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

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2
votes
1answer
604 views

Draw graph from limit

How do I draw the graph of a function when the limit is given? For example, how can I draw the graph of the following function: $$f(x)=\lim_{n\to∞ }\frac{x^{2n+1}}{x^{2n}+1}$$ if the value of x is ...
1
vote
2answers
75 views

convergence proof

Let $k$ be a natural number. If sequence $b_n$ is obtained by deleting the first $k$ members of the sequence $a_n$, then $b_n$ is convergent if and only if $a_n$ is convergent. I know that if $a_n$ ...
1
vote
1answer
200 views

A contradiction to do with continuity? (involves chain rule)

Suppose for each $t$, $S(t) \subset \mathbb{R}^n$ is a domain (hypersurface). We have a diffeomorphism $D^0_t:S(0) \to S(t)$ for each $t$ such that it solves the ODE $$\frac{d}{dt}D^0_t(\cdot) = ...
2
votes
1answer
472 views

integration of function equals zero

Let $f$ be a continues function in $[a,b]$ $\forall x \in [a,b] \ \ \ f(x)\geq 0$ $ \int_{a}^{b}f(x)dx \ = 0$ Proof that $ \forall x \in [a,b] \ \ f(x) = 0 $ So how do I do that ? What I know ...
2
votes
1answer
244 views

Any space $X$ with the indiscrete topology is compact.

Let $\tau_X = \{ \varnothing, X \} $. Let $A \subseteq X$, Let $O$ be an open cover of $A$. Since topology on $X$ is finite, then $O$ must finite too. Obviously, any subcover $O'$ of $O$ must be ...
1
vote
1answer
47 views

Integration problem help

I have this ODE : $$y'-2xy=e^{\frac{1}{x}}$$ So I tried multiplying the equation by the integrating factor : $e^{\int-2xdx}=e^{-x^2}$. $$\begin{align}\Rightarrow ...
5
votes
2answers
196 views

A tough inequality

After fiddling with things, I came to wonder whether the following holds for $0<x<1:$ $$\left(\frac{1}{x}-1\right)^x \geq (x-1)(4x^2-4x-1)$$ To get an idea of what I was dealing with I plotted ...
2
votes
2answers
1k views

How to integrate $\int_{3\sqrt{2}}^6 1/\big(t^3\sqrt{t^2-9}\big)\;dt$

Here's the full problem $$\int_{3\sqrt{2}}^6\frac{1}{t^3\sqrt{t^2-9}}\;dt.$$ The problem for me is what to do with the $-9$. I know that $1 - \sec^2(t) = \tan^2(t)$, so I'm trying to substitute ...
12
votes
1answer
228 views

Teaser or fun calc equation to surprise husband (physicist/EE) at work

I am a geneticist and unfortunately have not worked much with advanced calc since undergrad. In genetics, as you likely know, a male is denoted as XY and a female as XX. I plan to leave a riddle for ...
1
vote
1answer
83 views

Calculus: Area Between Two Curves

Find the area between the two curves $y = x^2 + ax$ and $y = 2x$ between $x=1$ and $x=2$, given that $a>2$ I found the antiderivative of $x^2+ax-2x$ and included the area between $x=1$ and $x=2$ ...
0
votes
2answers
234 views

calculating required deceleration for known velocity and distance

I want to apologize if you think my question is a duplicate but honestly I could not understand the answer I found in this question, nor any other answer I found on from searching. I have a robot ...
2
votes
3answers
89 views

Integrating $\int x^3 e^{-x^2}dx$ by parts

How would I solve the following $$\int x^3 e^{-x^2}\,dx$$ I set $u=e^{-x^2}(-2x)$ $du=e^{-x^2}(2x)$ $dv=x^3$ $v=\frac{x^4}{4}$ Then I did $$e^{-x^2}\frac{x^4}{4}+\frac{1}{2}\int x^4e^{-x^2}(-x) \, ...
0
votes
4answers
126 views

Limits, textbook error?

Why would the limit not exist? Shouldn't it be "The limit as x --> 1- is infinity" ?
2
votes
2answers
433 views

Maximum and minimum values for a curve

Considering the curve: $$ f(x) = x² $$ Does the curve $f$ have a maximum in the open interval $-1 < x < 1$? A minimum? I had a hard time interpreting this question. My first thought was: ...
0
votes
1answer
60 views

I-order ODE using substitution method

In my ODE class lecture notes there's a section on solving I-order ODE using substitution method and there's one step which I couldn't really figure out, I'd really appreciate it if somebody could ...
1
vote
2answers
159 views

Evaluating $\int{e^{x^{1/3}}dx}$

How can I get $$\int{e^{x^{1/3}}dx}$$ I think integrating by parts may work, but I can't figure out the exact way.
1
vote
2answers
345 views

A difficult differential equation $ y(2x^4+y)\frac{dy}{dx} = (1-4xy^2)x^2$

How to solve the following differential equation? $$ y(2x^4+y)\dfrac{dy}{dx} = (1-4xy^2)x^2$$ No clue as to how to even begin. Hints?
1
vote
3answers
280 views

What would the steps be to solve this limit?

