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

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1
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2answers
4k views

Find the equation of the tangent line to the curve $y = \sqrt{x}$ at the point $(9,3)$.

Find the equation of the tangent line to the curve $y = \sqrt{x}$ at the point $(9,3)$. I did $$\frac{f(x+h)-f(x)}{h}=\frac{\sqrt{9+h}-3}{h} \frac{\sqrt(9+h)+3}{\sqrt(9+h)+3} = ...
2
votes
1answer
148 views

Show $\lim\limits_{n\to\infty} \frac{2n^2-3}{3n^ 2+2n-1}=\frac23$ Using Formal Definition of Limit

I want to show that $a_n=\frac{2n^2-3}{3n^ 2+2n-1}$ is convergent. So I did the following: \begin{align*} \left|a_n-\frac23\right|&=\left|\frac{2n^2-3}{3n^ 2+2n-1}-\frac23\right|\\ ...
1
vote
1answer
51 views

projection of inner products

Update of question Let $V$ be the space of real polynomials in one variable $t$ of degree less than or equal to three. Define our inner product to be: $$ \langle p,q\rangle = ...
0
votes
3answers
70 views

Problem with change of variables

I have this integral: $$\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}xyz\,dx\,dy\,dz=\frac{1}{8}$$ But when I make this change of variables:$$x=t$$$$y=t$$$$z=t$$ I have ...
18
votes
2answers
808 views

Integral $\int_0^{\pi/2}\frac{x}{\sin x}\log^2\left(\frac{1+\cos x-\sin x}{1+\cos x+\sin x}\right)dx$

Please help me to evaluate this integral: $$\large\int_0^{\pi/2}\frac{x}{\sin x}\log^2\left(\frac{1+\cos x-\sin x}{1+\cos x+\sin x}\right)dx$$
0
votes
1answer
377 views

Area by integration of the finite region bound by the two curves.

Homework - Q Sketch the graphs of the curves $y = 16 - x^2$ and $y = x^2 - 5x + 13$ The first thing I did was to set them equal to each other: $16 = x^2 = x^2 - 5x + 13$ Then set that equal to ...
2
votes
1answer
129 views

Stochastic dynamic programming

I am making some homework exercises at the moment and I was wondering if what I did in the following exercise was correct. PROBLEM Solve $E(\sum_{k=0}^{N-1}(1-u_k)X_k + X_N) \rightarrow \max$, ...
3
votes
1answer
38 views

Problem derivating the inverse cossecant

$\DeclareMathOperator{\arccsc}{arccsc}$ I've tried to derivate the $\arccsc$ function but something seems to be wrong with my reasoning. $$\csc (\arccsc x) = x $$ $$\frac{d}{dx} \csc (\arccsc x) = ...
0
votes
4answers
2k views

What is a continuous extension?

The continuous extension of $f(x)$ at $x=c$ makes the function continuous at that point. Can you elaborate some more? I wasn't able to find very much on "continuous extension" throughout the ...
2
votes
0answers
57 views

Approximating a function using its integral

Question: Let $f:\Bbb R \to \Bbb R \in C^{1}, \forall \delta>0:$ $$F_\delta = \frac 1{2\delta}\int^{x+\delta}_{x-\delta} f(t) \, d(t)$$ in $[a,b]$ prove that $\forall \varepsilon>0 \exists ...
1
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1answer
125 views

Relative Minima and Maxima and Graph $f''(x)$ using below graph $f(x)$

Let f be a function that is continuous for all $x$ and differentiable for all $x$ other than $0$. figure below is the graph of its derivative $f'(x)$ (a) What are the critical numbers? Where do any ...
0
votes
1answer
60 views

Finding mass by integration

I need some help here: Find the mass of a straight wire of length $L$ [cm] with density $Q(s) = \sin\left(\frac{\pi s}{L}\right)$ at distance $s$ [cm] from one end. What I did was to ...
1
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1answer
488 views

A particle is moving along the curve $y= 2 \sqrt{2 x + 2}$

A particle is moving along the curve $y= 2 \sqrt{2 x + 2}$. As the particle passes through the point $(1, 4)$, its x-coordinate increases at a rate of $2$ units per second. Find the rate of change of ...
2
votes
2answers
87 views

Integral of a positive function is positive?

