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

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160 views

Volume of a solid of revolution

Let $[a,b]$ be an interval, $a\geq 0$ and $f:[a,b]\to \mathbb{R}_+$ continuous. I want to calculate the volume of the solid of revolution obtained by rotating the area below the graph of $f$ around ...
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1answer
76 views

The lower bound of the product between two variables

I wonder how I can determine the minimum of the product between variables $x$ and $y$ (in terms of $\theta$), given that both $x < 1 - \theta$ and $y < 1 - \theta$, and $x + y = 1$? So far I ...
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201 views

Examine the Maximum and Minimum Value

I have to do the following problem, and I need help. Examine the function $f(x,y) = \dfrac{-3x}{x^2+y^2+1}$ with respect to maximum and minimum.
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2answers
158 views

Does this equality always hold?

Is it true in general that $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x} \int_0^{x} f(u,x) \mathrm{d}u = \int_0^{x} \left( \frac{\mathrm{d}}{\mathrm{d}x} f(u,x) \right)\mathrm{d}u +f(x,x )$ ? Thank ...
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259 views

Sketching a polar curve

Continued off the question I asked earlier, I also have to sketch the curve. $r^2=−4\sin(2\theta)$ So I have to set up a table of values I'm assuming. How do I know what values to choose for ...
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221 views

How to solve a polar equation when $r$ is $r^2$ instead?

I have $r^2=-4\sinθ$ and I'm asked to set $r=0$, then find θ. If I just set $r^2=0$ then I'll get $\sin(2θ)=0$. That doesn't seem right. Then I'm asked to set $θ=0$ and then find $r$. If I use the ...
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1answer
115 views

Where does this 1 come from when balancing this integral equation?

$$ \int e^{ax}\cos(bx)\,\mathrm dx = \frac1{a}e^{ax}\cos(bx) + \frac{b}{a^2}e^{ax}\sin(bx) - \frac{b^2}{a^2}\int e^{ax}\cos(bx)\,\mathrm dx$$ $$\left(1 + \frac{b^2}{a^2}\right)\int ...
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1answer
2k views

Finding a one sided limit algebraically (not plugging in numbers)

I'm looking for a way to determine a one sided limit algebraically, such as $$\color{blue}{f(x) = \frac {|x|}{x} , x \neq 0}$$ I know that you can find the limit by plugging in numbers or graphing ...
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1answer
113 views

Solution for a PDE on $\Omega=[0,\pi]^3$

In Strauss's Partial Differential Equations, the eigenvalue problem $$-\triangle v=\lambda v,\qquad v|_{\partial \Omega}=0$$ is solved by separating the $x,y,z$ variables: $v=X(x)Y(y)Z(z)$, $$ ...
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4answers
136 views

A few things about about derivatives and integrals

I'd like to know the reason behind some notations used when handling derivatives and integrals. For example, why does $x' = 1 \frac{d}{dx}$ and not simply $1$? Related to this, why is integral of ...
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1answer
118 views

Find connected components of a a set Y

Let Y be the union of all the circles of center (1,0) radius 1-1/n in R^2. Then, we have circles of increasing radius, finally reaching r=1 as n goes to infinity. The connected components are the ...
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1answer
1k views

Transforming Trig Function for Easier Integration

$$\int_0^\frac{\pi}{4} \! \frac{1+\cos^2\theta}{\cos^2\theta} d\theta$$ I've been attempting to mix and match identities to make this equation easier to integrate. Mathematica has given me an ...
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1answer
79 views

Interpreting “lying on the parabolas”

Question 49 of chapter 30 of Schaum's calculus is: The section of a certain solid cut by any plane perpendicular to the x axis is a square with the ends of a diagonal lying on the parabolas ...
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162 views

Volume of a rotated region?

How can I find the volume of the solid generated when the region enclosed by $y=0, x=0, x=1$ and $(1+e^-2x)^0.5$ is rotated through 360 degrees about the x axis?
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2answers
483 views

Find the length of the given curve $x=3y^\frac{4}{3}-\frac{3y^{\frac{2}{3}}}{32} $

Find the length of the given curve $\displaystyle x=3y^{\frac{4}{3}}-\frac{3y^{\frac{2}{3}}}{32} $ where the bounds are given by $-8 \leq y \leq 343$. I can solve for the positive part (0 to ...
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3k views

Chain Rule applied to Trig Functions

Given $f(x)= \sin(\pi x)^{2}$, find the derivative. Using the chain rule my work is as follows: $(\sin(\pi x)^2)'$ becomes $$2 \sin(\pi x) \cdot \frac{d}{dx}(\sin(\pi x)$$ The derivative of sin ...
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3answers
765 views

$\sin(2\pi nx)$ does not converge for $x \in (0,1/2)$

How to show that $\sin(2 \pi nx)$ does not converge as n goes to infinity? $x \in (0,1/2)$
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3answers
176 views

