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

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3
votes
1answer
63 views

Problem on limits

I have the problem with the following limit: $$\lim_{n \to ∞} \frac{\sqrt{1}+\sqrt{2}+\sqrt{3}+\sqrt{4}+...+\sqrt{n}}{n\sqrt{n}}$$
0
votes
1answer
52 views

Logarithm inequality for specific range

I need to show that: $$ \ln(1+x)\left(\ln\left(\frac{1+x}{1-x}\right)+1\right)+\ln(1-x)\ge 0, $$ for $0\le x\le 2/3$. Thanks
-1
votes
3answers
61 views

Solutions of $2x\sqrt{1-x^2} \geq 1$ and $2x\sqrt{1-x^2} \leq 1$

I have to solve the following inequalities, but I'm a bit confused by the presence of the square root. What is the correct way to get rid of it and solve the problem? $$2x\sqrt{1-x^2} \geq 1$$ and ...
0
votes
1answer
47 views

Find for which value of a parameter $k$ a function is injective, surjective, or bijective

Let $$f(x)=\ln(x^4-x^2+1)-2\ln(x^2+1)+2\sqrt3\arctan(\frac{2x^2-1}{\sqrt3}) + kx -k\ln x;$$ $$f_{*}(x)=\ln(x^4-x^2+1)-2\ln(x^2+1)+2\sqrt3\arctan(\frac{2x^2-1}{\sqrt3}) + kx;$$ ...
5
votes
4answers
168 views

Convergence of $\sum^\infty_{n=1}\arctan(\frac 1 {\sqrt n}) $ and how to approach trigonometric expressions in sums

Does $$\sum^\infty_{n=1}\arctan\left(\frac 1 {\sqrt n}\right)$$ converge? The series probably diverges and I should probably use the comparison test, but I don't know what to use. Note: no integral ...
0
votes
1answer
52 views

Generating a number with an upper limit

We can generate an arbitrary real number by continuously adding randomly chosen digits after the decimal point. This number will have a certain value because we are only multiplying the (negative) ...
0
votes
1answer
20 views

How do I see that the complex ODE $z^{''} - 2iz = 0$ has a $2$-dimensional solution space?

How do I see that the complex ODE $z^{''} - 2iz = 0$ defined on $\mathbb R \times \mathbb C$ has a $2$-dimensional solution space ? I've already found the family of solutions given by $\{ ae^{t+it} + ...
0
votes
1answer
36 views

Determine whether a function possess the intermediate value property

Most of the books or online lecture notes that i have looked up always conclude whether a function possess the I.V.P without further explanation . But how do we know that? Take sin(1\x) for an ...
1
vote
2answers
51 views

finding the sum function of $\sum_ {n=1}^{\infty} \frac{n-2}{(n-1)!} z^{n+1}$

finding the sum function of $\sum_ {n=1}^{\infty} \frac{n-2}{(n-1)!} z^{n+1}$ So far i've substituted n-1 for m which gives me the following form: $\sum_ {m=0}^{\infty} \frac{m-1}{(m)!} z^{m+2}$. ...
0
votes
1answer
44 views

How to prove that multivariable function has a minima?

Consider for example the function $f:\mathbb{R}^3\rightarrow \mathbb{R}$ as: $$f\left(x,y,z\right)\:=\:z^2+3x^2+y^2-2xy+14$$ I need to prove the $f$ has a minima and to calculate it. So far i went ...
0
votes
3answers
44 views

Differentiability of the function$f(x)= x^2 \sin(1/x)$, $f(0)=0$ at the origin

It is easy to verify by definition that the function $f(x)=x^2 \sin(1/x), x\neq0,f(0)=0 $is differentiable at the origin ,that is, $f'(0)=0$.But by the formula we can not calculate ...
1
vote
1answer
199 views

Suppose $a_n$ converges and $b_n$ divergent. What can you tell about $a_n+b_n$? [duplicate]

Suppose $a_n$ converges and $b_n$ divergent. What can you tell about $(a_n+b_n)$? I know that if $(x_n)$ and $(y_n)$ converge, then both $(x_n+y_n)$ and $(x_n−y_n)$ converge also. But can't ...
1
vote
3answers
62 views