Question: Let $f:[a.b]\to \Bbb R \in R[a,b]$ s.t. $f(x)>0 \ \forall x \in \Bbb R.$ Is $\int _a^b f(x)\,dx>0$ ? What We thought: We know how to prove it for weak inequality, for strong ...
2
votes
1answer
124 views

Implicit derivative - $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Let $y$ be a function of $x$ determined by the equation $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ Find $\Large\frac{dy}{dx}$ and $\Large\frac{d^2y}{dx^2}$ I've obtained $\Large\frac{dy}{dx} = ...
2
votes
1answer
48 views

Problem book,typing error?

I have the following problem : Let $f:(0,\infty)\rightarrow \mathbb R$ be an arbitrary function satisfying the hypothesis, $\lim_{x \rightarrow 0} x(f(x)-1)=0$. Show that $\lim_{x \rightarrow 0 } ...
0
votes
1answer
213 views

evaluation of $\int\frac{x^2}{\left(1+x^4\right)\sqrt{1+x^4}}dx$

$\displaystyle \int\frac{x^2}{\left(1+x^4\right)\sqrt{1+x^4}}dx$ $\bf{My\; Try::}$ Let $\displaystyle I = \frac{1}{2}\int\frac{2x^2}{\left(1+x^4\right)\sqrt{1+x^4}}dx = ...
5
votes
1answer
215 views

Volume of a sphere with three holes drilled in it.

Suppose that the sphere $x^2+y^2+z^2=9$ has three holes of radius $1$ drilled through it. One down the $z$-axis, one along the $x$-axis, and one along the $y$-axis. What is the volume of the resulting ...
0
votes
1answer
27 views

Gaussian Algorithm

So here's what it looks like: x1 x2 x3 b 1 1 1 10 4 2 1 16 9 3 1 18 I understand that I'm supposed to have eliminate the 9 and 3 from the last row and the 4 from the ...
1
vote
0answers
57 views

How to know how many bounds an inequality has?

How can you know, before solving it, how many bounds an inequality should have? For example $$ \dfrac{x^2 + 2}{1-x^2} < 3$$ A priori to me it looks like it would have 2 bounds because it's a ...
5
votes
2answers
189 views

Infinite series $\sum_{n=0}^{\infty}\arctan(\frac{1}{F_{2n+1}})$

How can I find the value of the following sum? $$\sum_{n=0}^{\infty}\arctan(\frac{1}{F_{2n+1}})$$ $F_n$ is the Fibonacci number.($F_1=F_2=1$)
1
vote
2answers
124 views

prove if $(A_n)$ limit is $L$ then $(A_n)^2$ limit is $L^2$

Hello, What I need to prove is: if $(A_n)$ limit is $L$ then $(A_n)^2$ limit is $L^2$. I've added my attempt to prove it. I got stuck so I'm guessing I'm missing something here. help will be ...
4
votes
2answers
282 views

Evaluating $\lim\limits_{n\to \infty}\left\{(1+\frac{1}{n})(1+\frac{2}{n})\dots(1+\frac{n}{n})\right\}^{\frac{1}{n}}$

Question is to evaluate : $$\lim_{n\to \infty}\left\{ \left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)\dots\left(1+\frac{n}{n}\right)\right\}^{\frac{1}{n}}$$ I tried to do something like this ...
2
votes
1answer
135 views

Proving $\lim\limits_{x\to0}\left(\frac{1}{\log(x+\sqrt{1+x^2})}-\frac{1}{\log(1+x)}\right) =-\frac12$

How can I prove that $$\lim_{x\to0}\left(\frac{1}{\log(x+\sqrt{1+x^2})}-\frac{1}{\log(1+x)}\right)=-\frac{1}{2}$$
5
votes
2answers
1k views

