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

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3
votes
1answer
60 views

Is calculus not rigorous?

While studying single and multivariable calculus during my first year some people complained that calculus wasn't rigorous enough, when I asked about this no one seemed to be able to really specify ...
-4
votes
0answers
40 views

how do calculus this derivate? [on hold]

how do calculus this derivative? $$f(x) = (2x-x^2)^{1/2}$$ And how do I calculus this derivative? $$F(x) = (\sin{x}/(1+\cos{x}))^2 $$
1
vote
0answers
41 views

An equivalent for $\int_0^1\left(\frac{1}{\log x}+\frac{1}{1-x}\right)^n\;dx$

Set $$ I_n :=\int_0^1\left(\frac{1}{\log x} + \frac{1}{1-x}\right)^n \:\mathrm{d}x \qquad n=1,2,3,.... $$ We have $$I_1 =\gamma, \quad I_2 =\log (2 \pi) - \frac 32, \quad I_3 = 6 \log A - ...
1
vote
3answers
75 views

Evaluate the limit: $\lim_{x\to \infty}$

Evaluate the limit: $$\lim_{x\to\infty} \frac{(2x^2 +1)^2}{(x-1)^2(x^2+x)}$$ The answer is 4 and I don't understand why, but why can't I just do something like:$$\frac{(\infty)}{(\infty)(\infty)} = ...
-3
votes
0answers
26 views

Confusing pictures, Understand how each graph looks but not which matches with such bad pictures. Any Help? [on hold]

Guesses are appreciated, I understand this is poor quality imaging, I just don't see where some of them go because I keep getting it wrong. Any help, such as educated guesses are helpful! thanks.
-1
votes
2answers
44 views

What and how do derivate? [on hold]

How do I derive this function? $f(x) = x(e^{-x^2})$ I need the first and second derivative.
0
votes
1answer
24 views

Can someone help me, I cannot find the partial derivatives of this

Need help find the partial derivatives...I keep keeping cos(4x-3y+z)
1
vote
1answer
51 views

An integral representation for $\psi$

Let $\displaystyle \gamma$ denote the Euler constant defined by $\displaystyle \gamma := \lim\limits_{n \to \infty} \left(\frac11+\frac12+\cdots+\frac1n- \log n\right)$. Here is an integral for ...
0
votes
1answer
16 views

Calculating the area of a region using a mapping

The region: $\{{(x,y) \mid x^{2} < y < 2x^{2}, 2y^{2}<x<3y^{2}, x > 0, y > 0}\}$ The mapping: $u = y/x^{2}$, $v = x/y^{2}$ I calculated the jacobian to be $\frac 34$ which means ...
0
votes
2answers
37 views

How do you answer these questions regarding the Taylor series method?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the Taylor series method. (b) Assuming $f(x)\in C^3$, evaluate the ...
2
votes
1answer
52 views

Evaluate the area of the region bounded by the ellipse, where is my mistake?

$ (10x^2+6xy+y^2=2)$ => $ ((x/\sqrt2)^{2} + ((3x+y)/\sqrt2))^{2} = 1 $ so if I change the variables to $u$ and $v$, $u = x/\sqrt2$ $v= (3x+y)/\sqrt2) $ Then my bounds of integration become $-1 ...
0
votes
0answers
13 views

How do you obtain the version of Simpson's rule required as well as deduce the composite integration rule? [on hold]

Consider the function $$g(x)=f(a+(x−1)h)$$ and obtain a version of Simpson’s rule applicable to an integral $$\int_{a+h}^{a−h}f(x)dx.$$ Then deduce the composite integration rule ...
7
votes
4answers
111 views

Meaning behind differentials

So I think I understand what differentials are, but let me know if I'm wrong. So let's take $y=f(x)$ such that $f: [a,b] \subset \Bbb R \to \Bbb R$. Instead of defining the derivative of $f$ in ...
-3
votes
1answer
67 views

How to calculate these limits using L.Hopital Rule [on hold]

