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

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1
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2answers
41 views

Area under the curve $f(x) = \sin x$

Find the area under the curve $f(x) = \sin x$ on the interval $[0, \pi]$ if $\sin x \ge 0$ My handbook give this as $$\int_0^\pi \sin x \space dx = (\cos \pi) - (\cos 0) = (-1) - (-1) = 2$$ what ...
2
votes
1answer
34 views

Inverse trig and trigh in integration?

I have just done part (iii) of this question and can get the right answer but am a bit confused why do we take arcosh i.e. just the principle value of cosh and not the other value. I presume this is ...
0
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1answer
34 views

Chain Rule - Calculus II

If $w=f(x,y)$, where $x=r \cosθ$ and $y=r \sinθ$, then how to find $\dfrac{\partial}{\partial r}\left(\dfrac{\partial w}{\partial x}\right)$ and $\dfrac{\partial}{\partial r}\left(\dfrac{\partial ...
-5
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1answer
32 views

Applied Max/Min [on hold]

Please show work, thank you. A) Find the dimensions of the rectangle with the greatest area that can be built so the base of the rectangle is on the $x$-axis between $0$ and $1$ ( $0\leq x \leq 1$ ) ...
4
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1answer
31 views

Unsure with second order complex differential equations

Solve $$y'' - 4y' + 5y = 0 $$ Where $y(0) = 0 \ , \ y'(0) = 2$. So I solve this as a second degree polynomial (no idea why) $$\frac{4 \pm \sqrt{16-20}}{2} = 2 \pm 2i$$ So the CASE III solution as ...
2
votes
1answer
106 views

LogSine Integral $I=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) d\theta$

These are known as LogSine integrals at $\pi/3$, so I will call the integral Ls as this is common in the literature. I am trying to prove $$ Ls=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) ...
5
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1answer
197 views

Prove or disprove this argument

Let $L>0$ and let $\Omega$ be the set of all integrable functions from $[0,L]$ to $]0,+\infty[$. For all $\varphi, \psi \in \Omega$ define $\left \langle \varphi,\psi \right ...
0
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1answer
55 views

Integrate square root of 4th grad polynomials

During some calculations for a program I came upon this Integral which I am not able to solve. I already tried Matlab but it didn't help me. Here is the Integral: $$\int\left(\sqrt{\sum_{0}^{5} 9 ...
4
votes
4answers
211 views

Prove that these two curves have the same length

My midterms are approaching, and I was going through some of our past Calculus midterms when I stumbled upon this question from 1996: Show that these two curves, $$(\Gamma) : \frac ...
0
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2answers
30 views

Finding numbers $a$ and $b$ for a complex number

Problem. Given a complex number $$z=2-2i$$ Find numbers $a$ and $b$ such that $$a+ib = \frac{1}{z}$$ I tried multiplying both sides by $z$ and got $$(a+ib)(2-2i)$$ $$= 2a-2ai+2bi-2bi^2$$ ...
0
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1answer
69 views

Why isn't $i$ affected by powers?

When finding roots of complex functions we can write for example: $$z=2-2i$$ Let's find complex numbers $w$ such that $$w^4 = 2-2i$$ $$\large z = \sqrt{8} e^{ \frac{- \pi }{4} i}$$ This reads: ...
0
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0answers
34 views

Line Integral and Green's Theorem

I have been working on a simple line integral: line integral of $x\,dy+y\,dx$ (I don't know how do write this properly, I'm sorry!) over the closed curve enclosed by the the ellipse $x^2+5y^2=4$ and ...
5
votes
1answer
1k views

Area Bounded by Polar Curves

I am answering sample exams for my Calculus class and my attention was caught by the following item. Set-up the definite integral or sum of definite integrals equal to the area of the region above ...
10
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1answer
86 views

Why is an equation necessarily dimensionally correct?

I have just read a fascinating proof of the value of the integral $$ \int_{-\infty}^\infty e^{-ax^2} dx, $$ which proceeds by dimensional analysis, as follows: we know that we can write $$ ...
2
votes
1answer
27 views

functions of two variables with one variable defined on a compact set uniformly converge to zero

Let $f$ be a holomorphic function on $[0,1]\times \mathbb{R}$. If for each $x\in [0,1]$ fixed, $\lim_{y\to\infty}f(x,y)=0$, prove that $f$ is bounded. My idea: I do not know how to prove and I also ...
3
votes
5answers
336 views

Jumping back into Calculus III

At the age of 30 I am going back to school for Electrical Engineering. Because of the way higher education works, all of my previous college coursework is being transferred, which does not allow you ...
0
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0answers
25 views

confusion with ds

Let $\hat n$ be the unit vector in the normal direction of a surface $S$. Let $\vec p$ be any vector in the normal direction of surface $S$. Let $d\vec S$ be a vector and $dS$ not a vector. In my ...
0
votes
2answers
42 views

Multiplying and Dividing Series

For example, how do you compute the taylor series for $$e^x \sin x=\sum_{n=0}^{\infty} \frac {x^n}{n!} \sum_{n=0}^{\infty} (-1)^n\frac {x^{2n+1}}{(2n+1)!}$$ Of course I want the result to contain ...
3
votes
5answers
106 views

