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

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0
votes
1answer
36 views

2nd Order Differential Equation, particular integral query.

What would the form of the particular integral be of the following differential equation: $$\frac{d^2y}{dx^2} -4 \frac{dy}{dx} +5y=8 \sin x$$ Should the particular integral be of the form; ...
0
votes
1answer
38 views

Is the gap between successive real roots of $x(t) = \frac{1}{30000 e^t} + \frac{2 e^{t/2} \cos (\sqrt{3}t/2)}{30000} $ eventually less than $5$?

Consider the function $x : \mathbb{R} \to \mathbb{R}$ given by $$x(t) = \frac{1}{30000} \frac{1}{\mathrm{e}^t}+ \frac{2}{30000} \mathrm{e}^{\frac{t}{2}} \cos \left(\frac{\sqrt{3}}{2}t\right), \quad ...
1
vote
4answers
72 views

Minimum distance between the curves $f(x) =e^x$ and $g(x) =\ln x$ [on hold]

What is the minimum distance between the curves $f(x) =e^x$ and $g(x) = \ln x$? I didn't understand how to solve the problem. Please help me.
1
vote
2answers
40 views

Finding the First Derivative ( 1 question)

Using the Definition of a limit: [ Of form $\lim_{x\to a} \frac{f(x)-f(a)}{x-a}$] Find $f'(x)$ when $x=9$ for $f(x)=\frac{2}{\sqrt{x}}$ I tried simplifying it but got jumbled when trying to multiply ...
3
votes
1answer
68 views

Circle is similar to a polygon with infinite number of sides

It is know from the time of Euclid, that a circle is similar to a polygon with infinite number of sides. But this ^^ is informal. Do you know any formalization where it appears that a circle is a ...
1
vote
1answer
42 views

solve $\frac{\partial u^2}{\partial x\partial y}=0$

I need to solve $$\frac{\partial u^2}{\partial x\partial y}=0$$ with the boundary conditions: $u(x,y=x^3)=\sin(x^6)$ and $\frac{\partial u}{\partial x}(x,y=x^3)=0$. I got a particular solution, I ...
6
votes
3answers
301 views

Is there a formula for the area under $\tanh(x)$?

I understand trigonometry but I've never used hyperbolic functions before. Is there a formula for the area under $\tanh(x)$? I've looked on Wikipedia and Wolfram but they don't say if there's a ...
1
vote
1answer
47 views

Weird indefinite integral homework questions

I'm solving a couple of integration problems using the method of changing variables, and would like assistance with two particular problems that I can't seem to solve. I completed rest of the problems ...
2
votes
4answers
17k views

Finding parametric equations for the tangent line at a point on a curve

Find parametric equations for the tangent line at the point $(\cos(-\frac{4 \pi}{6}), \sin(-\frac{4 \pi}{6}), -\frac{4 \pi}{6}))$ on the curve $x = \cos(t), y = \sin(t), z=t$ I understand that in ...
0
votes
0answers
51 views

Surface Integral over Ellipse

Let $$E=\left\{(x_1,x_2)~\middle|~\frac{x_1^2}{ a^2}+\frac{x_2^2}{b^2}=1\right\}=\left\{X(\theta)=(a\cos(\theta),b\sin(\theta))\,\middle|\,0\leq \theta\leq 2\pi\right\},$$ be an ellipse. Let ...
0
votes
0answers
27 views

Range of Derivative

Let $g(x) = f(x)/(x+1)$, where $f(x)$ is differentiable on $x\in[0,5]$, such that $f(0)=4$ and $f(5)=-1$. What is the range of values $g'(c)$ for a $c$ belonging to $[0,5]$? Considering values of ...
1
vote
1answer
23 views

Finding a limit on multiple square roots in a row?

