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

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2
votes
2answers
156 views

Is there an easy way to expand $\arctan(1+x)$ at point $x=0$

I want to expand $\arctan(1+x)$ into Taylor series at the point $x=0$, but I do not want to calculate its higher order derivatives. Is there an easy way to do this, just as we done for $\arctan(x)$ ...
0
votes
1answer
632 views

How to solve the recurrence $T(n)=3T(n/2)+n$

The exercise stated that i have to solve the recurrence using the Recursion-Tree Method. I have already finished the base part, which is $\Theta(n^{\lg3})$ But for the recursive part I'm having ...
2
votes
1answer
170 views

Alternative ways to show $\int_{0}^{\infty}f(x)\, dx = \int_{0}^{1}f^{-1}(y)\, dy$

Let $f$ be a continuous, strictly decreasing, real-valued function such that $\int_{0}^{\infty}f(x)\, dx$ is finite and $f(0) = 1$. In terms of $f^{-1}$, we see that $$\int_{0}^{\infty}f(x)\, dx = ...
2
votes
5answers
87 views

Why am I not getting an infinite limit?

I am trying to solve $\displaystyle\lim_{x\to3^-}\frac{3}{x - 3} - \frac{3}{x} - 9$. Here's what I have tried. $$ \lim_{x\to3^-}\frac{3}{x - 3} - \frac{3}{x} - 9 \\ \lim_{x\to3^-}\frac{3x}{x(x - 3)} ...
8
votes
1answer
795 views

Integral $\int_0^1\sqrt{1-x^4}dx$

I am asked to show $\int_0^1\sqrt{1-x^4}dx=\frac{\{\Gamma(1/4)\}^2}{6\sqrt{2\pi}}$. I know the gamma function is defined by $\Gamma(n)=\int_0^\infty x^{n-1}e^{-x}dx$. I tried to substituted ...
0
votes
2answers
699 views

Maximum Area of a Rectangular Trough

I didn't learn how to do this in my class, and the examples in my book do not apply to this type of problem. To make a rectangular trough, you bend the sides of a 32-inch wide sheet of metal to ...
1
vote
0answers
73 views

Help in differentiating equation

I need to derive Euler-Lagrange equations and natural boundary conditions for a given model. I've worked out and broken down the model into the following $5$ parts: $$J_1 = \int_{\varphi>0}|f(x) − ...
1
vote
1answer
210 views

Advanced integration problem

The solution to Schrodinger's equation are wave functions $\Psi (r,\theta ,\phi )$ of the form, $\Psi (r,\theta ,\phi )= R(r)\Theta(\theta)\Phi(\phi)$ Where, the probability of finding an electron in ...
3
votes
2answers
3k views

Solve the differential equation using Taylor-series expansion

Solve the differential equation using Taylor-series expansion: $$ \frac{dy}{dx} = x + y + xy \\ y (0) = 1 $$ to get value of $y$ at $x = 0.1$ and $x = 0.5$. Use terms through $x^5$.
3
votes
2answers
134 views

Solution of a differential equation that would be a generalized mean?

I am trying to solve this differential equation on which I've been stuck for several days now. $$\frac{d X}{d t}=\frac{\int_{-\infty}^{\infty}\frac{\partial f}{\partial t}\frac{\partial f}{\partial ...
8
votes
1answer
434 views

“Cut” (hexagon-like) Reuleaux triangle area

Let me start by giving the reason my question: as part of a 3D printer I'm building (Rostock), I'm trying to figure out the work area of the printer. The printer consists of 3 arms, each attached at ...
1
vote
1answer
122 views

How to compute $\int\frac{x^7}{\sin(x)} dx$ efficiently?

How to compute $\int\frac{x^7}{\sin(x)} dx$ efficiently ? We need $Polylog$ for this.
2
votes
1answer
152 views

How to calculate $\int_{0}^{+\infty} {\sin {x^2}\mathrm{d}x}$? [duplicate]

Possible Duplicate: Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods? How to calculate $\displaystyle\int_{0}^{+\infty} {\sin {x^2}\mathrm{d}x}$ ?
4
votes
4answers
161 views

Evaluating $\int_0^1 x \sinh (x) \ \mathrm{dx}$

I am looking to evaluate $$\int_0^1 x \sinh (x) \ \mathrm{dx}$$
2
votes
3answers
138 views

Proving the inequality $e^{-k-1} \leq \frac{1}{k^2}$

I'm trying to prove that $e^{-k-1} \leq \frac{1}{k^2}$ for $k > 1$ but I feel I'm missing something (maybe an standard inequality?) Could anyone give me a pointer like "Use such fact"? Context: ...
3
votes
3answers
204 views

Proof the logarithmic identity $\log_{b+c} a+\log_{c-b} a=2\log_{b+c} a\cdot\log_{c-b} a$

Please help me proof $\log_{b+c} a+\log_{c-b} a=2\log_{b+c} a\cdot\log_{c-b} a$, for $a,b,c>0$ and $a^2+b^2=c^2$. Thanks.
1
vote
2answers
105 views

calculus limit inequality

How can I elegantly show that: $(1 + \frac{1}{k})^k \leq 3$ For instance I could use the fact that this is an increasing function and then take $\lim_{ k\to \infty}$ and say that it equals $e$ and ...
0
votes
0answers
48 views

update plot with new data

i have several thousand Cartesian coordinated separated into two groups. The location of the two groups are relative to each other. I have created a new model for one of the groups and I would like to ...
2
votes
3answers
124 views

Calculus and Physics Help!