Determine if the given differential equation is separable

I tried to solve this equation $dy/dx-\sin(x+y)=0$ but I don't know if is separable or not. I did alternate form $dy/dx = (\sin x\cos y)+(\cos x\sin y)$. Then i used trigonometry identities and ...
1
vote
0answers
42 views

Integral over a product

How the following integral can be computed: $I = \int_0^1 (x-a_1)^{b_1}(x-a_2)^{b_2}...(x-a_n)^{b_n} dx$? Here, $a_i,b_i$ are real numbers and $n$ is a natural number. Are there any techniques for ...
0
votes
2answers
49 views

Epsilon-Delta Proof of $\lim_{x\to 0}\sqrt[3]{x}=0$

Just need to check if $$\lim_{x\to 0}\sqrt[3]{x}=0$$ using the epsilon-delta definition of limits is something like $\delta < \epsilon/3$. Thank you
1
vote
2answers
78 views

Find $\lim\limits_{n \to \infty} \frac{\log(1+2^n)}{\log(1+3^n)}$

How to calculate this limit? $$\lim\limits_{n \to \infty} \frac{\log(1+2^n)}{\log(1+3^n)}$$
1
vote
4answers
2k views

Help finding turning points to plot quartic and cubic functions

My teacher assigned us some graphing homework. I know how to graph all of them manually all except these two $f_1(x)=x^4-x^3-4x^2$ $f_2(x)=\frac{1}{2}x^3+2x^2-8x-2$ When i asked him how he said it ...
0
votes
2answers
40 views

Delta notation in Thermodynamics

Assume we want to calculate the finite Enthalpy change for a process. $$H=U+pV$$ $$\Delta H=\Delta U + \Delta(pV) $$ Everything clear so far, but I do not understand how my teacher consecutively ...
0
votes
1answer
56 views

Differential equation $(\cos x - \sin x) \dfrac{dy}{dx}+(\cos x + \sin x)y=(\cos x + \sin x)$ given that $y=-1$ when $x=\dfrac{\pi}{2}$

Differential equation $(\cos x - \sin x) \dfrac{dy}{dx}+(\cos x + \sin x)y=(\cos x + \sin x)$ given that $y=-1$ when $x=\dfrac{\pi}{2}$ I have done some rearranging and worked out the integrating ...
0
votes
1answer
50 views

Volume of a parallelepiped depending on $\lambda$

I've got a relatively simple calculus problem here but it has an unknown variable that I am not sure how to deal with. Find the volume of the parallelepiped depending on $\lambda$ with; $a = ...
1
vote
0answers
79 views

Integrating over a symmetric-group function (elements being permutations)

I would like to integrate a permutation of a function. Namely I have the following: $\sum_{\sigma, \sigma'\in S_{n+1}}\int_{-A}^A dz_1dz_2 ... dz_{n+1} ...
0
votes
1answer
116 views

Rate of Heat Flow Across the Surface of a Sphere

This is cal 3-4 and I assume I need a surface integral of some sort. But, all this time I have been working with field vectors and fluid. Therefore, I don't even know what formula to use.
1
vote
1answer
39 views

How to solve this differential equation??

Good morning (or evening) to everybody. I would like to know how may I work to solve this differential equation: $$\dot{R}^2 = \alpha\dot{r}^2 - \beta\dot{r}^4$$ Where $R$ is $R(t)$ and $r$ is also ...
1
vote
1answer
62 views

Is this function continuous? Polar coordinates

Is the function $f:\mathbb R^2\to \mathbb R^2$ given in polar coordinates by $f(r,\theta)=(1,\theta)$ continuous? How would one prove it? My guess would be yes, since geometricly it simply change ...
0
votes
1answer
17 views

Finding critical points of a multivariable

I'm a bit confused on the procedures for finding the zeros of the partial derivatives of these kind of functions. If you could correct what I am doing wrong it would be much appreciated. $$ ...
0
votes
0answers
152 views

