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

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2
votes
1answer
51 views

Convergence of a series that looks similar to $e^x$

Suppose I have some $\epsilon > 0$ and some constant $c > 0$. Does the series $$ \sum_{n=1}^{\infty} \frac{c^{n^{\epsilon}} }{[n^{\epsilon}]!}, $$ where $[r]$ is the integral part of a real ...
1
vote
1answer
29 views

General Solution to a Differential EQ with complex eigenvalues.

I need a little explanation here the general solution is $$x(t)=c_1u(t)+c_2v(t)$$ where $u(t)=e^{\lambda t}(\textbf{a} \cos \mu t-\textbf{b} \sin \mu t$ and $v(t)=e^{\lambda t}(\textbf{a} \sin \mu t ...
0
votes
1answer
26 views

Calculus implicit differential

So i am here with my question , Question asks $(x-y)^2 = x + y - 1$. $2 (x-y) (1- y') = 1 + y'$ I am a little confused when it comes to right here, can anyone clarify a little, I know the step that ...
1
vote
1answer
80 views

A Cubic Equation

$2x^3+ax^2+bx+4=0$, $(a,b \in R^+)$ has three real roots. Then : A. $a\geqslant 4.2^{\frac 1 3}$ B. $a\geqslant 1.2^{\frac 1 3}$ C. $a\geqslant 6.2^{\frac 1 3}$ D. $a\geqslant 2.2^{\frac 1 3}$ ...
0
votes
1answer
29 views

Progressive Linear functions

I have a problem and I'm not sure how to calculate it or write it. It uses Linear functon as: y = am + b So the problem is that evrey time the unknown-m increases by 1, The unknown-a will add to ...
7
votes
2answers
617 views

A question about limits at infinity

I got this question: Let $f, g$ be functions defined on the interval $[0,\infty)$ and let $L \in \mathbb{R}$ be a real number such that $$\lim_{x \to \infty}\left(f(x)\cdot g(x)\right)= L.$$ (1) If ...
0
votes
1answer
28 views

When finding the frequencies of normal modes, can you have a negative frequency?

Do you simply just consider the positive solutions? I tried a google search but didn't find anything quickly. The work I am studying is Lagrangian systems.
1
vote
1answer
60 views

Finding angle of sector which forms a cone

To find the angle where it is in rad, am i right to say that (10)(angle in rad)=(2pi)*(radius of cone)
1
vote
2answers
78 views

How do I prove the value of $\lim_{x \to 3} {{x^3 - 27} \over {x - 3}}$ is 27?

Find the value of $\lim_{x \to 3} {{x^3 - 27} \over {x - 3}}$ and use an $\epsilon-\delta$ proof to show your answer is correct. This is a review problem, I always forget how to do this. All I need ...
0
votes
1answer
58 views

Optimization question about box

An open box is to be made from a rectangular 30cm x 18cm cardboard piece by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box ...
2
votes
2answers
57 views

Reducing to first-order differential systems

Hello, in order to do this, I am aware I need to substitute $x''$ for say $a'$ and $y''$ for say $b'$, but I'm unsure of how this will yield 4 equations? Furthermore, in terms of wording of the ...
0
votes
1answer
29 views

Tangent to a function in 2 points

I have a function that looks like this- the function is a parametric function $x=x(t) y=y(t)$ And I need to find the line tangent to it. Would it be right to do $(y(t2)-y(t1))/(x(t2)-x(t1))=a$ ...
3
votes
5answers
2k views

What is the difference between Cauchy and convergent sequence?

I am really confused. I will appreciate if somebody can help me to define the difference between Cauchy and convergent sequence. Many thanks.
4
votes
3answers
397 views

A definite integral­

$$\int_{-\frac\pi2}^{\frac\pi2}\frac{\ln(1+b\sin x)}{\sin x} dx \\|b|<1$$ I tried putting $-x$ using properties of definite integral, but that doesn't really help. ...
0
votes
0answers
39 views

Tangent line to a vector function

Find the tangent line to the curve of intersection of $S_{1}: x^{2} + y^{2} = 2$ and $S_{2}: z = -7x + 5y - 3$, at the point $P(1,1-5)$. I'm having trouble with this exercise at the moment. I have ...
1
vote
1answer
142 views

consequences of Schwarz lemma of holomorphic functions of unit disk

Let $D$ be the open unit disk centered at $0$ in the complex plane. Let $f:D\longrightarrow D$ be holomorphic such that $f(0)=0$. Use the Schwarz lemma to prove that $|f(z)+f(-z)|\leq 2|z|^2$ for any ...
1
vote
3answers
29 views

Using U Substitution on 1/(3x)

Say I want to find the indefinite integral of 1/(3x). I can pull out the (1/3) so now I just have 1/x to integrate and I get (1/3)(lnx) as my final answer. This is the correct answer. But now I'm ...
0
votes
1answer
642 views