If a particle's position is given by $x = 4-12t+3t^2$ (where $t$ is in seconds and $x$ is in meters): a) What is the velocity at $t = 1$ s? Ok, so I have an answer: $v = \frac{dx}{dt} = -12 + 6t$ ...
1
vote
1answer
607 views

Pursuit Curve. Dog Chases Rabbit. Calculus 4.

(a) In Example 1.21, assume that $a$ is less than $b$ (so that $k$ is less than $1$) and find $y$ as a function of $x$. How far does the rabbit run before the dog catches him? (b) Assume now that ...
2
votes
1answer
44 views

What are $\partial/\partial f^j$ in Jost's definition of the differential mapping?

Let $M$ be a $d$-manifold and $x_0=(x^1,x^2,\cdots, x^d)\in M$, Jost defines the tangent space at $x_0$ to be \begin{equation}\{x_0\}\times \operatorname{span}\left\{\frac{\partial}{\partial ...
0
votes
3answers
180 views

Question about solving 2nd order linear differential equations

This question might take a while. Bear with me. I'm going to explain how I solve these up to the point where this textbook does something unusual, then I'll explain why it's unusual to me and ask ...
7
votes
2answers
1k views

If a function is continuously differentiable and bounded, is its derivative also bounded?

Two questions: Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuously differentiable and bounded. Is its derivative $f'$ bounded as well? What about in the case of ...
2
votes
4answers
4k views

Find all values of $a$ and $b$ that make the following function differentiable for all values of $x$

Problem Find all values of $a$ and $b$ that make the following function differentiable for all values of $x$: $$ f(x) = \begin{cases} ax + b, x > - 1\\ bx^2 - 3, x \leq -1\\ \end{cases} ...
1
vote
2answers
48 views

Generalization of monotonicity and condition

Consider a function $f: \mathbb{R} \to \mathbb{R}$. As usual, $f$ is non-increasing if $f(x) \geq f(y)$ for all $x < y$. We also have the condition $f'(x) \leq 0$ $\forall x$, provided that $f$ is ...
1
vote
1answer
234 views

Calculus the n'th derivative of $y_{n}=x^{n-1}e^{1/x}$

Without using Mathematical Induction to calculus the n'th derivative of the following function. $y_{n}=x^{n-1}e^{1/x}$ , $n\in\mathbb{N}$ Find : $\frac{d^n}{dx^n}y_n$ I tried to finish the question ...
1
vote
0answers
227 views

Pursuit Curve. Dog chases rabbit.

(a) In example 1.18, assume that $a\lt b$ (so that $k\lt 1$) and find y as a function of $x$. How far does the rabbit run before the dog catches him? (b) Assume now that $a=b$, and find $y$ as a ...
2
votes
0answers
55 views

Can a rate be proportional to a shape?

This question may be a little vague, but it has a point. I woke up this morning with an idea. Let's say I wanted to design a projectile that has a velocity proportional to its 'shape'. When the ...
1
vote
0answers
71 views

A function has at least 4 critical points on $[0,1]\times[0,1]$

Define a smooth function $f(x,y)$ on $\mathbb{R}^2$,which satisfies the following condition:$$f(x,y)=f(x+1,y);\quad f(x,y)=f(x,y+1)$$ (i.e.$f$ can be defined on a torus)then hotw to proof that there ...
1
vote
1answer
146 views

Gaussian approximation

Consider the following function of a variable $\theta\in[0,\frac{\pi}{2}]$ ...
1
vote
1answer
184 views

is $\exp \left( - 1 \over x^2 \right)$ differentiable at $x=0$

is $\exp \left( - 1 \over x^2 \right ) $ differentiate at $x=0$. Wolframlapha says it is. But is it continuous since we have $1 \over x^2$ and $x=0$? Can we really do this $f(0) = \exp \left( - {1 ...
1
vote
2answers
45 views

$\lim_{x\to 0} \big(\lim_{y\to0} f(x,y)\big)$ doesn't exist.

Here is a problem in calculus: Let $$ f(x,y)= \begin{cases} y+x\sin\bigg(\frac{1}{y}\bigg),& y\neq0\\\\0,& y=0 \end{cases}$$ show that : $\lim_{(x,y)\to (0,0)} f(x,y)$ ...
1
vote
1answer
168 views

Calculus 4. Area under arc.

What curve lying above the x-axis has the property that the length of the arc joining any two points on it is proportional to the area under the arc?
0
votes
4answers
151 views

Find $\lim_{x \to 0^{+}} \frac{\ln(x+1)}{x}$ without using L'Hopital's Rule?