How to evaluate the limits, using L'Hospital rule? $$\lim_{x \rightarrow 0} \frac{x-\sin x}{x^2}$$ $$\lim_{x \to 0^+}\frac{\ln(x^2 + 2x)}{\ln x} $$
1
vote
0answers
20 views

Let $m \geq 1$ be an integer. Evaluate $\int_R \frac{\sin t}{t}J_m(t)\,dt$

Let $m \geq 1$ be an integer. Evaluate $\int_R \frac{\sin(t)}{t}J_m(t)\,dt$ We know that $\hat{\chi_{S^{n-1}}}(rx)=(2\pi)^\frac{n}{2}\frac{J_\frac{n-2}{2}(r|x|)}{(r|x|)^\frac{n-2}{2}} \iff ...
1
vote
2answers
78 views

Evaluation of $\int\frac{\sqrt{\cos 2x}}{\sin x}dx$

Evaluation of $$\displaystyle \int\frac{\sqrt{\cos 2x}}{\sin x}dx$$ $\bf{My\; Try::}$ Let $\displaystyle I = \int\frac{\sqrt{\cos 2x}}{\sin x}dx = \int\frac{\cos 2x}{\sin^2 x\sqrt{\cos 2x}}\sin xdx ...
2
votes
0answers
43 views

Calculating work [on hold]

How much work is required to lift a $1000\mathrm{kg}$ satellite from the surface of the earth to an altitude of $2\times10^6$ meters? The gravitational force is $F=\frac{GMm}{r^2}$, where $M$ is the ...
1
vote
1answer
36 views

Finding radius of convergence using root test

Find the radius of convergence of the following power series $$\sum_{n=1}^{\infty} \frac{2^n + 1}{n} x^n.$$ Using the ratio test, I have found that the radius of convergence is $R = \frac{1}{2}$. I ...
0
votes
0answers
31 views

Help with math steps, chain rule.

I'm trying to to understand the math steps to go from Eqn. (1) to Eqn. (2). $$\tag{1} q(x,t)=\frac{-V_t(1+\delta f(c,g))}{P(x,t)}\cdot \left(\frac{dP_o}{dt}\right)$$ $$\tag{2} \frac{-V_t ...
0
votes
1answer
26 views

How do you solve the second part of the question where i am required to derive Simpson’s integration rule?

When $v(x) = A + Bx + Cx(x − 1)$ show that $$\int_0^2v(x)dx= 2A + 2B + \frac23.$$ By choosing A,B and C so that $y = v(x)$ fits a given curve $y = g(x)$ at $x = 0$, $x = 1$ and $x = 2$ derive ...
1
vote
3answers
57 views

For What Values Of $x$ Is $f$ Continuous

For what values of $x\in\mathbb{R}$ is $f$ continuous? $f(x) = \left\{ \begin{array}{lr} 0 & \text{if}\, x \in \Bbb Q\\ 1 & \text{if}\, x \notin \Bbb Q \end{array} ...
1
vote
1answer
26 views

What function to use to get geometric mean in trapezoidal rule?

When deriving a trapezoidal rule an integral of $f(x)$ is switched to integral of new function $g(x)$ approximating the first one $$\int_a^b {f(x)dx}\approx \int_a^b {g(x)dx}$$ where $g(x)$ is a ...
4
votes
1answer
110 views

How can I evaluate this indefinite integral? $\int\frac{dx}{1+x^8}$

How do I find $\displaystyle\int\dfrac{dx}{1+x^8}$? My friend asked me to find $\displaystyle\int\dfrac{dx}{1+x^{2n}}$ for a positive integer $n$. But looking up I am getting pretty noisy answer for ...
4
votes
1answer
37 views