Mean Value Theorem: Continuous and Differential

Mean Value Theorem. If $f:[a,b] \rightarrow R$ is continuous on [a,b] and differentiable on (a,b), then there exists a point $c \in (a,b)$ where $$f'(c) = \frac{f(b)-f(a)}{b-a} $$ I am having a ...
0
votes
2answers
40 views

Trig. identity question

I was wondering if $-(r^2\sin^2{\theta}+r^3\cos^2{\theta}) = -r^2-r^3$. If not, how would you simply the above function? Thank you.
1
vote
1answer
44 views

First Order Lagrange Remainder Using Ordinary MVT

I would like to show my Calc I class that $f(x)=f(a)+f'(a)(x-a)+(f''(c)/2)(x-a)^2$ for some $c$ in $(a,x)$ (for $f$ smooth). Just from the form of the statement, it seems as though this should be ...
0
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1answer
68 views

Proof of a condition in a proof that $\sqrt{5}$ exists

As a continuation of Prove that $\sqrt{5}$ exists I came across a different proof of $\sqrt{5}$ exists that is different from the answers on the original post and the proof goes something like this: ...
0
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0answers
39 views

LogSine Integral $\int_0^{\pi/3}\ln^n\big(2\sin\frac{\theta}{2}\big)d\theta$

I am trying to integrate the Log Sine Integral: $$ Ls_{n+1}=-\int_0^{\pi/3}\bigg[\ln\big(2\sin\frac{\theta}{2}\big)\bigg]^nd\theta $$ where n is a non-negative integer. This problem is strongly ...
2
votes
1answer
95 views

Determining k: $\int_{6}^{16} \frac{dx}{\sqrt{x^3 + 7x^2 + 8x - 16}} = \frac{\pi}{k}$

I have a calculus II final coming up and this question came up in a past final exam: $$\int_{6}^{16} \frac{dx}{\sqrt{x^3 + 7x^2 + 8x - 16}} = \frac{\pi}{k},$$ where $k$ is a constant. Find $k$. My ...
0
votes
2answers
39 views

Higher Order Partial Derivatives

If i have 3 times differential function $ z= f(x^3 / y^4) $ how can i get: a) ${\partial z \over \partial x}$ b) ${ \partial ^2z \over \partial x^2}$ c) ${\partial^2z \over \partial x \partial ...
0
votes
4answers
86 views

Prove that 1 is the only real number which satisfies $|x-1|<\frac{1}{n^2}$ for every $n \in N$

Prove that 1 is the only real number which satisfies $|x-1|<\frac{1}{n^2}$ for every $n \in N$ Here's how I would do this problem: First I noticed that $|x-1|\geq 0$ for all $x \in R$ So suppose ...
16
votes
1answer
465 views

Fourier Transform $J^3_0(x)$.

Hi I am trying to evaluate the integral $$ \mathcal{I}(\omega)=\int_{-\infty}^\infty J^3_0(x) e^{i\omega x}dx $$ analytically. We can also write $$ \mathcal{I}(\omega)=\mathcal{FT}\big(J^3_0(x)\big) ...
1
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0answers
29 views

LogSine Generating Fn $ \int_0^\pi \big(2\sin\frac{\theta}{2}\big)^x e^{\theta y} d\theta$

This is related to generating functions for Ls (Log Sine Integrals.) I am trying to calculate $$ \int_{0}^{\pi}\left[2\sin\left(\theta \over 2\right)\right]^{x} {\rm e}^{\theta y}\,{\rm d}\theta. $$ ...
8
votes
3answers
286 views

Prove that $\sqrt{5}$ exists

Prove that $\sqrt{5}$ exists; in other words prove that there exists a positive number $x\in \mathbb R$ satisfying $x^2=5$ Here's what I've done: I let $A= \{x>0:x^2\leq 5\}$ We know that $A$ is ...
2
votes
2answers
122 views

Cool Integral $\int_0^{\pi/2}dx\ln \sinh x$

$$ I_1=\int_0^{\pi/2}dx\ln \sinh x,\quad I_2=\int_0^{\pi/2}dx\ln \cosh x, \quad I_1\neq I_2. $$ I am trying to calculate these integrals. We know the similar looking integrals $$ \int_0^{\pi/2}dx\ln ...
0
votes
2answers
41 views

Length of an Astroid

I am having trouble with this question: Calculate the length of the astroid of $x^{\frac23}+y^{\frac23}=1$. s = ? I approached it by doing the following: setting $x^{\frac23}=1$ because then ...
3
votes
1answer
119 views

Integrate $ \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx $

Hi I am trying to come up with a closed form expression for $$ \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx,\quad p\geq 0. $$ I am interested in this general case in terms of p. For small p, we can ...
0
votes
1answer
22 views

Marginal Profit?