Without using L'Hopital's Rule, how does one find $\lim_{x \to 0^{+}} \frac{\ln(x+1)}{x}$? I was hoping to find a way using basic calc I, pre-differentiation knowledge and not knowing the definition ...
1
vote
1answer
159 views

Computing a Differential Form

Apologies in advance, I don't know TeX, so this might look a bit gross... I'm given a 1-form $A=f_1dx_1+...+f_ndx_n$, infinitely differentiable and closed on $R^n$. I want to show that $dg=A$ for ...
1
vote
1answer
291 views

How do you solve differential equations in the form of $ay'' + by' +cy = d$?

I've only recently started learning how to work with differential equations. I'm about to start on linear 2nd order DE, but I wondered - what approach is taken to solve equations of the form: $ay'' ...
-2
votes
2answers
43 views

Explain these functions Part2

I have another problem: We are given the function $$ g(t) = \int_{-\infty}^{\infty} f_X(u)f_X(t-u)du $$ And we have the piecewise function $$ f_X(t) = \begin{cases} 0 & \text{if }t<0\\ 1 ...
2
votes
2answers
70 views

Explain these functions.

We are given two functions of moments of a random variable X: $${\bf E}(X)=\int_{-\infty}^{\infty} tf_X(t)\,dt.$$ $${\bf E}(X^2)=\int_{-\infty}^{\infty} t^2f_X(t)\,dt.$$ Then we are given a ...
6
votes
4answers
614 views

Strictly increasing function

If $f(x)$ is a continuous function on $\mathbb R$, and $|f(-x)|< |f(x)|$ for all $x>0$. Does it imply that $|f(x)|$ is strictly increasing on $(0,\infty)$? I tried to use the definition: let ...
11
votes
1answer
1k views

Is there error in the answer to Spivak's Calculus, problem 5-3(iv)?

I'm puzzled by the answer to a problem for Spivak's Calculus (4E) provided in his Combined Answer Book. Problem 5-3(iv) (p. 108) asks the reader to prove that $\mathop{\lim}\limits_{x \to a} x^{4} ...
2
votes
0answers
49 views

linear DE - Where did I go wrong?

I'm trying to find the general solution for $(x+2)y' = 3-\frac{2y}{x}$ This is what I've done so far: $y'+\frac{2y}{x(x+2)}=\frac{3}{x+2}$ $(\frac{x}{x+2}y)'=\frac{3x}{(x+2)^2}$ ...
7
votes
6answers
2k views

Example where $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective

Can anyone come up with an explicit example of two functions $f$ and $g$ such that: $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective? I tried the following: $$f:\mathbb{R}\rightarrow ...
2
votes
0answers
42 views

How do we use Galois theory to show that an integral has no closed form? [duplicate]

Possible Duplicate: How can you prove that a function has no closed form integral? How do we use Galois theory to show that an integral has no closed form ? I know this is called ...
4
votes
2answers
114 views

Compute $\sum_{k=0}^{\infty} \frac{3}{(3 k)!}$

These days I came across this series and I'm trying to figure out how to compute it $$\sum_{k=0}^{\infty} \frac{3}{(3 k)!}$$ I thought to combine some elementary functions, but it doesn't work. Some ...
5
votes
1answer
258 views

Could Residue theorem be seen as a special case of Stokes' theorem?

Residue theorem in complex analysis is seems like Stokes' theorem in real calculus, so a question arose that could Residue theorem be seen as a special case of Stokes' theorem?
2
votes
2answers
1k views

Proving Green's Theorem for Computing Area

I'm having a hard time understanding where exactly the formula for computing area using Green's theorem comes from. It is typically: \begin{equation} \int_C x\,dy = \int_C -y\,dx = Area ...
0
votes
2answers
295 views

Defining an upper/lower bound in lexicographically ordered C

If I have a lexicographic ordering on $\mathbb{C}$, and I define a subset, $A = \{z \in \mathbb{C} : z = a + bi, a, b \in \mathbb{R}, a < 0\}$. I have an upper bound, say $\alpha = 0 + di$. My ...
0
votes
1answer
100 views

maclaurin series and induction over binomial theorem

How can we show for any $\alpha\in\Bbb R$, the Maclaurin series of the function $p(x) := (1 + x)^\alpha$ is $$1+\alpha x + \frac{\alpha(\alpha-1)}2+\ldots = \sum_{n = 0}^\infty ...
0
votes
2answers
97 views

Integral Problem with Partial Fractions

The problem is: $$\int\frac{7x}{(2x+5)^2}\;.$$ I got and am fairly confident in: $$\frac74 \ln|2x+5| + \frac{35}{8x + 20}$$ However this is apparently not the correct answer. Im not being graded ...
0
votes
0answers
72 views

relative error relation

Let $x$ be a non-null quantity. Let $\hat{x}$ be its approximation. I am trying to find the relation between: $\frac{\left | x-\hat{x} \right |}{\left | x \right |}$ and $ \frac{\left | x-\hat{x} ...