Existence of improper integral

Prove that $$\int_{0}^{\infty} \frac{(\arctan x)^2}{x^2} dx$$ converges. This is my attempt: The above integral is equal to $$\int_{1}^{\infty} \frac{(\arctan x)^2}{x^2} dx + \int_{0}^{1} ...
3
votes
5answers
159 views

if $\frac{1}{(1-x^4)(1-x^3)(1-x^2)}=\sum_{n=0}^{\infty}a_{n}x^n$,find $a_{n}$

Let $$\dfrac{1}{(1-x^4)(1-x^3)(1-x^2)}=\sum_{n=0}^{\infty}a_{n}x^n$$ Find the closed form $$a_{n}$$ since $$(1-x^4)(1-x^3)(1-x^2)=(1-x)^3(1+x+x^2+x^3)(1+x+x^2)(1+x)$$ so ...
0
votes
0answers
28 views

Expand trigonometric expression

I am supposed to expand this expression $${\frac {\sin \left( x \right) b \left( 4\,b\cos \left( x \right) + \sqrt {16\,{b}^{2}+1}+5 \right) }{4\,b\cos \left( x \right) +\sqrt {16 \,{b}^{2}+1}+1}} $$ ...
1
vote
1answer
22 views

$|\int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t}| dt \leq G(x,w), G\in L^{1} ? $

Put $\lambda >0,$ and we define, $$F_{\lambda}(x, w)= \int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t} dt;(x,w) \in \mathbb R^{2}$$ we note that, $F_{\lambda} \in ...
1
vote
3answers
55 views

Proving a limit exists - solving for epsilon with absolute values

I have the equation that I want to prove the limit goes to 1: $$\lim_{n \to \infty} \frac {(n+8)(n+1)}{n(n-10)} = 1$$ Using definition of limit, I get this equation: $$ \left | \frac ...
2
votes
1answer
52 views

Does a function that is twice weakly differentiable have a version that is classically differentiable?

I have been wondering about the idea of functions that are weakly differentiable. My intuition tells me that the weak derivative allows one to differentiate functions that either have a removable ...
-1
votes
3answers
61 views

Find $\frac{dG}{dx}$ of $G(x)=\int_0^{x^2}\frac{dt}{t^2+4}.$ [on hold]

Define $$G(x)=\int_0^{x^2}\frac{dt}{t^2+4}.$$ What is $\displaystyle\frac{dG}{dx}$? How do I approach this question? What are the steps? What is the solution?
1
vote
0answers
20 views

an inequality derived from conformal automorphisms of unit disk

Let $f$ be a holomorphic function on $D(0,1)$ such that $|f(z)|<1$ for all $z\in D(0,1)$. I have obtained $$ \frac{|f(0)|-|z|}{1+|f(0)||z|}\leq |f(z)|\leq \frac{|f(0)|+|z|}{1-|f(0)||z|}. $$ Is it ...
0
votes
3answers
59 views

How to solve for $x$ for $\frac{1}2 x^{-1/2}- \frac14x^{-3/4}$

This is a derivative and I am trying to find the max and min. Right now I am trying to solve for x. $$\frac{1}2 x^{-1/2}- \frac14x^{-3/4}$$ $$\frac{1}{2 x^{1/2}}- \frac1{4x^{3/4}}$$ $$\frac{1}{2 ...
3
votes
1answer
39 views

Use a double integral in polar coordinates to find the area

So the area is just an intersection of two circles Converting the two circles to polar coordinates, I get: $r(r-2\sin\theta)=0$, and $r(r-2\cos\theta)=0$ Ummm so $r =0$ and r = $2\sin\theta$ ...
1
vote
1answer
43 views

Existence of a function with certain integral properties

Does there exist a non-negative Borel-measurable function $g:\mathbb [1,\infty)\to[0,\infty)$ such that \begin{align*} \int_1^{\infty}g(y)^2\,\mathrm dy<&\,\infty,\\ ...
0
votes
2answers
19 views

Determining the best way to compute a double integral

The question is: When graphed, this is what it looks like: I thought that the best way to do it would be with respect to y first, then x. The bounds: x/sqrt3 < y < sqrt(4-x^2) 1 < x ...
0
votes
3answers
19 views

Definite integral fractional exponent in the denominator

I have come across this question and I cannot understand the step highlighted. I would have expected that the fractional exponents of the terms in the numerator would have a negative value after ...
0
votes
3answers
55 views

How to find the values of m and b?