Here are basically my two problems, which I have the answer from WolframAlpha: $$ \lim_{n\to\infty}(1-\sqrt 2-\sqrt{n+1}+\sqrt{n+2})=1-\sqrt 2 $$ $$ \lim_{n\to\infty}(\sqrt n-2\sqrt{n+1}+\sqrt{n+2})=0 ...
0
votes
1answer
47 views

Convert Riemann sum to definite integral: $\sum_{i = 1}^n \frac{n}{n^2 + i^2}$

I am having trouble with this problem. Basically, I am given a Riemann sum and I have to rearrange it so that I can deduce the definite integral that it is equivalent to. Thank you. $$\lim_{n \to ...
1
vote
2answers
690 views

Explain why graph of f lies below the $x$-axis in interval $[4\pi/9,5\pi/9]$

$f(x)=(x+1)\sin(3x)$ Explain why the graph of f lies below the $x$-axis for values of $x$ in the interval $[4\pi/9, 5\pi/9]$ From what i know/understand I'd have to look at the function in two ...
-5
votes
2answers
45 views

Why is $ \lim_{x\to 25}(25−x)/(\sqrt x−5)= -10$? [on hold]

I know the answer is $-10$ but I don't know where the negative sign is coming from. This is what I ended up with. $$\frac{(x-25)(\sqrt{x}+5)}{x-25} = (1)\sqrt x+5 = 10 $$ ...
-5
votes
0answers
41 views

Give a clear and lucid description of the function $f(x,y,z)=\sin(x)\sin(y)\sin(z)$. [on hold]

Give a clear and lucid description of the function $f(x,y,z)=\sin(x)\sin(y)\sin(z)$.
0
votes
3answers
40 views

$\lim_{(x,y)\to(0,0)}\frac{x^2y^2}{\sin(x)\cos(y)}$ is this done correctly?

$\lim_{(x,y)\to(0,0)}\frac{x^2y^2}{\sin(x)\cos(y)}$ is it allowed to split a multi-variable limit into its component variables as in the next step? $= ...
0
votes
5answers
63 views

find the area of a kite with integration

A stunt kite has the shape in the diagram below: How can I find the area using calculus integration. Can anyone help me start this question, I am not looking for the full answer. I assume I only ...
1
vote
1answer
51 views

How to compute $\lim\limits_{x \rightarrow 0} \frac{1}{x^2}\int_0^{G(x)} \arctan(s+2s^2) ds$

Suppose $g$ is a function that has its derivatives everywhere and $G(x)=\int_0^x g(t)dt$. How to compute $\lim\limits_{x \rightarrow 0} \frac{1}{x^2}\int_0^{G(x)} \arctan(s+2s^2) ds$? To start ...
1
vote
2answers
60 views

Memorizing Formulas for Differentiation

Once upon a time, I memorized the following formula out of laziness. Let $k(x)=\frac{f(x)^{g(x)}h(x)+i(x)}{j(x)}$. Then $k'(x)$ is as follows. ...
-4
votes
1answer
47 views

Integration CALCULUS [on hold]

I need help on problem based on integration calculus. Q: how to integrate $$\int\frac{dx}{1+\sin(x)\tan(x)}$$ Wolfram and integrate calculator does not help me.
0
votes
0answers
30 views

Find required degree of Maclaurin polynomial to estimate the cosine to two decimal places

I have a question where I am asked to find the amount of terms required in a Maclaurin polynomial to estimate $\cos(1)$ to be correct to two decimal places. So far what I have done is used Taylor's ...
0
votes
1answer
29 views

Finding length of curve $y^2 = 64(x+3)^3$ for $0 \le x \le 3$

Not getting the right answer for this, can someone point me to where I'm going wrong?
2
votes
1answer
2k views

Calculating an integral derived from the convolution of two Fourier transforms

Let $\sigma>0$ , $1<\alpha\leq 2$, and $-1\leq \beta \leq 1$. I am looking for a closed-form solution (or something near) for the following integral. $$\frac{1}{2 \pi } \text{PV}\int_{-\infty ...
-4
votes
0answers
36 views

How to prove the following questions by IBP? (Integrated By Parts) [on hold]