Series $\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^2}$

Is it true that for $x\in[0,2\pi]$ we have $$\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^2}=\frac{x^2}{4}-\frac{\pi x}{2}+\frac{\pi^2}{6}$$ How can I prove it? For other intervals what is the value of above ...
10
votes
2answers
134 views

Prove $\frac{1}{b-a}\int_a^b\frac{x}{\sin x}dx\leqslant\frac{a+b}{\sin a+\sin b}$ $a,b\in(0,\frac{\pi}{2}),a<b$

For $a,b\in(0,\frac{\pi}{2}),a<b$, prove $$\frac{1}{b-a}\int_a^b\frac{x}{\sin x}dx\leqslant\frac{a+b}{\sin a+\sin b}.$$ By mean value theorem, there is $c\in(a,b)$ s.t. $$\frac{c}{\sin ...
3
votes
2answers
168 views

Evaluate the following definite integral

Evaluate $$\int_0^{\frac{\pi}{4}}\arctan\sqrt{\frac{\cos 2x}{2\cos^2x}}dx$$ It seems that the anti-derivetive of the integrant cannot be written as elementary functions. How to evaluate this?
0
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1answer
22 views

limit of sequence of functions

Suppose $$ f_n = \frac{1}{(1 + \frac{x}{n})^n x^{\frac{1}{n}}} $$ What is $\lim_{n \to \infty } f_n $ ?? I am having hard time with this sequence which seems like it is going to be something like ...
0
votes
3answers
59 views

Limit of bounded sequence

Prove that: if $x_n \gt 0\forall n\in \mathbb N$ and if $\lim_{n\rightarrow \infty}\frac{x_{n +1}}{x_n} \lt 1$ then $\lim_{n\to \infty}x_n =0$ I don't know how to start, please help me.
1
vote
1answer
41 views

Help with determining trigonometric limit

Use the relation $\lim_{\theta \to 0}\frac{\textrm{sin}\theta}{\theta}=1$ to determine the limit of $f(x)=\frac{\textrm{tan}(2x)}{x}$ I understand the identity ...
0
votes
3answers
58 views

Inner Product of Real Polynomials

Updated improved question: Let $V$ be the space of real polynomials in one variable $t$ of degree less than or equal to three. Define $$ \langle p,q\rangle = ...
2
votes
3answers
2k views

Find the point on the line $x + 4y − 7 = 0$ which is closest to the point $(-2, -2)$

Find the point on the line $$x + 4y − 7 = 0$$ which is closest to the point $(-2, -2)$ First I used the distance formula and found I need to minimize $$(x+2)^2 + ...
0
votes
0answers
47 views

How to find x with this equation

I have a problem when deriving this equation to find x : $(\frac{A}{B}+x)\rho^x=(\frac{C}{B})$ Previously I use lambert W function to solve this however the result is weird for x maybe some ...
0
votes
1answer
44 views

proving $\int_{a}^{b}\frac{x^{n-1}\left((n-2)x^2+(n-1)(a+b)x+nab\right)}{(x+a)^2(x+b)^2}dx = \frac{b^{n-1}-a^{n-1}}{2(a+b)}$

How can we prove:: $\displaystyle \int_{a}^{b}\frac{x^{n-1}\left((n-2)x^2+(n-1)(a+b)x+nab\right)}{(x+a)^2(x+b)^2}dx = \frac{b^{n-1}-a^{n-1}}{2(a+b)}$ I did not understand How can I start the above ...
6
votes
2answers
326 views

proving of Integral $\int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}dx = \ln\left(\frac{a}{b}\right)$

How can we prove $\displaystyle \int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}dx = \ln\left(\frac{a}{b}\right)$ $\bf{My\; Try}::$ Let $\displaystyle I(a,b) = \int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}dx$ ...
1
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2answers
1k views

Finding derivative of three variables

Consider a box with dimensions x, y, and z. x is changing at a rate of 1 m/s, y at -2 m/s and z at 1 m/s. Find the rate that the volume, surface area and diagonal length ($s = \sqrt{x^2+y^2+z^2}$) are ...
2
votes
1answer
708 views

Integration by Substitution of a fraction with the numerator having a square root

I can't seem to see how to split or simplify this before integrating: $$\int \frac{\sqrt{3x+5}}{x} \, dx$$. Any ideas where to start from?...using integration by substitution that is
1
vote
1answer
503 views

How to differentiate CDF of Gamma Distribution to get back PDF?