We are currently learning how to evaluate limits in my Calculus class. I'm starting to get a lot better at them however I got this one wrong. I know the answer is 12 but I'm not sure how to get ...
2
votes
1answer
75 views

convergent of series problem

So I have this question which I do not understand: The series $ \sum_{n=1}^{\infty} |a_n| $ is convergent. The series $ \sum_{n=1}^{\infty} b_n $ is convergent, but the series $ \sum_{n=1}^{\infty} ...
2
votes
2answers
1k views

How to evaluate this limit as x approaches zero?

How do I evaluate the limit of $$\lim_{x\to 0}\frac{\sqrt{x+1}-1}{x} = \frac{1}{2}$$? As $x$ approaches $0$, I know the answer is $\frac{1}{2}$, but I got this question wrong. I think you have to ...
2
votes
1answer
48 views

If $f_x(x,t)$ is continuous in $t$, does this imply $f$ is continuous in $t$?

Let $f(x,t)$ be a function with $f_x \neq 0.$ If $f_x(x,t)$ is continuous in $t \in [0,T]$, does this imply $f$ is continuous in $t \in [0,T]$?
2
votes
1answer
85 views

$X,Y$ equipped with discrete topology $\implies$ any $f:X \to Y$ is continuous

My attempt: Pick an open set $O \subseteq Y$ open in $Y$. So, $f^{-1}(O) \in X$. But since all elements of $X$ are open sets since $X$ enjoys the discrete topology, then $f^{-1}(O)$ must be open. ...
0
votes
0answers
76 views

Conservative field and potential functions

How can I prove that the field $F=\frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy$ is a conservative vector field, and isn't local conservative on the domain $U=(x-5)^{2/3}+(y-7)^{2/3}<1$? I tried to ...
3
votes
0answers
84 views

Working with projection of areas?

I was recently solving a physics problem which had to do with the momentum imparted by a photon beam to a perfectly absorbing sphere and a perfectly reflecting one. Considering the former and Putting ...
4
votes
2answers
121 views

Integrating $\int (\ln(x))^2$

How would I integrate the following? $\int (\ln(x))^2 dx$ I did $u=(\ln(x))^2$ $du=2ln(x)\frac{1}{x}$ $dv=1$ $\ln(x)^2(x)-2 \int \ln(x)\frac{1}{x}(x)$ $\int-2\ ln(x)1$ $u\ln(x)$ $dv=1$ ...
1
vote
1answer
484 views

Equivalent Cauchy sequences.

Hi everyone I'm having a bad time with two questions in the Analysis book of Terry Tao. I finally finished one of the exercises and I'm wondering if the next reasoning is correct or maybe needs some ...
1
vote
1answer
263 views

Help with the proof of the formula for the derivative of the implicit function of two variables.

My question is at the end. It's about the proof of the following theorem in my calculus textbook. Theorem: If an equation $F(x,y,z)=0$ determines an implicit differentiable function $f$ of two ...
1
vote
1answer
111 views

Find the point where the area of a graph is split evenly

I searched and found one other answer that was similar to my question, but it's still not enough detail for me to understand. I need to find $a$ such that the line $x = a$ evenly divides the region ...
0
votes
1answer
705 views

Maximizing and Minimizing a function

Let $f(x,y)$ be a function such that $f:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$. Now we have to maximize $f$ over $x$ and minimize it over $y$ $i.e.\ $ $$\underset{x}{\text{max}}\: ...
1
vote
2answers
230 views

Find the derivative of this function at a specific point [closed]

What is the derivative of $$f(t)= \frac{t^3 +2} t$$ at point $(-2,3)$.
0
votes
1answer
288 views

Integration by parts Green's formula

I would like to do some kind of integration by parts to $$\int{f(\nabla g \cdot \nabla h)}$$ We know Green's identity holds with $f \equiv 1$. Is there a nice expression with general $f$? Let's say ...
1
vote
1answer
175 views

Using the precise definition of limits, show that $\lim\limits_{x\to \infty} \frac{(x+\cos x)}{x}=1$

Use the precise definition of $\lim\limits_{x\to \infty} f(x)=L$ to establish the following limit. $$\lim_{x\to \infty} \frac{(x+\cos x)}{x}=1$$ What I did is: I assumed $\cos x\ge -1$, then ...
0
votes
2answers
385 views

Question on limits and infinity

Just to clarify, the limit of $x \nearrow 0$ from the left of $1/x$, would be $-\infty$, and the limit of $x \searrow 0$ from the right of $1/x$, would be $+\infty$ right? This is only true when its ...
1
vote
3answers
126 views

The maximum and minimum of a function

How can I find the maximum and minimum of $$L=\sqrt{1-x^2}+\sqrt{1-y^2}+\sqrt{1-z^2},$$ given that $x^2+y^2+z^2=1$? I worked on some basic algebra, but it doesn't get me anywhere, so I'm stuck on ...
1
vote
1answer
81 views

Can we have $\displaystyle \lim_{x \to f(x)}$?

