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

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0
votes
3answers
149 views

Use Lagrange Multipliers to find the absolute extrema

Use Lagrange Multipliers to find the absolute extrema (if any) of: $f(x,y) = 4x^2 + 9y^2$; subject to $2x +3y = 6$. Using Lagrange I end up with one point: $(\frac{3}{2}, 1)$ I'm just not sure how ...
1
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1answer
42 views

simple dumb vector question

so I've started vector calc otherwise known as calc 3 and simple question for the image above does the vector $\parallel v \parallel$ or $\overrightarrow{PQ}$ change as the initial point moves ...
1
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2answers
68 views

Product of Gamma functions I

What is the value of the product of Gamma functions \begin{align} \prod_{k=1}^{8} \Gamma\left( \frac{k}{8} \right) \end{align} and can it be shown that the product \begin{align} \prod_{k=1}^{16} ...
3
votes
1answer
67 views

ODE standard form.

I noticed that whenever mathematicians talk about Legendre polynomials they bring the ODE to the form $(1-x^2)f''(x)-2xf'(x)+n(n+1)f(x)=0$. When solving Poisson's equation, this form is not the most ...
1
vote
2answers
304 views

Product of Gamma functions II

What is the value of the product of Gamma functions \begin{align} \prod_{k=1}^{10} \Gamma\left(\frac{k}{10}\right) \end{align} and can it be shown that \begin{align} \prod_{k=1}^{20} ...
1
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2answers
91 views

compute taylor series about $x=0$ of $\arctan(e^x -1 )$

hello I am having some issue and need a little guidance with this taylor expansion $$f(x)=arctan(e^x -1)$$ the terms i should get are $x+\frac{x^2}{2}-\frac{x^3}{6}-\frac{11 x^4}{24}-\frac{5 ...
0
votes
0answers
30 views

I was wondering if there was a way to plot a vector field and a 3d surface in the same window

I want to plot the vector field $$\textbf{F}(x, y, z) = \sin(xyz)\textbf{ i }+ x^2y\textbf{ j }+ z^2e^{x/5}\textbf{ k }$$ And the surface (part of elliptic cylinder) $$\textbf{r}(u, v) = u\textbf{ i ...
1
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3answers
120 views

Why rotating a function around line $y=x$ gives an inverse of this function?

So I'm trying to read through a book on calculus on my own and there is a statement that if we have a graph of some function $y=f(x)$ and this is an injective function, then rotating it around the ...
5
votes
1answer
84 views

Show that $f$ is bounded at $(0,+\infty)$.

Show that a function $f\in C^{1}\bigl((0,+\infty)\bigr)$ which satisfy $$f'(x)=\frac{1}{1+x^{4}+\cos f(x)},\, x>0$$ is bounded at $(0,+\infty)$. My attempt : I would like to prove that ...
0
votes
2answers
37 views

simple vector question

In a book I'm reading it mentions 'For motion along a straight line, i.e. in a 1-dimensional space, the velocities are also contained in that 1-D space since they are just numbers. For general motion ...
0
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1answer
24 views

Surface integral over the plane $x+y+z=2$

Evaluate $$\iint_S x\;dy \times dz + y \; dz \times dx + z \; dx \times dy$$ where $S$ is the part of the plan $x+y+z=2$ in the first octant, with normal $n$ such that $n . (0,1,0) \geq 0$ My ...
0
votes
5answers
95 views

Guys, hard limit, please help.

Here is the limit I'm struggling with: $$\lim_{x\to0}\cfrac{x\tan x-x\sin x}{x\sin^2x/\cos x}.$$ Worked so hard to find it, but couldn't.
0
votes
0answers
57 views

Finding criteria for a household financial budget falsification

I’m working on a financial problem about budget of households. Households in a state fill a form about their net budget in every year and our insurance company investigate their financial status and ...
1
vote
1answer
84 views

Can this polynomial have two distinct roots in $[-1,1]$? [closed]

how can I prove that $f_m(x) = x^3 + 3x +m$ can not have two distinct roots in $[-1,1]$? I tried Rolle's theorem but this hasn't worked for me. Help, please.
1
vote
2answers
63 views

Short question about the proof of Cantor nested sets theorem

I have a question about a part of this proof I have: We have two sequences such that: $\forall n : a_n\le a_{n+1}\le b_{n+1}$ $a_n$ is a monotone increasing sequence bounded above by $b_{n+1}.$ ...
0
votes
1answer
59 views

Evaluate $\lim\limits_{x\to\color{red}{-1^{-1}}}\sum\limits_{i=2000}^{2009}|x-i|$

$$\lim_{x\to\color{red}{-1^{-1}}}\sum_{i=2000}^{2009}|x-i|$$ I could not able to properly evaluate it. Can anyone help?
1
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0answers
28 views

Additive like function representation [duplicate]

Let's $f:R\rightarrow R,~r\gt0.$ And we have that for every $x,y\in R\Rightarrow |f(x+y)-f(x)-f(y)|\le r.$ Prove that there are $h:R \rightarrow R$ additive and $g:R \rightarrow [-r,r]$ functions such ...
2
votes
1answer
59 views

