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

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11
votes
2answers
538 views

Showing that $\lim_{n\to\infty}\sum^n_{k=1}\frac{1}{k}-\ln(n)=0.5772\ldots$

How to show that $$\lim_{n\to\infty}\left[\sum^n_{k=1}\frac{1}{k}-\ln(n)\right]=0.5772\ldots$$ No clue at all. Need help! Appreciated!
2
votes
2answers
88 views

For $0 \le x \le 1 , p\gt 1$, prove $1 \over2^{p-1}$ $\le x^p +(1-x)^p \le 1$.

For $0 \le x \le 1 , p\gt 1$, prove $1 \over2^{p-1}$ $\le x^p +(1-x)^p \le 1$.
1
vote
2answers
97 views

What are the number of greatest/least possible maxima for $f(x) = \frac{x^n}{x-1}$?

I have a math contest question that I found in my textbook and I have no idea where to start. Please provide some hints as to how to go about solving this: Consider the function $f(x) = ...
0
votes
1answer
40 views

Elementary inequalities for polynomial functions

Let $$f(t) = \sum_{j= 1}^{n} jb_jt^{j-1}\,\,,\,\;t \in \Omega \subset \mathbb{R}\;\;\; \text{an open bounded set.}$$ Is it true that there exists a constant $D$ such that $|f(t)| \le D(1 + ...
2
votes
2answers
70 views

How to prove limit of $2$ variables?

I know that the following limit exists, and is $1$, but I don't know how to prove it. I know of the definition of a limit "If $0 < (x^2 + y^2)^{1/2} < \delta$ Then $f-L < \epsilon$" but I am ...
3
votes
1answer
56 views

Help me create formula for such sequence

:) I'm web-developer, and currently I'm looking for formula to automate "discount" percent discovery. Our old formula is min(10, 5 + ordersDone), which generally ...
2
votes
3answers
3k views

find local maximum and local minimum

ser my question is related to how to find local maximum and local minimum(relative maximum or relative minimum of given function).as i know for this first we should find derivative of this ...
-3
votes
1answer
105 views

Compute a derivative [closed]

$$\mathcal{F}(x)=\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}$$ What is $\mathcal{F}'(x)$?
3
votes
4answers
110 views

Derivative of $x$ with respect to $xy$

I know that $\dfrac{d(xy)}{dx} = y$ but what does $\dfrac{dx}{d(xy)} =\, ?$ I know this is an odd equation, but it comes from some ugly change of variables and I am stuck with it.
8
votes
2answers
117 views

Making both bounds of integration zero

I came across a question while evaluating the integral: $$ \int_{0}^{\pi}\frac{\cos{t}}{1+9\sin^2{t}}\, dt $$ If you substitute $u=3\sin{t}$, you get: $$ \int_{0}^{0}\frac{1}{3+3u^2}\, du $$ Which ...
1
vote
2answers
74 views

Lipschitz sufficiency condition

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a function and $0 < M < \infty$, prove that $f$ is Lipschitz with constant $M$, if for each $x \in \mathbb{R}$ we have $Df(x) \subset [-M,M]$. ...
3
votes
1answer
638 views

Maclaurin Series Question: $\frac{x}{\sqrt{4+x^2}}$

How do I obtain the Maclaurin series representation of: $$\frac{x}{\sqrt{4+x^2}}$$ I know I should use a well known Maclaurin Series and work my way back to the representation of this problem, but ...
15
votes
2answers
434 views

Interesting log sine integrals $\int_0^{\pi/3} \log^2 \left(2\sin \frac{x}{2} \right)dx= \frac{7\pi^3}{108}$

Show that $$\begin{aligned} \int_0^{\pi/3} \log^2 \left(2\sin \frac{x}{2} \right)dx &= \frac{7\pi^3}{108} \\ \int_0^{\pi/3}x\log^2 \left(2\sin\frac{x}{2} \right)dx &= ...
8
votes
4answers
316 views

What's wrong in this equation? [duplicate]

What's wrong in this equation? $$\underbrace{x+x+x+x+\cdots+x}_{x \textrm{ times}}=x^2$$ now differentiate w.r.t. 'x' both sides $$\underbrace{1+1+1+1+\cdots+1}_{x \textrm{ times}}=2x$$ So, $$x=2x$$ ...
0
votes
1answer
44 views

Is it true that $(\ \forall S \ \ \ \ \int_{S} A dS = 0 )\implies A \equiv 0 $

Is it true that $(\ \forall S \ \ \ \ \int_{S} A dS = 0 )\implies A \equiv 0 $ where $S$ is a surface and $A$ is some function which takes values on $S$? Is there a requirement on the ...
3
votes
3answers
251 views

The distance between a pair of skew diagonals on two adjacent faces of a cube.

Say we're interested in the distance between the diagonals $u=(0,0,0)+(1,0,1)t$ and $v=(0,0,1)+(1,1,0)s$ of a unit cube. The standard formula for the distance between two skew lines $$d=|\mathbf ...
1
vote
1answer
68 views

How to graph this problem?