$f(2x)-f(x)<f(3x)-f(2x)$ using Lagrange theorem

Let $f$ be a continuous function, with the initial condition: $f''>0$. I need to prove that $f(2x)-f(x) < f(3x)-f(2x)$. By $f''>0$ I can learn that $f'$ is monotonous ...
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1answer
106 views

More help with vertical asymptotes

Of the six questions regarding finding vertical asymptotes of graphs, I've had problems with two. The second is using the graph of $$g(x)= \frac{3+x}{x^{2}(3-x)}$$ Now, looking at the function, it ...
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1answer
431 views

Equation for calculating the volume of liquid in an ellipsoid

I am looking for an equation to calculate the volume of liquid in an underground tank based on a depth reading. The tank is an ellipsoid shape with the following dimensions: ...
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2answers
73 views

For every $y\in \mathbb{R}$ there is some $c$ in $(a,b)$ with $f'(c) = yf(c)$

Suppose $f\colon [a,b] \to \mathbb{R}$ is continuous and has a finite derivative $f'$ everywhere on $(a,b)$. If $f(a)=f(b)=0$ prove that for every $y\in\mathbb{R}$ there is some $c$ in $(a,b)$ ...
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816 views

how to find bounds of the series $\sum_{i=1}^n \frac1{4i-1}$

$$\sum_{i=1}^n \frac1{4i-1}$$ I know I have to integrate the function but from what to find lower and upper bound.
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95 views

Parametric Equation Question

Ok this is a really silly question and I should know this, but I can't seem to figure something out: for the last step, how do they know that $0 \leq x \leq 4$? If we use the minimum value of theta, ...
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780 views

How to calculate the expected distribution of results using random numbers?

See this SO thread: Calculating which item is next in a percentage distribution. To summarize, you calculate ...
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1answer
262 views

Infimum of some quantities

I am not sure if this question is appropriate here but would appreciate any help. Let $n>2$ and $x_1,\cdots,x_n$ be real numbers. What is the infimum of: $$A = \sum_{i=1}^n \frac{1}{(1+x_i)^2} $$ ...
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44 views

What is $\int \frac{1}{\sqrt{25y^2-10y-3}}dy$

$= \int \dfrac{1}{\sqrt{(5y-1)^2-4}}dy$ $=\int \dfrac{1}{\sqrt{u^2-4}}\dfrac{du}{5}, \quad U$ substitution $=\int \dfrac{1}{10\cos(\theta)} 2\cos(\theta) d\theta, \quad$ Trig substitution $= ...
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2answers
45 views

Integration of the following

What is the definite integral of $$ \int_0^1 \left(\frac{g(x)}{f(x)}\right)'\cdot\frac{1}{g(x)}\,dx, $$ where the conditions are as follows: $f(0) = 2 $ $f(1) = 3 $ $f'(x) $ is continuous For all ...
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1answer
21 views

Maximum and minimum of a fractional function

Let $x, y \in \mathbb{R}$, $a, b, c$ are three real parameters with $c\neq 0$. Find the maximum and minimum of $\dfrac{ax+by+c}{\sqrt{x^2+y^2+1}}$ This is quite complicated if I calculate the ...
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26 views

How would I start to solve this?

I need to calculate the derivative of $F(x)=\int_{f(x)}^{f^2(x)}f^3(t)dt$. Usually for a derivative of an integral I would plug the upper bound and lower bounds into $f(t)$ then multiply each by their ...
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1answer
37 views

Evaluate an integral quickly

Evaluate the integral $$\int \sqrt{x} \ln(1+x)dx $$ so we should start with the substitution: $t=\sqrt{x}$ $$ \int t\ln(1+t)dt2t = 2\int t^2\ln(1+t)dt $$ From here, it seems reasonable to ...
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2answers
23 views

What are the derivative, differentiable and differentiation? [on hold]

If the question is $f(x)=\frac{2x+1}{1-x}$? The derivative is $f'(x)=\frac{(1-x)(2)-(2x+1)(-1)}{(1-x)^2}$?
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25 views

Find all singularities of a function and determine its types

Find all singularities of a function and determine its types $$f(z)=\frac{e^{iz}-1}{\sin{z}}e^{\frac{1}{z}}$$ I already showed, that $f$ has poles at points $z=\pi n$ where ...
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1answer
40 views

Prove that $\lim_{n \rightarrow \infty} \int_0^1 f_n(x)dx \ne \int_0^1\lim_{n \rightarrow \infty}f_n(x) dx$

If $f_n(x)=nxe^{-nx^2}~\forall~n=1,2,\cdots$ and $x$ real, show that $$\lim_{n \rightarrow \infty} \int_0^1 f_n(x)dx \ne \int_0^1\lim_{n \rightarrow \infty}f_n(x) dx$$ Attempt: By the $Mn$ Test, it ...
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31 views

Does this supremum equal infinity?