So this is the question that I have to solve. I know this is related to IBP, but Have no idea how to start and prove... need help
2
votes
2answers
55 views

Finding roots of an equation wich involves floor function

I'm trying to solve this equation $$ \left \lfloor{x +\frac{1}{100}}\right \rfloor + \left \lfloor{x +\frac{2}{100}}\right \rfloor + ... + \left \lfloor{x +\frac{223}{100}}\right \rfloor = 521 $$ I ...
1
vote
2answers
539 views

Any tips to solve this integral : $I_1 = \int \ln(x^2)e^{\sin(x)}\sin(x^{\cos(x)}) dx$

Background: I was making new expressions to see whether I could efficiently find their derivatives... After having done that, I've started trying to integrate most of them; obviously most of them ...
1
vote
4answers
73 views

What is the difference between sum and integral?

I am a beginner in calculus and I want to know what is the difference between sum and integral. More specifically I came across this example: Compare $$\sum^\infty_1\frac1x\space \text{and} \space ...
2
votes
1answer
36 views

Complex derivative numerically using real $h$ and imaginary $h i$?

I want to find numerically (the functional expression might become too complicated) the derivative of a complex function (to use it in a Newton-algorithm). Can I simply do something like $$ \frac ...
3
votes
3answers
44 views

A simple problem on first order differential equations

An ODE (Ordinary Differential Equation) of order $n$ becomes a relation: $$F(x,y,y^{(1)},...,y^{(n)})=0$$ Then $F(x,y,y^{(1)})=0$ defines an ODE of order one. In "basic standard texts", for purposes ...
0
votes
1answer
43 views

Path to Self Study Calculus 1-4 and Linear Algebra [on hold]

For the past year I've taken up self studying mathematics. My initial intent was to study so that when I entered college (currently a junior) I would have most of the basic mathematics for studying ...
9
votes
3answers
311 views

Developing Mathematic Intuition

I'm an engineering student, currently working my way through the fundamental mathematics courses. I've done reasonably well so far—mostly A's and a couple of B's in Algebra, Statistics, ...
0
votes
0answers
42 views

Finding the integral of a 1/variable*radical function

I'm trying to find the integral of $$\int\frac{1}{x* (\sqrt{4x^4 - 9})}$$ Attempt: I assumed that the integral would be some sort of inverse trigonometric function. Because of this, I did the ...
0
votes
2answers
28 views

Showing the set $A = \{ x \in l_1 : |x_n| \leq 1/n^2 ,\forall n\}$ is closed

Showing the set $A = \{ x \in l_1 : |x_n| \leq 1/n^2 ,\forall n \}$ is closed. I had to show it is compact, and I am done showing it is relatively compact, but now I am stuck showing it is closed. ...
0
votes
1answer
22 views

Finding volume of a solid of revolution

I need to find the volume of the solid that is formed when the (x>0, y< -1) region of the curve y= -1/x is rotated about the y-axis. If I'm correct, this volume can be calculated by: Evaluating ...
0
votes
1answer
51 views

Explain a physical problem by mathematics

I am not very good at physics, so I don't understand how to solve the following problem with vector calculus. A perfect incompressible fluid moves steadily under gravity around the outside of a fixed ...
0
votes
1answer
42 views

Finding twice-differentiable function of x of a parametric curve when dx/dt = 0

We're working on finding tangents of parametric curves and I feel like this problem isn't as hard as I'm making it out to be, but I am completely stumped. I am given this information: Given ...
0
votes
2answers
32 views

How do I find the equation for a semicircle with a radius of 2 on the x axis?