Trigonometric equation - calculus or algebra

Find number of roots of $x^2 - x\sin x -\cos^2 x =0 $ In original IIT problem it was $\cos x$ instead of $\cos^2 x$ and then it is pretty easy. you have to find number of roots and not prove that ...
1
vote
1answer
82 views

Differentiation question find the normal to the curve

Hi I was struggling to this question, can anyone please help me :P The curve $C$ has equation $2x^2+y^2=18$. Determine the coordinates of the four points on $C$ at which the normal passes through the ...
3
votes
3answers
78 views

Computation of $\int{\sqrt{c^2+b^2t^2}\,\mathrm dt}$.

How to solve $$\int{\sqrt{c^2+b^2t^2}\,\mathrm dt}$$ I tried substituting $t$ with $\sin{x}$ but it doesn't work, since $b^2$ creates a problem.
2
votes
1answer
112 views

This double integral

$$ \int_0^1\int_0^1x^3y^2\sqrt{1+x^2+y^2}\hspace{1mm}dxdy$$ We have to compute this up to 4 decimal places
1
vote
2answers
78 views

Prove that a differentiable function $f$ with $f(x+1)=f(x)$ has at least two points in $[0,1]$ such that $f ' (x) =0$.

Prove that a differentiable function $f$ with $f(x+1)=f(x)$ has at least two points in $[0,1]$ such that $f ' (x) =0$. I used Mean value theorem to obtain one such point in $[0,1]$ , but i am not ...
3
votes
1answer
182 views

Integration of $\int_0^\infty \sqrt x e^{-x^2} dx$

I am having a really hard time trying to figure out what to do with this. I feel like I've tried everything but I'm obviously missing something. Any suggestions? $$\int_0^\infty \sqrt x\ e^{-x^2}\ ...
5
votes
4answers
371 views

Prove that $\int_a^c f(t)dt - \int_c^b f(t)dt = f(c)(a+b-2c) $, for some $c\in(a,b)$

Let $f$ be a continuous on $[a,b]$ then prove that there exist some $c$ that lies in $(a,b)$ such that $$\int_a^cf(t)\,dt - \int_c^b f(t)\,dt = f(c)(a+b-2c) $$ and hence prove that $\int_a^c ...
0
votes
1answer
20 views

A question about Integrability and Uniform Continuity

I got this questions: Prove or disprove by a counterexample the following statements: Let $f:\mathbb{R}\to\mathbb{R}$ be a function that is integrable on every closed interval and let $F(x)=\int_0^x ...
2
votes
4answers
1k views

Why do differentiating and integrating 'work'?

Why do you get a function's (changing) slope when you take its derivative and why do you get the area under the function when you take its integral? What is the easiest reasoning behind this?
0
votes
2answers
127 views

Differential Equation for cooling speed

Suppose that a cup of boiling hot coffee (ie its temperature is $100^oC$) is left in a $20^oC$ room and sits until it has cooled to $60^oC$. The cup of coffee is then taken outside where the temperature ...
1
vote
2answers
99 views

Stuck on finding the value of $\sum_{n=0}^\infty {n(n+1) \over 3^n}$

I am trying to find the value of the series $$ \sum_{n=0}^\infty {n(n+1) \over 3^n} $$ Here's what I have done so far: $$ \sum_{n=0}^\infty {n(n+1) \over 3^n}=\sum_{n=0}^\infty {n^2 \over ...
0
votes
1answer
43 views

Some questions about integrals

I got those questions: Prove or disprove the statement by a counterexample: (1) If $f$ is a continuous function on $[a,b]$, Then there exist $c\in(a,b)$ such that $\int_a^b f(x)\;dx = f(c)(b-a)$. ...
2
votes
2answers
53 views

Limits and Continuity- Sandwich Theorem

$$\lim_{x\to0} \sinh x \tanh \left(\frac1x\right)$$ I am sure that the Sandwich theorem applies here but how do I go about it? Do I separate the terms or just put in one range of values for both?
3
votes
3answers
71 views

Evaluating e using limits

What algebraic operations can I use on the $RHS$ to show $RHS = LHS$ $$e=\lim_{k\to\infty}\left(\frac{2+\sqrt{3+9k^2}}{3k-1}\right)^k$$
1
vote
0answers
31 views

Local extrema given the graph of a function's derivative

I am given a graph of the derivative of a function and answered most of the questions, but am still stuck at answering where the local extrema are. I had a sample question to reference from and it ...
3
votes
0answers
59 views

Does this integral variable change makes sense to you?

I was Reading a book about calculus when I've found this part about variable substitution in integrals: Consider $f$ defined in na interval $I$. Suppose that $x =\phi(u)$ is inversible, and its ...
0
votes
4answers
40 views

Differentiate $4x^3\ln(2x+1)$ with respect to $x$. [closed]

Differentiate $4x^3\ln(2x+1)$ with respect to $x$. Please help me...i can't find any examples in my book! I've got exams...3days left!
1
vote
1answer
73 views

The rate change of the radius of a coil.