CDF of a gamma distribution ($X \sim \mathcal{G}(n, \lambda)$) looks like $$F(x) = \frac{\Gamma_x(n)}{\Gamma(n)}$$ Where $\Gamma_x(n) = \int_0^x t^{n-1} e^{-t} \, dt$ the incomplete gamma function. ...
0
votes
2answers
51 views

Integration by Substitution problem

I was given an integration problems sheet...with answers too but how a certain answer is to be obtained is obviuosly not stated. Using integration by substitution integrate the following: $$ \int ...
3
votes
4answers
150 views

What other definite integrals can be computed in a manner similar to $\int_{-\infty}^\infty e^{-x^2}dx$?

The technique for computing $\int_{-\infty}^\infty e^{-x^2} dx=\sqrt{\pi}$ by computing the integral squared using polar coordinates is well known. Are there any other integrals that can be computed ...
1
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2answers
235 views

How to compute the integral $\int_{-\infty}^\infty e^{-x^2}\,dx$? [duplicate]

How to compute the integral $\int_{-\infty}^\infty e^{-x^2}\,dx$ using polar coordinates?
1
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1answer
52 views

real polynomials

Improved part to this question Let $V$ be the space of real polynomials in one variable $t$ of degree less than or equal to three. Define $$ \langle p,q\rangle = ...
2
votes
2answers
116 views

Help with this integral? I can't figure out the substitution.

I've been struggling to figure out this integral. $$\int \frac {1}{x\sqrt{5-x^2}}$$ I'm almost certain it has something to do with this fact: $$\int \frac 1{\sqrt{1-x^2}} = \sin^{-1}(x) + C$$ But ...
0
votes
2answers
121 views

angles of polynomials

Here is an improved question that was asked before. Let $V$ be the space of real polynomials in one variable $t$ of degree less than or equal to three. Let our inner product be defined by: $$ \langle ...
0
votes
1answer
71 views

What exactly is a stationary point?

I am asked to find the stationary points of the function $y=5+24x-9x^2-2x^3$. When I looked on the wikipedia page for the definition of stationary points, I read that a stationary point is a point ...
0
votes
1answer
90 views

Why does the limit exist on this interval?

Does the $\lim_{x \to x_0} f(x)$ exist at every point $x_0$ in $(-1,1)?$ I answered False, but the correct answer is True. Why? My thoughts: $f(x)$ is not the same number as $x \rightarrow 1$ from ...
2
votes
5answers
282 views

Find $\lim_{x\rightarrow 0} \frac{\cos x - 1}{x}$

I'm trying to find the following limit: $$\lim_{x\rightarrow 0} \frac{\cos x - 1}{x}$$ I tried to use squeeze theorem but it's not making much sense. I did the following: $$\begin{align} ...
2
votes
2answers
50 views

solving differential equation $(x+y)=\frac{dy}{dx}(4x+y)$

we've got this differential equation $$(x+y)=\frac{dy}{dx}(4x+y)$$ now what should be substituted for $x,y$ i can always do $$x+y=k$$ and then $$1+\frac{dy}{dx}=\frac{dk}{dx}$$ but this does't suite ...
2
votes
2answers
58 views

solving differential equation $\frac{dy}{dx}=(x+y)\ln(x+y)-1$

we have got differential equation and need to solve it $$\frac{dy}{dx}=(x+y)\ln(x+y)-1$$ my attempt $$x+y=k$$ $$1+\frac{dy}{dx}=\frac{dk}{dx}$$ $$\frac{dk}{dx}=k\ln(k)$$ ...
1
vote
3answers
66 views

Calculus Question pertaining to volume

I'm having a lot of trouble with this problem, so I need some help. A company plans to make aluminum can, each with a lid and containing a volume of 2,000 cubic centimeters. a) Find the dimensions ...