Can we have a limit where $x$ approaches a variable like this one:$\displaystyle \lim_{x \to f(x)}$ or $\displaystyle \lim_{x \to \cos(x)}$ ? And why? Thank you!
1
vote
1answer
71 views

question about deriving the midpoint method

http://en.wikipedia.org/wiki/Midpoint_method $y'(t) \approx \frac{y(t+h) - y(t)}{h} \qquad\qquad (3)$ For the midpoint method, one replaces (3) with the more accurate $ y'\left(t+\frac{h}{2}\right) ...
1
vote
1answer
108 views

u-substitution integral $\int \frac{e^{-x^2}}{2x}dx$

Integrate$$\int \frac{e^{-x^2}}{2x}dx$$ using $u$ substituion My main problem is what to use as my $u$. This is a question given by a professor in a class I am TAing for. Specifically, I do ...
1
vote
2answers
68 views

Comparison between Bessel's coefficients

The spatial solution is written as $$\Phi_k(r) = r^{1-\frac{d}{2}} \left(c_1 J_{1-\frac{d}{2}}(k r) + c_2 Y_{-1+\frac{d}{2}}(kr)\right).$$ In the case $d=3$, the solutions can be written as ...
0
votes
2answers
57 views

What are the steps to solve this limit?

I'm reviewing for a quiz but this one has got me stumped. I know the answer is $8$, but I'm confused about how to get there. What is $$\lim_{x\to 1} \frac{4x^2 - 4}{x-1}?$$ I know when you plug in ...
1
vote
1answer
57 views

Find $y''$ in terms of $x$ and $y$ when when $xy=3x+8y$.

The question is in the title (could not post it here for some reason). Came up with an answer $(y-3)/(x-8)^2$, but I am not sure if I have done it right.
0
votes
3answers
32 views

Help with maximization problem

In a book I'm reading, a claim is made that $f(a) = r \cos(a) - m \sin(a)$ has the maximum $\sqrt{r^2+m^2}$ (where a, r and m are real numbers). But I'm not sure how to prove it or even if ...
0
votes
0answers
97 views

Which side does $f(x)=\tan\sqrt{x-\pi/2}$ continues if any

Determine whether $f(x)=\tan\sqrt{x-\pi/2}$ is continuous at $\pi/2$, continuous only from one side (left or right) or continuous from neither side. I'm little confused with this problem. Am I ...
1
vote
2answers
47 views

Making $R(x)=\frac{x^3+8x^2+18x+15}{x+5}$ continuous from $x = -5$ point

The function $R(x)=\frac{x^3+8x^2+18x+15}{x+5}$ is not defined at the point $x = -5$. How should it be defined to make it continuous at this point. I'm pretty sure you have to factor the top but for ...
4
votes
3answers
155 views

non-archimedean in lay terms

I've dabbled with studying infinitesimals off and on for years ... Robinson, Keisler, Bell ("Smooth Worlds"), etc., even a bit of category theory. But I'm not a mathematician and tend to jump in way ...
1
vote
0answers
58 views

Does existence of weak spatial derivative imply existence of classical time derivative in this situation?

Let $f(x_1,...,x_n,t)$ be a function, where $(x_1,...,x_n) \in \mathbb{R}^n$ and $t \in [0,T].$ Denote by $f_{x_i}$ the weak (partial) derivative of $f$ wrt. $x_i.$ Is it possible for ...
0
votes
3answers
204 views

At what velocity v do you have to throw the stone

Suppose you are standing on the ground and throwing a stone straight up in the air. Let assume, for simplicity, that the gravitational constant is 10 meters per second per second. Neglecting air ...
0
votes
3answers
5k views

Finding the values of A and B in a continuous function

Consider the piecewise-define function $f(x)=2x^2+5$ if $x < -1$, $f(x)$=$Ax+B$ if $-1\le x\le 2$ and $f(x)=8x$ if $x\ge 2$. Given that $f$ is continuous everywhere, determine the values of $A$ ...
0
votes
1answer
211 views

Determining if ODE is linear

Determine whether the given equation is linear in the dependent variable $v$ and the dependent variable $u$: $$u dv+(v+uv-ue^u)du=0$$ I'm confused by this because I've never seen $dv$ and $du$ ...
2
votes
2answers
78 views

Is it true that $ \sum \limits_{i=1}^{\infty} f(i) = \lim_{n \to \infty} \sum \limits_{i=1}^{n} f(i) $?

Is the equality below true? $$ \sum \limits_{i=1}^{\infty} f(i) = \lim_{n \to \infty} \sum \limits_{i=1}^{n} f(i) $$
1
vote
3answers
350 views

Convergence of recursive sequence

I have tried to do this exercise. Do you think my solution is ok? Is it possible to get more information about the convergence? Is there a better way to do it? Let $ f:[0,1]\mapsto[0,1] $ be a ...