How do I find the values of m and b when: a) the function is continuous in $x = \pi$ b) the function can be derivated in $x =\pi$ $$y=\begin{cases} \sin x & x<\pi \\ mx+b & x\ge ...
2
votes
1answer
92 views

Evaluate the limit $\lim \limits_{x \to \infty} \frac{1}{x(x+1)}$ [on hold]

How can I evaluate the limit $$\lim_{x \to \infty} \frac{1}{x(x+1)}$$
-1
votes
1answer
14 views

Curve sketching from derivative to the original [on hold]

The graph of the derivative function f'(x) of a function f(x) is shown below. Determine: i) the intervals where f(x) is increasing; ii) the intervals where f(x) is decreasing; iii) the ...
3
votes
3answers
49 views

Finding Fourier transform of $\frac{x}{(x^2 + 4)^2}$

So I have this function $$ f(x) = \frac{x}{(x^2 + 4)^2} $$ and I have to find its Fourier transform. This is however much harder than what I have done before so I don't have a clue where to start. I ...
2
votes
2answers
77 views

Exactly How Does This Proof Mean That The Cosine Function Is Continuous

The question is: Prove that cosine is a continuous function. To give some context in what way this must be answered, this question is from a sub-chapter called Continuity from a chapter introducing ...
1
vote
1answer
34 views

Calculus - Curve Sketching

The graph of $f(x)$ has the following properties: i) $f(x)$ is increasing when $x < -1$ or $x > 1$; ii) $f(x)$ is decreasing on the interval $-1 < x < 1$; iii) $f(x)$ has a local ...
3
votes
2answers
35 views

Area of spherical cap with integrals

Given a sphere $S$ of fixed diameter $D$ (or radius $R=D/2$, it will be convenient to have both, I suppose), and a point $P$ on its surface, let's create a ball $B$ of variable radius $r$ with its ...
2
votes
2answers
56 views

Compute limit of a function

Compute: $$\lim_{x \rightarrow 0^+} \frac{\arctan(e^x+\arctan x)-\arctan(e^{\sin x}+\arctan(\sin x))}{x^3}$$ WolframAlpha tells me it's 1/6. Any nice idea how to rewrite that expression? Thanks!
11
votes
0answers
101 views

Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\left\{\frac{1}{x_{1}\cdots x_{n}}\right\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

Here is my source of inspiration for this question. I suggest to evaluate the following new one. $$ I_{n}:= \int_{0}^{1} \! \cdots \! \int_{0}^{1} \left\{\frac{1}{x_{1}x_{2} \cdots ...
2
votes
0answers
20 views

Derivative of terminal state w.r.t. the inital conditions.

Let $x\in R^n$ and consider the system $$ \dot{x}=f(t,x) \;\;\mbox{with}\;\; x(0)=x_0 $$ and suppose that we know it's exact or very accurate solution $x(t)$ for the time interval $[0,T]$. I'm ...
2
votes
3answers
37 views

Integration Trig Substitution

After making the correct trig substitution what does the integral of $\dfrac{1}{\sqrt{9-x^2}} dx$ reduce to without solving the equation? I reduced it down to the integral of ...
3
votes
2answers
25 views

Basic limit question to understand the methods

I have a very basic question about proving limits with the epsilon-delta method. So i want to prove $\lim _{x\to 0}\left(\frac{1}{1-2x}\right)\:=\:1$ . first, i write it like that: ...
1
vote
1answer
95 views

Evaluation of $ \int\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$ [duplicate]

Evaluation of $\displaystyle \int\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$ $\bf{My\; Try::}$ Given $\displaystyle \int\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx = \int ...
1
vote
5answers
57 views

How to get the following limit into indeterminate form?

I am struggling to get the following limit into its indeterminate form so that i can apply the l'Hopitals rule: $$\lim_{x\to 0^+}(\sin x)^x$$ A solution would be greatly appreciated, been struggling ...