Proving that a rational function with two variables is continuous

for some reason i'm struggling with this very basic propsition: Let $f:\mathbb R \times \mathbb R \to \mathbb R$ be a rational function. for that matter we can every assume $f(x,y)=\frac{1}{p(x,y)}$ ...
0
votes
2answers
37 views

Prove that there is exactly one pair of reals $a, b$ such that $x^{2} = x\sin x + \cos x$ for $x=a,b$

Some observations. Let $$f(x) := x^{2} - x\sin x - \cos x.$$ Then $$f'(x) = 2x - x\cos x.$$ On setting $f'(x) := 0,$ I obtain $$x = 0$$ (because either $x=0$ or $x \neq 0$ and if $x \neq 0$ then ...
0
votes
1answer
99 views

Difference between Double and triple integral?

Hi all I am going to be starting multivariable calc and I am trying to read up but I can't seem to quite grasp this exactly yet. What are the differences between double and triple integrals? I am ...
0
votes
2answers
56 views

Volume of a Solid, $x^2 - y^2 = a^2$

The question is Find the volume of a solid rotated around the y axis, bounded by the given curves: $$x^2 - y^2 = a^2$$ $$x = a + h$$ I am lost by the number of variables in this question ...
0
votes
1answer
100 views

Find $\lim_\limits{n\to \infty}\underbrace{{\sin\sin\cdots\sin(x)}}_{n\text{ times}}$. Why am I wrong? [duplicate]

Find $\lim_\limits{n\to \infty}\underbrace{{\sin\sin\cdots\sin(x)}}_{n\text{ times}}$. It is known that after the first sine, we get something in $[-1,1]$. If it is $0$ then it is constant and ...
0
votes
1answer
33 views

Taking the derivative of a vector and non-vector that represent the same thing?

In a cylindrical coordinate system, the following formula is used derived for theta acceleration by taking the derivative of the position function $r=r \vec{u_r}$, ($\vec{u_r}$ is a unit vector) ...
59
votes
2answers
5k views

Compute $ \lim\limits_{n \to \infty }\sin \sin \dots\sin n$

I need your help with evaluating this limit: $$ \lim_{n \to \infty }\underbrace{\sin \sin \dots\sin}_{\text{$n$ compositions}}\,n,$$ i.e. we apply the $\sin$ function $n$ times. Thank you.
0
votes
0answers
73 views

Question on Partial Derivative with Constraints

I am having a little trouble understanding partial derivatives with constrained variables. I believe I am starting to get it but there are a few things I am unsure about. Here is the problem I am ...
1
vote
1answer
49 views

Using the limit of $ (1+1/n)^n$ to find the limit of $((n^2+2)/(n^2+1)) ^ {3n^2+1/n}$

Use: $ \lim_{n \to \infty} (1+1/n)^n = e$ To find: $ \lim_{n \to \infty} (\frac{n^2+2}{n^2+1}) ^ {3n^2+1/n}$ Any help is appreciated. Not looking for the complete answer since it's looked down upon ...
0
votes
1answer
39 views

Find potential function for the vector field $\vec F(x)=\left \| x \right \|^px$

Define a vector field $\vec F$ in $\Bbb{R}^n \setminus 0$ by $\vec F(x)=\left \| x \right \|^px$, where $p$ is a real constant. How to find a potential function for $\vec F$? Shall I just directly ...
0
votes
1answer
185 views

Show that the path followed by the boat is the graph of the function.

The problem I am trying to figure out is as follows: A man initially standing at the point O walks along a pier pulling a rowboat by a rope of length L. The man keeps the rope straight and taut. The ...
1
vote
3answers
56 views

For what values of $k$ is $g(x)=x^3+kx^2+x$ one-to-one?

I need to find for what values of $k$ $g(x)=x^3+kx^2+x$ is one-to-one. I tried finding for what values it is strictly increasing and got the derivative to be $3x^2+2kx+1>0$, but I'm not really sure ...
2
votes
0answers
238 views

Evaluating $\int e^{\Gamma(x)} dx $ and $\int \pi^{\Gamma(x)} dx $

I don't know how to solve these integrals: $$I_1 =\int e^{\Gamma(x)} dx $$ $$I_2 =\int \pi^{\Gamma(x)} dx $$ As a tenth grader I have no idea what the solutions could be. How would one go about ...
0
votes
2answers
82 views

When are there not min/max values of a function subject to a constraint?