Here is what I am working on; ! I am really confused because there is not a great explanation of this in my book, and from what I can find in videos and examples, to find marginal profit you take the ...
0
votes
2answers
27 views

Double integral with integration by parts

This is for an online web assignment with multiple choice. $ \int_1^2 \int_0^1 xye^{(x^{2}+1)y} $ I solved the inner integral with respect to y: $\int_0^1 xye^{(x^{2}+1)y} $ = ...
0
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1answer
30 views

If $\{f(b + x) - f(a - x)\}/(b - a + 2x) = f'(c)$ then show that $c$ is an increasing function of $x$

Let $f$ be twice differentiable on $\mathbb{R}$ and also assume that $f, f', f''$ are increasing. Let $a\leq b$ and $x > 0$. By mean value theorem there is a $c \in (a - x, b + x)$ such that ...
0
votes
1answer
21 views

basic question about inequalities and supremum and infimum

Let $f: A \subseteq \mathbb{R}^n \to \mathbb{R}$ and suppose for any $x,y \in A , |f(x) - f(y)| \leq M $ Can we conclude that $$ \sup_{x \in A} f(x) - \inf_{x \in A} f(x) \leq M $$
2
votes
1answer
34 views

Differential question word problem?

Joules law is a physical law expressing the relationship between heat generated by a current flowing through a semi conductor. It states $Q=I^2Rt$ where q is heat in J joules, generated by a constant ...
4
votes
2answers
46 views

Taylor series of an integral function

Problem $$I(x) = \int_{1}^x \frac{e^t - 1}{t}$$ Find $I'( \sqrt{x} )$. Solution We know that $F'(x) = f(x)$ by the fundamental theorem of calculus so $$I'(x) = \frac{e^t -1}{t}$$ And so $$I'( ...
1
vote
2answers
61 views

How to Evaluate as a Riemann Sum

In the given sum ... $\lim_{n\to \infty} \frac{1}{n} (\sqrt{\frac{1}{n}} + \sqrt{\frac{2}{n}} + \sqrt{\frac{3}{n}} + ... + \sqrt{\frac{n}{n}})$ It states in the book that it is recognized as a ...
1
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0answers
28 views

Total differentiation

For each of the functions below use the total diferential to approximate the change in $Y$ due to the given changes in $X$ and $Z$: $Y= X^2 + 4X -Z^2 -2XZ$, where $X=1$ and $Z = 4$ , and $\Delta X=2$ ...
0
votes
0answers
39 views

Mean value theorem mindset

So I am to learn to use the mean value theorem to prove these types of problems that I will list. I would really like for someone to provide some visual/intuitive information on how I can imagine the ...
0
votes
1answer
24 views

Derivative of fraction

How to find the deravitive of $\frac{x}{1+y}$ with respect to $y$? If it is with respect to $x$ it is no problem, but this is the harder case. Thanks
0
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2answers
54 views

Why are the derivatives not treated the same?

It seems to me that derivatives are treated differently in certain places, but I do not understand why. Here is an example, if \begin{align} \frac{d}{dx} (\sqrt{1 + 4x^2}) & = \frac{1}{2\sqrt{1 ...
1
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1answer
36 views

Definite Integral

Solve the following : $$\int_0^1 x^5 \sqrt{1 - x^2}dx $$ Any ideas ?I haven't been able to make any progress on this exercise , it's driving me insane , since there's probably no big deal to the ...
0
votes
2answers
31 views

Finding complex roots

Problem Find the roots of $$z^3 = -1 - i$$ And calculate $$ \sqrt[3]{-1-i}$$ I'm looking at the solution outlined in my book but I'm having problems understanding it. I can find the length and ...
1
vote
1answer
37 views

Let $a_n = \frac{2n}{3n+1}$, determine if {$a_n$} is convergent.

Let $a_n = \frac{2n}{3n+1}$, determine if {$a_n$} is convergent. So what I did is I followed the definition of a convergent sequence which states that for any $\epsilon >0$ there is an $N>0$ ...
0
votes
2answers
16 views

Find extrema on the interval

Problem Find the extrema of the function $$f(x) = cos^2(x)$$ on the interval $ [-4,4]$ I can differentiate and get $$f'(x) = -2 \sin(x) \cos(x)$$ And set that to zero, but I'm pretty sure that's ...
0
votes
1answer
18 views

Help finding inflection points & concavity

I'm trying to find the inflection points and concavity of the graph of $$ x^4 - 50x^2 + 2 $$ The first derivative is $$ 4x^3 - 100x $$ and the second derivative is $$12x^2 - 100 $$ Trying to ...
2
votes
2answers
42 views

Reduction formula for $\int \frac{dx}{x^n \sqrt{ax+b}}$

I want a reduction formula for $$I_n=\int\frac{dx}{x^n \sqrt{ax+b}}$$ in terms of $I_{n-1}$. I have tried various substitutions but I just can't seem to find the right one. Any help or hints will ...
0
votes
2answers
51 views

Evaluate the integral $\int \frac{1}{2+3\sin x}dx $

Please evaluate the integral $$\int \frac{1}{2+3\sin x}dx $$ What I have tried is to substitute $\sin x = \sqrt(1-x^x)$ but I was stuck in a maze. Also, I did look a the wolfram solution. Can anyone ...