Let $R$ be the region in the first quadrant bounded by the graph of $y=2\sqrt{x}$, the horizontal line $y = 6$, and the $y$-axis. Region $R$ is the base of a solid. For each $y$, where $0 \le y \leq ...
2
votes
0answers
171 views

Personal project help needed [closed]

This is the first time i have posted on this site. If the people here judge post as brutal as the programming forums. I apologize ahead of time my math background isn't deep enough to make a thesis ...
1
vote
1answer
516 views

Help finding critical numbers.

Find the critical numbers of the function: $$f(x)=x^\frac{4}{5}(x-4)^2$$ Help would be so greatly appreciated :]
-1
votes
1answer
218 views

Maximizing area and finding coordinates of a rectangle constrained by a curve

Consider rectangles located in the first quadrant and inscribed under an increasing curve, with the upper right hand corner vertical line x = 3 and the upper left hand corner on the curve y = ...
0
votes
2answers
195 views

Maximizing the volume of a box.

Suppose that the paper is 45 cm wide and 60 cm long. how much should you cut from the corners to form the box with maximum capacity, and what would be the width, length and volume of the resulting ...
1
vote
2answers
126 views

Finding the length and width of a house that maximize its area

A house is built in the shape of a rectangle, with $3$ rectangular interior sections separated by parallel walls, using fencing. The owner has $900$ feet of fencing, and he wants to enclose the ...
1
vote
4answers
480 views

Finding the point on $f = \sqrt x$ closest to $(1,0)$

Find the point on $y = \sqrt x$ closest to $(1,0)$. I tried setting it $-x=x^{1/2}$ and then solve but I really am confused. please help.
0
votes
1answer
283 views

Sketching a function given certain properties

The following is the last problem on a practice exam and it is giving me trouble. Given the following information about a continuous function, $f$, how do I sketch a possible graph of $f$: ...
0
votes
1answer
46 views

Help with optimization

The rectangle located in the first quadrant and is basically inscribed under a decreasing curve.The lower left hand corner is at the origin and the upper right hand corner on the curve. the equation ...
0
votes
1answer
51 views

Gravel off of a conveyor belt problem.

Gravel is being dumped from a conveyor belt at a rate of $30\, \mathrm{ft^3/min}$ . It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast ...
0
votes
2answers
61 views

Find $T$, $N$, and $k$ for vector

$$ x(t) = ( t , \sin(4t), \cos(4t) )$$ I am unsure on what they are referring to here. would $T$ be the tangent and therefore: $$ x'(t) = ( 1 , 4\cos(4t) , 4\sin(4t) ) ?$$ Thanks :) Also, ...
2
votes
2answers
411 views

Water leaking out of a conical tank and water pumped in

Water is leaking out of an inverted conical tank at a rate of 0.0068 m^3/min . At the same time water is being pumped into the tank at a constant rate. The tank has height 14 meters and the diameter ...
1
vote
1answer
82 views

Please help solve this calculus calculation

If I was trying to design a printed billboard using a minimal area of plywood. The printed area must be $2000$ sq ft.Here are the margins. side margins $10$ ft top margin $8$ ft bottom margin ...
1
vote
1answer
49 views

Extreme values and monotonicity

So, I am given that $$f'(x)=x^2(x-2)(x+3)$$ and asked to find where the original function $f$ is increasing and decreasing. The question also asks for the $x$ coordinates of all extreme points and ...
-1
votes
1answer
76 views

If $\lim_{n\to\infty} x_n = \infty$ how does the $\lim_{n\to\infty} = a$ with $ a \in R$?

Let $g$ be a real continuous function over $[0,\infty)$. Let $H$ be the set of all real numbers $h$ such that can be expressed as $\lim_{n\to\infty} g(x_n) = h$ for some sequence $(x_n) \subset [0, ...
0
votes
3answers
123 views

For which $a$ does $\lim_{x\to a}f(x) = \lfloor x \rfloor$ exist?

How do I go about finding for which values of $a$ the $\lim_{x\to a}f(x) = \lfloor x \rfloor$ doesn't exist? I've tried using $x-1 \lt \lfloor x \rfloor \le x$, but it seems I'm lacking some ...
6
votes
2answers
158 views

$\lim_{n \rightarrow \infty} \sqrt{1+\left(\frac1{2n}\right)^n}$

Find the limit of the sequence as it approches $\infty$ $$\sqrt{1+\left(\frac1{2n}\right)^n}$$ I made a table of the values of the sequence and the values approach 1, so why is the limit $e^{1/4}$? ...
1
vote
1answer
37 views

Integration in $\mathbb{R}^3$

I'm reading a text and find the following integral $\int_{S_{r}}f(X,Y,Z)d{S_{r}}$ where $f$ a function of three variables and $S_{r}=\partial B_{r}(x_{1},y_{1},z_{1})$ is the sphere of radius $r$. ...
0
votes
6answers
97 views

Integration issue

I am trying to solve $\int^{+\infty}_{-\infty}\frac{1}{x}dx $. I read that it is a contour integral along the semi-circle of large radius in the lower complex plane. First, is there any justification ...
4
votes
2answers
242 views