How do I know if there are no extreme values of a function subject to a constraint? For example, if $f(x,y,z)=xy+3xz+2yz$ subject to the constraint $5x+9y+z=10$. Why does it not have min/man values?
0
votes
3answers
73 views

Find $\lim_{n \to \infty} \sqrt[n]{n^2+1} $

I am struggling to figure out $$\lim\limits_{n \to \infty} \sqrt[n]{n^2+1} .$$ I've tried manipulating the inside of the square root but I cannot seem to figure out a simplification that helps me find ...
2
votes
2answers
233 views

Calculate limit for continuous function

I am having trouble with this question: if f(x) is a continous function and $$ 1 - x^2 \leq f(x) \leq e^x $$ for all values of x, calculate $$ \lim_{x\to0} f(x) $$. Do I have to calculate the limit ...
1
vote
1answer
89 views

Uniform limit in definition of second order directional derivatives

If $f:E\rightarrow F$ is twice differentiable at $x\in E$, do we then have $$\lim_{h,k\rightarrow 0}\frac{A_x(h,k)-f''(x)(h)(k)}{\|h\|\|k\|}=0$$ where $A_x(h,k):=f(x+h+k)-f(x+h)-f(x+k)+f(x)$? This is ...
0
votes
1answer
644 views

Using partial fractions to integrate dy/(y(y-2))

This is a solution given for a practice exam I'm working through. However, I don't get where the 1/(-2)-(-0) (the part with the circled numbers in the denominator) came from. For instance, when I ...
4
votes
2answers
132 views

How to find the definite integral $\int_0^\infty \frac{x}{\sinh ax}\;dx$

I'm trying to prove that $$I:= \int_0^\infty \frac{x}{\sinh(ax)} dx = \frac{\pi^2}{4a^2}$$ Attempt: $$\sinh (ax) = \frac{1}{2}(e^{ax}-e^{-ax}) = \frac{1}{2}e^{-ax}(e^{2ax}-1)$$ Now I have ...
1
vote
2answers
256 views

How to solve the differential equation $y'=(x+y)/(x-y)$?

I have a question, $y'=\frac{x+y}{x-y}$ is asking to be differentiated, but isn't $y' =\frac{dy}{dx}$ so I'd have to integrate.
0
votes
1answer
17 views

show that $c′(5)$ is orthogonal to $\nabla f(1,4,2).$

I need some help here. Let $f(x, y, z)$ be a differentiable function and suppose that $c(t)$ is a path which lies on the surface $f(x, y, z) = 17.$ If $c(5) = <1, 4, 2>$ show that $c′(5)$ is ...
3
votes
2answers
57 views

How to find $y$ from $y' = e^{2x}-e^x y$?

The problem asks me to find $y(x)$ from the equation $$y' = e^{2x}-e^x y$$ The $y'$ is $dy/dx$ right, so wouldn't the correct step be to integrate right away? If not, should I change some terms ...
0
votes
4answers
128 views

How to solve linear differential equations such as $y''-7y'+10y=0$?

The question asks to solve the linear equation $$y''-7y'+10y=0$$ I'm not sure where to even start. I'm new to this so it would be nice if I got an in depth explanation ( with rules too ) Thank ...
1
vote
2answers
304 views

Related Rates Problem involving two runners on a circular path

Problem: There was a typo in the original statement. I fixed it now!! Two runners start running (from the same point) in opposite directions along a circular path of radius $100\ m$ at a speed of ...
5
votes
1answer
51 views

Prove that $ \int_0^1 f(x)dx - \frac{1}{n}\sum_{k=1}^n f(\frac{k}{n}) = O(\frac{1}{n}) $

Let the function $ f(x) $ is bounded and monotone on $[0,1]$. Prove that $ \int_0^1 f(x)dx - \frac{1}{n}\sum_{k=1}^n f(\frac{k}{n}) = O(\frac{1}{n}) $ It is clear that $ \int_0^1 f(x)dx - ...