differential? equation in car dashboard problem [on hold]

I stumbled upon this question while I was driving my car. On my dashboard I have fuel gauge and engine temperature gauge next to each other, look at the pic: http://i.stack.imgur.com/aDgKj.png Fuel ...
6
votes
2answers
95 views

Anti-derivative of continuous function $\frac{1}{2+\sin x}$

I use tangent half-angle substitution to calculate this indefinite integral: $$ \int \frac{1}{2+\sin x}\,dx = \frac{2}{\sqrt{3}}\tan^{-1}\frac{2\tan \frac{x}{2}+1}{\sqrt{3}}+\text{constant}. $$ ...
1
vote
1answer
35 views

3rd degree polynomial fraction decomposition

Was solving some differential equations and came upon this integral: $$\int\frac 1{x(x+1)^2} dx$$ Looked it up on wolframalpha and it can be decomposed to: $$\frac ...
1
vote
1answer
25 views

Can every separable differential equation be rewritten to potentially be exact (or NOT exact)?

Let's say an ordinary linear DE is separable. Then $$\frac{dy}{dx} = P(y)Q(x) \Leftrightarrow \frac{1}{P(y)}dy = Q(x)dx \Leftrightarrow Q(x)dx + R(y)dy = 0$$ is in exact form where ...
-2
votes
3answers
91 views

Solving $x \ln x=25$ [on hold]

Can someone help me solve the following equation? $$x \ln x = 25$$
0
votes
0answers
63 views

Solving an indefinite integral problem [on hold]

The given problem is $$ \int \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx$$ please help me solving this indefinite integral problem...thank you very much, actually I have solved an definite integral ...