Is sin 1/x Riemann integrable

Is the function $\sin \frac1x$ Riemann integrable on an interval containing $0$?
1
vote
2answers
489 views

conformal map/Mobius transformation from annulus to $\mathbb{C}\setminus \overline{D(0,1)}$

Does there exist a conformal bijection/Mobius transformation from the open unit disk to the whole complex plane? Does there exist a conformal bijection/Mobius transformation from the annulus $\{z\in ...
0
votes
2answers
77 views

Show that $\lim_{n \to \infty} a_n b_n = ab$?

alright so I have to assume that $\lim_{n\to \infty} a_n=a$ and $\lim_{n\to \infty} b_n=b$ And then I have to show that $\lim_{n\to\infty} a_n \ b_n=ab$ by using the definition of convergence of ...
0
votes
1answer
37 views

Can a value $c$ satisfying the Mean Value Theorem be on the interval $[a,b]$?

So the question asks to find the values of c that satisfy the conclusion of the Mean Value Theorem on the closed interval $[0,3]$. after doing the $\frac{f(b)-f(a)}{b-a} $, I got $0$, and set the ...
0
votes
1answer
16 views

Reduction formula for the tangent function squared

Find the reduction formula : $\tan^{2n}x dx, n ∋ ℤ+$ So I was thinking that I would break it into three parts then use integration by parts, but then there would be two terms with nth power still. If ...
0
votes
2answers
63 views

Finding concavity:$ f'(x)=2\cos x+\sin2x$

"Suppose there is a function $f$ defined on $[0,2\pi]$ where the first derivative, $f'(x)=2\cos x+\sin2x$" a. State intervals where $f$ is concave up. b. Find inflection pts. So far, I found the ...
0
votes
2answers
45 views

How does $(x^3-1)$ factor out? (With application to calculus)

So the problem is: $f(x)=2x^5-5x^2+1$ Find critical numbers determine intervals where function increases/decreases I know you have to take the first derivative which is $10x^4-10x$ However, when ...
0
votes
2answers
58 views

LaPlace transform of the delta function

I am having difficulty taking the laplace transform of $$\delta(t-2\pi)\cos(t)$$ I know that if we have the delta function it is just $e^{-cs}$ but what about the product?
0
votes
1answer
55 views

Finding local extrema using both single derivative test and double derivative test

I get these 2 so confused so I am trying to use both of them with the same problem. I am able to solve it with single derivative but not double: $f(x) = x^5 - 5x + 3$ single Derivative calculations ...
1
vote
2answers
43 views

The MacLaurin series of $e^{\frac{x^2}{2}}$

Could someone please explain to me how you derive the MacLaurin series for $e^{ \frac{x^2}{2}}$? I understand how it is derived from the MacLaurin series for $e^x$ where it is $e^x = 1 + x + ...
0
votes
0answers
50 views

Heat equation form $u_t = u_{xx} + f(x)$.

What is the Heat equation form $u_t = u_{xx} + f(x)$.What is this $f(x)$ means. In the question here it is used. (That is it hasn't asked to change $u_t = \alpha^2 u_{xx}$ as $ u_t = u_{xx} + f(x)$) ...
1
vote
1answer
34 views

Finding the General Solution to the system of equation

Find the General solution of $\textbf{x}^{'}=\begin{pmatrix} 2&2+i\\-1&-1-i\\ \end{pmatrix}\textbf{x}$ I started out by finding the eigenvalues. ...
0
votes
2answers
48 views

Using the definition of a limit

Suppose I want to calculate $\displaystyle\lim_{x \rightarrow 3} x^2$ using the definition of the limit. i.e $\forall \epsilon > 0, \exists \delta > 0$ s.t. when $|f(x) - f(a)| < \epsilon, ...
3
votes
2answers
69 views

Integration Problem with a Trig substitution

Okay I am a little stuck on this problem. $$\int \tan^5(x)\sqrt{\sec(x)} \; dx$$ What should be my first step for a u sub or a trig sub? I have tried to use $u=\sec(x)$ and then $u=\tan(x)$, but I ...
0
votes
2answers
93 views

Power Series Representation at a given value a

Find a power series representation for the function centered at a given value of a and determine the radius of convergence. $$f(x)=\frac{1}{(1-x)^3}; a=0 $$ How would i begin with this type of ...
0
votes
1answer
46 views

Borsuk–Ulam theorem for $n=2$

How one can intuitively prove the following statement: At any moment there is always a pair of antipodal points on the Earth's surface with equal temperatures. What about a rigorous proof?
5
votes
3answers
226 views

$\sin^2(x)+\cos^2(x) = 1$ using power series

In an example I had to prove that $\sin^2(x)+\cos^2(x)=1$ which is fairly easy using the unit circle. My teacher then asked me to show the same thing using the following power ...
0
votes
1answer
44 views

Length of a very basic exponential curve

I have the beginning points (0,1) and end points (180, 141.732) of a curve. The function I am currently using is f(x) = Ae^kx. However, when deriving the original function, I end up with 0 (from ...
14
votes
5answers
643 views