Suppose I have a tube of radius $r_0$ that I want to wrap a sheet of length $l$ and thickness $\Delta x$. Assuming the radius changes only when the paper overlaps the where the previous section ...
1
vote
3answers
33 views

Integral with trig functions and substitution [duplicate]

How to integrate: $\int_0^T\cos(T-s)\sin(s)ds$? I was trying to use $\cos(a+b)=\cos a\cos b-\sin a\sin b$ and substitute $\cos(s)=u \Rightarrow \sin(s)ds=du$ but it does't help.
0
votes
0answers
43 views

Multivariable optimisation for pipe insulation

I have two functions: Heat Loss Cost $(C_h)$ $$C_h = 0.23 \times 8.67q$$ where $q$ is the heat loss of the fluid through an insulated pipe (in watts): $$q = \frac{2 \pi L(T_1 - ...
2
votes
2answers
92 views

Why can't we substitute in limits for other limits?

I apologize if this is a stupid question, but I don't really understand this part of limits. I know that for all functions $f(x)$ and $g(x)$ and all numbers $a$, $$\lim_{x \to a} f(x)g(x) = \lim_{x ...
2
votes
3answers
110 views

Implicit differentiation question

Given that $x^n + y^n = 1$, show that $$\frac{d^2y}{dx^2} = -\frac{(n-1)x^{n-2}}{y^{2n-1}}.$$ I found that $\displaystyle nx^{n-1}+ny^{n-1}\frac{dy}{dx} = 0$ so that $\displaystyle ...
2
votes
1answer
57 views

Question about Differentials

I am reading the book "Advanced Calculus" written by Kaplan, and here is what I have: Suppose that $y(x)$ is a differentiable function at $x = x_0$. Then, we can write $y(x_0+\Delta x) = y(x_0) + ...
-1
votes
1answer
49 views

Paths (Calculus). [closed]

Find a path to the origin so that lim (x,y) --> (0,0) x^y is any nonnegative value? I've tried a lot of things but i need help!
3
votes
1answer
73 views

Moving average as ODE

Is it possible to represent or approximate the moving average $m(t) = \frac{1}{w}\int_{t-w}^t x(\tau) d\tau$ of a function $x(t)$ as a set of ordinary differential equations $\dot{y} = \ldots$? I am ...
2
votes
1answer
68 views

$\mathcal{C}^\infty$ strictly monotone function $\lim f(x) = 0$, but $\lim f^\prime(x) \ne 0$

Does there exist a strictly monotone function $f\colon \Bbb R\to\Bbb R$ that is $\mathcal{C}^\infty$, and $\lim\limits_{x\to+\infty} f(x) = 0$, but $$\lim_{x\to+\infty} f^\prime(x) $$ and ...
4
votes
4answers
405 views

How to calculate the sum of the infinite series

Please help me with the sum of the infinite series: $$ \Large \frac{1+\frac{\pi^4}{5!}+\frac{\pi^8}{9!}+\frac{\pi^{12}}{13!}+...} ...
1
vote
2answers
971 views

Finding the volume by Shell method: $y=x^2, y=2-x^2$ about the line x=1

Finding the volume by Shell method: $y=x^2, y=2-x^2$ about the line x = 1 what I get from this after graphing is: $2\pi \int (1-x)(2-2x^2)dx $ which becomes: $2\pi \int (2-2x^2-2x+2x^3)dx $ ...
2
votes
1answer
116 views

Evaluation of tricky integral

I want to evaluate the integral $$\int _ {b} ^ {\infty} \mathrm{d} x \, \frac{e ^ {x ^ {2} / s} (b^2 + 3 x ^ 2) ^ {2}}{x (x^2 + b^2)}$$, where $b$ and $s$ are positive real numbers. I thought of ...
0
votes
1answer
110 views

Monotonic, surjective function

I have tried to answer the following question from my textbook: Suppose the following function $f: [a,b] \rightarrow [c,d]$ with $a,b,c,d\in\mathbb{R}$ and $a<b$, $c<d$. The function is ...
8
votes
6answers
281 views

$\int_{0}^{+\infty} \frac{1-\cos t}{t} e^{-t}dt=?$

I have a question: $$\int_{0}^{\infty} \frac{1-\cos t}{t} e^{-t}dt=\ ?$$ Thanks for your any help. Thanks ahead.
1
vote
3answers
82 views

Taylor expansion at $x=0$ of $\ln(1/(1-x))$

Hello I am having some trouble with the taylor expansion of $$f(x)= \ln \frac1{1-x}$$ Would it be correct to treat the inner part as the following geometric series? ...
0
votes
1answer
63 views

Find the arc length of a curve. Problem integrating

The question is find the arc length of the parabola $y^2 = 4ax$ cut by the line $3y = 8x$ I applied this formula $\int(1+ (dx╱dy)^2) dy $. However by substituting the value of $dx/dy I$ obtain an ...