This is a generalization of the previous question Does this infinum tend to infinity? Let $f:\mathbb{R}^2\to\mathbb{R}$ be a continuous function satisfying $$\sup ...
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1answer
34 views

nth derivative of ${1\over x}$. A problem. [on hold]

$f(x)=f^{(0)}(x)=x^{-1}$, $f^{(1)}(x)=-x^{-2}$, $f^{(2)}(x)=2x^{-3}$. Therefore, $f^{(n)}(x)=(-1)^{n}n!x^{-n-1}$. Except I see in some places that the expression is different, using, for example, ...
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1answer
37 views

Show how $\frac{\partial}{\partial x} \left[\int_0^x (x-t)g(t)\,\mathrm{d}t\right] = \int_0^x g(t)\,\mathrm{d}t$

It has something to do with the second part of the Fundamental Theorem of Calculus right? I've always had trouble with this theorem ever since I learned it several years ago :\ Would somebody please ...
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55 views

Explanation for $\lim_{x\to2} e^{\frac{1}{x-2}}$

I can't find out why is the limit from the left side = 0 and from the right = Infinity?
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3answers
77 views

How $\int_{-\infty}^{\infty}\frac{dx}{1+x^2}$ exists?

How $$\int_{-\infty}^{\infty}\frac{dx}{1+x^2}$$ exists? It is difficult question to me. i have tried to evaluate by using fact that $$\int_{-\infty}^{\infty} f(x) \ dx =\int_{-\infty}^{0} f(x)\, dx ...
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2answers
62 views

How can I prove this integral?

I have to use the identity $b^4-a^4=(b-a)(b^3+b^2a+ba^2+a^3)$ to prove that: $\int_b^ax^3dx=\frac{b^4-a^4}{4}$. I know that you can just do $F(b)-F(a)$ and since the integral of $x^3$ is ...
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1answer
58 views

Computing a strange integral

Prove that $(-1)^n \int_{-1}^1 (x^2 - 1)^ndx = \frac{2^{2n+1}(n!)^2}{(2n+1)!}$ This one has me stumped. I've tried the obvious (using binomial theorem and then integrating termwise, or computing the ...
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2answers
51 views

Level curves for “unsolvable” integral

Problem: Sketch the level curves of g defined by $$g(x,y)=\int_x^y{e^{-t^2}dt}$$ Attempts at solution: (1) Apparently we could take $y=x$, then for $z=0$ the level curves will be just that. However, ...
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1answer
21 views

Transformation of an equation

How do you get from the left side to the right side in this equation? $$\frac{1+\sqrt{5}}{2} + 1 =\left(\frac{1+\sqrt{5}}{2}\right)^2$$
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55 views

Evaluate $\lim_{t \rightarrow 0} \int_{0}^t \frac{1}{f(u)}du$

Let $f(u)$ be a function such that $\lim_{u \rightarrow 0} f(u)=0$ e.g. $f(u)={u}$. How would I evaluate $$ \lim_{t \rightarrow 0} \int_{0}^t \frac{1}{f(u)}du $$ Is this always equal to zero? My ...
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2answers
45 views

Convergence of $\sum_{n=1}^\infty \frac {n^3[\sqrt 2 + (-1)^n]^n}{1.05^n} $

Test convergence of $\sum_{n=1}^\infty \dfrac {n^3[\sqrt 2 + (-1)^n]^n}{1.05^n} $ Attempt: By using the the $n^{th}$ root test : $\lim_{n \rightarrow \infty} a_n^{1/n} = \lim_{n \rightarrow \infty} ...
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1answer
14 views

Composed of non differentiable functions

It will be possible to find a function $f:\mathbb{R}\rightarrow \mathbb{R}$ non-differentiable at zero such that $f\circ g$ is differentiable at zero where $g:\mathbb{R}\rightarrow \mathbb{R}$ is ...
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1answer
64 views

Pullback of a differential form

My question is in regards to a proof in Lee's 'Introduction to Smooth Manifolds'. He proves a lemma about the pullback of a differential form on a manifold $N$, where $F:M\rightarrow N$ is a smooth ...
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1answer
86 views

Convergence of $\sum_{n=2}^\infty \frac {1}{(\log n )^3}$

Test Convergence of $\displaystyle\sum_{n=2}^\infty \dfrac {1}{(\log n )^3}$ Attempt: I haven't been able to find a suitable comparator for $\dfrac {1}{(\log n )^3}$ . The integration test also seems ...
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2answers
29 views

Integration of rational function

Please help me in integrating the following: $$\int \frac{r^2}{(r-n) (-ar^2 + r (1 - an) + n - an^2 - 2m)}dr$$ I have done it with the help of Mathematica, I just want to confirm that result. So how ...
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1answer
27 views

Use the Newton-Raphson algorithm to find all roots accurate to $10$ decimal places of the polynomials

Use the Newton-Raphson algorithm to find all roots accurate to $10$ decimal places of the two polynomials $p(x)=5ix^4-(9+2i)x^3+7x+6-i$ and $q(x)=9x^5-x^3+7x+6$. The roots, with accurate to $10$ ...
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2answers
51 views

Profound Calculus Theorems list

Is there a website or a book with a calculus theorems list? Or what are the ways remembering calculus theorems list?