Putnam Exam Integral

I am trying to evaluate$$ \lim_{n\to \infty} \int_0^1 \int_0^1...\int_0^1 \cos^2\big(\frac{\pi}{2n}(x_1+x_2+...x_n)\big)dx_1 dx_2...dx_n. $$ This is from an old Putnam mathematics competition. Either ...
0
votes
2answers
54 views

Prove that these two integrals are equal [closed]

Prove that: $$\int\limits_{ - \infty }^\mu {{e^{{{ - 1} \over 2} \cdot {{\left( {{{x - \mu } \over \sigma }} \right)}^2}}}} dx = \int\limits_\mu^\infty {{e^{{{ - 1} \over 2} \cdot {{\left( {{{x - \mu ...
3
votes
3answers
209 views

Approximate $|x|$ with a smooth function

I am trying to get the derivative of $|x|$, and I want that derivative function, say $g(x)$, to be a function of x. So it really needs the |x| to be smooth (ex. $x^2$); I am wondering what is the ...
1
vote
1answer
37 views

Derivatives of conditionally defined functions

I was asked in an exercise to show on what intervals of $\mathbb{R}$ a function $f(x)$ is solution to certain differential equations. The function is defined as: $$f(x) = \left\{ \begin{array}{rl} ...
1
vote
1answer
52 views

How to integrate reciprocal of square root?

How do you integrate the following? $$ \int \frac{1}{\sqrt{1-2x+x^2}} $$ Just a hint should suffice.. thanks!
2
votes
2answers
60 views

Integrate $\int\frac{5x-7}{x^2-3x+2}$

I want to integrate $\int\frac{5x-7}{x^2-3x+2}$ but my result differs from the one on Wolframalpha http://www.wolframalpha.com/input/?i=integrate+%285x-7%29%2F%28x%5E2-3x%2B2%29 I did the following ...
2
votes
1answer
46 views

Find the supremum $\sup_{x>a} |(1+\frac{x}{n})^n-e^x| $

I want to find the supremum: $\sup_{x>a} |(1+\frac{x}{n})^n-e^x| $,where $a$ is a random number.. $e^x-(1+\frac{x}{n})^n>0$ at the interval $(a,+\infty)$..But why is it like that? Wouldn't this ...
0
votes
2answers
45 views

Find the function $h(x) = g(2g^{-1}(x))$

Show that the function $g(x) = x^4 + x^3 + 1$ is one-to-one on [0, 2]. In addition, for the function $h(x) = g(2g^{-1}(x))$, find h′(3). For the first part, I manage to prove that g(x) is increasing ...
2
votes
1answer
43 views

I need help showing this inequality

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a twice differentiable function such that $f'>0$, $f''<0$, and $f(0)=0$. I need to show, that for every $x>0$: $\frac{f(x)}{f'(x)}>x$ Thanks ...
3
votes
2answers
89 views

Series Word Problem

So the questions is: A ball dropped from a height of 13 feet begins to bounce. Each time it strikes the ground, it returns to $\frac 34$ of its previous height. What is the total distance traveled by ...
0
votes
4answers
114 views

Limit: $\lim_{x\to 0} \frac {x-\sin x}{x-\tan x}$

Here's the question: $$\lim_{x\to 0} \frac {x-\sin x}{x-\tan x}$$ I've used l'Hospitals to get $$\lim_{x\to 0} \frac {1-\cos x}{1-\sec^2x}$$ I then tried to use it again, resulting in $= ...
2
votes
1answer
85 views

Alternating series, does the series converge or diverge?

Does the series $\sum _{n=0}^{\infty }{\frac { \left( -1 \right) ^{n}}{\sqrt [3]{n+1}}}$ converge or diverge? The series can be written as $\sum _{n=1}^{\infty }{\frac { \left( -1 \right) ...
1
vote
1answer
50 views

Is the function $\frac{1}{\sqrt{|x_1|}}$integrable on the unit sphere $S^{n-1}\subset\mathbb{R}^n$?

Is the function $\frac{1}{\sqrt{|x_1|}}$integrable on the unit sphere $S^{n-1}\subset\mathbb{R}^n$? That is, is the integral $$\int_{S^{n-1}}\frac{1}{\sqrt{|x_1|}}d\sigma(x)$$ finite? Where $\sigma$ ...
6
votes
1answer
55 views

Alternating Series and Convergence

The question is: Approximate the value of the series within an error of at most $10^{-4}$. $$ \sum_{n=1}^\infty \frac{(-1)^{(n+1)}}{(n+79)(n+73)} $$ According to $$|S_N - S| ≤ a_{N+1}$$ what is ...
1
vote
1answer
27 views

How do I do Linearization at a point that lies on a curve?

I keep applying the formula to the info given but I keep getting lost/weird answers. Can someone please help me? I know $L(x)=f(a)+f'(a)(x-a)$ question Y(x) satisfies $x^2y^2 + xy = 6$. Point (x,y) ...