Limit of $f(x) = \lfloor x \rfloor + \sqrt{x - \lfloor x \rfloor}$

I'm doing Spivak's Calculus book and one of the exercises from 5th chapter says 4. For each of the functions [...], decide for which numbers a the limit $lim_{x \to a}\ f(x)$ exists. i) $f(x) ...
1
vote
2answers
67 views

4-th derivative of $(1+x+x^2) / (1-x+x^2) $ using Taylor polynomial for $1/(1-x)$

Using $n$-th Taylor polynomial for $f_1(x)=\frac{1}{1-x}$ with center in $0$, find $4$-th derivative of $f_2(x)=\frac{1+x+x^2}{1-x+x^2}$ in the point $0$ without calculating it's $1$,$2$ or $3$ ...
0
votes
5answers
129 views

Solve: $\lim_{n \rightarrow \infty}\left(\frac{1^3+3^3+5^3+7^3+…+\left(2n-1\right)^3}{\left(1+3+5+7…+\left(2n-1\right)\right)^2} \right)$

Solve the following limit $$\lim_{n \rightarrow \infty}\left(\frac{1^3+3^3+5^3+7^3+...+\left(2n-1\right)^3}{\left(1+3+5+7+...+\left(2n-1\right)\right)^2} \right)$$
1
vote
2answers
94 views

How to transform the coefficients of a summation of cosine functions? (cos(kw)=?)

I know the title is confusing. Here is the problem and thanks in advance. $f(w)= r0 + r1*\cos{w} + r2*\cos(2w) + ... + rn*\cos(nw)$, $r0,r1,...,rn$ is given. Since $\cos(nw)$ can be transformed to a ...
0
votes
1answer
108 views

The shortest distance between a point and a figure

I noted subjects speaking about point and surface but my question is more global. Let a set of figures (polygons, cercles,...) in a plan, does exist a formula for computing the shortest distance ...
1
vote
1answer
286 views

How to find the delta of a graph with a limit that approaches infinity?

I thought that in order to find $\delta$, given $\epsilon$, you would need to first subtract epsilon from the limit. How would you do that if the limit is infinity? The exact problem is $(2x+4)^{-1}$ ...
0
votes
3answers
57 views

Integrate $f(x) = g(x)\cdot{}h(x)$

I need to find the anti-derivative of $f(x)=g(x)\cdot{}h(x)$. How do I do that?
3
votes
3answers
76 views

differentiate f(x) using L'hopital and other problem

Evaluate: $\ \ \ \ \lim_{x\to1}(2-x)^{\tan(\frac{\pi}{2}x)}$ Show that the inequality holds: $\ \ \ x^\alpha\leq \alpha x + (1-\alpha)\ \ \ (x\geq0, \,0<\alpha <1)$ Please help me with ...
2
votes
4answers
181 views

What function has derivative $\log(x)$?

I have the derivative of a function $f(x)$: $f'(x) = \log(x)$, where $\log(x)$ is the natural logarithm. What's the original function $f(x)$ and what is that calculation called in English?
2
votes
3answers
114 views

Prove that for any real numbers $a,b$ we have $\lvert \arctan a−\arctan b\rvert\leq \lvert a−b\rvert$.

Prove that for any real numbers $a,b$ we have $\lvert \arctan a−\arctan b\rvert\leq \lvert a−b\rvert$. This should have an application of the mean value theorem.
0
votes
2answers
247 views

Hard contour integral problem

With a difficult integral is it possible to split it up and then apply the Cauchy principal value theorem to the first integral to turn it to $0$, then to use contour integration on the other. It ...
3
votes
3answers
322 views

Prove that if $a>b>0$, then $\frac{a-b}{a}>\ln a - \ln b >\frac{a-b}{b}$.

Prove that if $a>b>0$, then $$ \frac{a-b}{a}>\ln a - \ln b >\frac{a-b}{b}. $$ This might/should be an application of the Mean Value Theorem
0
votes
1answer
170 views

Use the mean value theorem to prove that

Use the Mean Value Theorem to prove that $$\frac{\pi}{6} + \frac{(2x-1)}{\sqrt3} < \arcsin x < \frac{\pi}{6} + \frac{(2x-1)}{2\sqrt{1-x^2}} \qquad \text{for} \ \frac{1}{2}\leq x< 1.$$ How ...
1
vote
0answers
44 views

Taylor expansion of an integral in spherical co-ordinates

I've some difficulty deriving this equation from jackson electrodynamics (The equation after 1.30) $\nabla^2 \Phi_a\left({\textbf{x}}\right)=-\frac{1}{\epsilon_0}\int_{0}^{R} ...
1
vote
3answers
148 views

How do I Use L'Hopitals rule to find all values of K and M such that $\lim_{x \to 0}\left (K + \dfrac{\cos(mx)}{x^2}\right) = -4$

Use L'Hopitals rule to find all values of $K$ and $m$ such that $$ \lim_{x \to 0} \dfrac{K+\cos(mx)}{x^2} = -4 $$ I have no idea how to solve this please help?