I'm doing a calculus project where we have to make a model of some graph rotated about the y axis. I am doing a fish bowl and I have most of it understood and ready. The only thing I'm not sure of is ...
0
votes
1answer
18 views

Using Rodrigues' formula to show a result

use the formula $P_n(x) = \dfrac{1}{2^nn!}\dfrac{d^n}{dx^n}((x^2-1)^n)$ to show that $P_{2n}(0) = \dfrac{(-1)^n(2n)!}{4^n(n!)^2}$ and odd terms are 0. I first subbed in 2n to the formula and got ...
0
votes
2answers
54 views

Evaluate the definite integral $\int_{0}^{a}\frac{dx}{(a^2+x^2)^{3/2}}$

I'm trying to solve this integral with trigonometric substitution but am having a ton of trouble: $$\int\limits_{0}^{a}{\frac{dx}{(a^2+x^2)^{\frac{3}{2}}}}$$ I tried $x=a\tan{\theta}$ and thus ...
9
votes
1answer
110 views

Solve the given Differential Equation

Solve the equation $$\frac{dy}{dx}=\frac{x+2y-5}{2x+xy-4} $$ I tried substituting $x=X+h$ and $y=Y+k$ but the $xy$ term is creating problem. How to solve it?
-3
votes
1answer
29 views

Notational problem [on hold]

Please, how do I write the following as a combination of a sum and a product: $$ (c-a_1)(c-a_2)(c-a_3)b_1 + (c-a_2)(c-a_3)b_2+(c-a_3)b_3 ?$$ Also, how can I generalize it?
-1
votes
2answers
46 views

$\lim_{n\to\infty}{a_{n}} =\lim_{n\to\infty}{b_{n}}$ iff $\lim_{n\to\infty}{(a_{n}-b_{n})=0}$ [on hold]

I need to proof or disproof that: $$\lim_{n\to\infty}{a_{n}} =\lim_{n\to\infty}{b_{n}}$$if and only if $$\lim_{n\to\infty}{(a_{n}-b_{n})=0}$$
1
vote
0answers
23 views

limit of $\frac{(k+1)[x^k(a+sin(x^{-k-1}))-x^{-1}cos(x^{-k-1})]}{k[x^{k-1}(a+sin(x^{-k}))-x^{-1}cos(x^{-k})]}$ as $x \to 0$

Limit of equation: $\frac{(k+1)[x^k(a+sin(x^{-k-1}))-x^{-1}cos(x^{-k-1})]}{k[x^{k-1}(a+sin(x^{-k}))-x^{-1}cos(x^{-k})]}$ as $x \to 0$ and k=0,1,2,... My calculation steps: $=\frac{k+1}{k} ...
28
votes
2answers
804 views

Function that is the sum of all of its derivatives

I have just started learning about differential equations, as a result I started to think about this question but couldn't get anywhere. So I googled and wasn't able to find any particularly helpful ...
4
votes
2answers
136 views

Prove that $7<e^2<8$

I was asked by my teacher to prove that $7<e^2<8$ using only algebraic methods and knowing that $2<e<3$. I don't know how to do this, where to start from, but I guess that I would need ...
-1
votes
1answer
17 views

How to find $x$ such that $f(x)$ takes a prescribed value [on hold]

Find $x$ such that \begin{equation} x\tanh(x\sqrt{2\alpha})=\frac{2}{\sqrt{2\alpha}} \end{equation}
5
votes
4answers
1k views

Proving that $\lim_{h\to 0 } \frac{b^{h}-1}{h} = \ln{b}$

Is there a formal proof of this fact without using L'Hôpital's rule? I was thinking about using a proof of this fact: $$ \left.\frac{d(e^{x})}{dx}\right|_{x=x_0} = e^{x_0}\lim_{h\to 0} ...
0
votes
2answers
25 views

How do I use the ratio test to determine convergence or divergence in this problem?

I have the problem: $$a_{n} = \frac{e^{n+5}}{\sqrt{n+7}(n+3)!}$$ I am told to use the ratio test to determine convergence or divergence (or the test could be inconclusive). So I take the limit: ...
54
votes
0answers
1k views
+200

Generalization of Liouville's theorem

As proposed in this answer, I wonder if the answer to following question is known. Let $E = E_0$ be the set of elementary functions. For each $i > 0$, inductively define $E_i$ to be the closure ...