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

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1
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2answers
26 views

What is the potential function of the field $\left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right)$

The vector field is obviously conservative on every closed domain that doesn't encompass the point $(0,0)$, so there must be a potential function. I've got $\arctan(\frac{x}{y})$ for $x$ unequal to ...
0
votes
4answers
56 views

If $a>1$, prove that $\lim_{n \rightarrow \infty } a^n = \infty$

I want to know a rigorous method to prove that If $a>1$, $\displaystyle\lim_{n \rightarrow \infty } a^n = \infty$
-1
votes
1answer
49 views

Advice on helping a talented highschool student

I've watched an interview of the famous mathematician Pierre Deligne. In it he says that the family had given him 'Set theory' by Bourbaki, which is known for its difficulty, and I happen to know a ...
3
votes
2answers
44 views

Finding $\lim_{x\to0} \frac{\sin(x^2)}{\sin^2(x)}$ with Taylor series

Evaluate $$\lim_{x\to0} \frac{\sin(x^2)}{\sin^2(x)}.$$ Using L'Hospital twice, I found this limit to be $1$. However, since the Taylor series expansions of $\sin(x^2)$ and $\sin^2(x)$ tell us that ...
-2
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1answer
21 views

Books to get started on mathematics

I'm studying grammar and I feel a based mathematics would help me. What you recommend to start considering I'm not familiar with well developed therms and etc?
1
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1answer
40 views

Evaluate Derivative Using $\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ Definition

Evaluate the derivative of $x^3 - 3x +1$ using the $\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ definition to find the tangent of the curve at the point $(2, 3)$. I already calculated this derivative ...
1
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1answer
114 views

Comparing $\pi^e$ and $e^\pi$

Comparing $\pi^{e}$ and $e^{\pi}$ I read the answer there but I didn't understand one thing. How I should know to put $\dfrac{π}e-1$ instead of $x$? If I had this question on a test, I had no idea ...
1
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4answers
285 views

Why don't we indicate the variable to summed as we do for integrals?

When integrating over a certain variable $x$, we make sure to end the integral with $dx$, like so: $$\int_{1}^{\infty}\frac{1}{x^2}dx$$ The reason for this of course becomes more clear as one goes ...
6
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2answers
212 views

A problem on Mean Value Theorem

If $f''(x)$ exists on $[a,b]$ and $f'(a)=f'(b)$, then : $$f(\frac{a+b}{2})=\frac 1 2[f(a)+f(b)]+\frac{(b-a)^2}{8}f''(c)$$ for some $c\in(a,b)$. I tried but was unable to think of a function and was ...
0
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2answers
15 views

Average Value - Graphs

long method: Determine an equation for each and solve using average value formula alternative methods? How could you prove the average value to be C over an interval [a,b] if you are given a ...
0
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1answer
8 views

Left & Right Area Approximation Using Y-Axis - Method Alternatives

Is there a simpler way of solving this then calculating x1(h)+x2(h)+x3(h)+x4(h) by using the given y values (in this case h, the height is one, because the length of each rectangle is one) ...
0
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1answer
8 views

Related Rates of Change - Cylinder Question

A cylindrical tank with radius 5 cm is being filled with water at rate of 3 cm^3 per min. how fast is the height of the water increasing? I dont want this question solved, but please help me correct ...
-2
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0answers
21 views

Grade 12 Calculus Planes Intersections Question [on hold]

I am stuck on these 3 questions, help with all or any 3 of these questions is greatly appreciated! Give the equation of a plane and three lines, one of which is parallel to the plane, one of which ...
2
votes
2answers
115 views

solving the inequality

I'm looking for hints on how to efficiently solve this inequality: $$\left( \frac {|x|-|1-x|}{|x|} \right)^{2x-1} \gt \left(\frac {|x|-|1-x|}{|x|} \right)^{8-x} $$
0
votes
1answer
36 views

What is the graph of $y = \sin n$ and why is it different from the graph of $y = \sin x$?

I have downloaded a book about Calculus from MIT OCW. In that book, there is a section "A Thousand points of Light". (You can download the relevant section from here.) In that section, it is written ...
1
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1answer
36 views

Differentiable functions and examples

can someone give me an example of Differentiable function at x=4 and funcstions who dont Differentiable function at x=4? $f(x) = 2x-7$ $k(x) = 100x^7-55x^5+10000x^2$ $g(x) = 23$ Those are ...
0
votes
0answers
11 views

Find upper bound $\sum_{n=m}^{k} \left[ \binom{b-a}{n-m} \sum ^m_{j=0} …\right]$

I would like to find the upper bound of this expression: $$S_n = \binom{a}{m+1}\sum_{n=m}^{k} \left[ \binom{b-a}{n-m} \sum ^m_{j=0} \frac{\binom{a-m-1}{j} \binom{b-a-n+m}{n-j} } {{b \choose ...
1
vote
9answers
128 views

Find $\lim_{x\to0}\frac{\sin5x}{\sin4x}$ using $\lim_{\theta\to0}\frac{\sin\theta}{\theta}=1$.

I am trying to find $$\lim_{x\to0}\frac{\sin5x}{\sin4x}$$ My approach is to break up the numerator into $4x+x$. So, $$\begin{equation*} ...
1
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4answers
57 views

Why does $\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$ converge conditionally?

Why does it converge conditionally? $$\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$$
1
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3answers
52 views

Proving that $\lim\limits_{ x \rightarrow 0}{f(g(x))}=f(\lim\limits_{ x \rightarrow 0}g(x))$ when $f$ is continuous.

$$\lim\limits_{ x \rightarrow 0}{f(g(x))}=f(\lim\limits_{ x \rightarrow 0}g(x))$$ I have seen this step in a derivation of a result which is not the point of interest here. The book wrote the ...
0
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2answers
47 views

If the series $\sum_{k=1}^{\infty} a_k3^k$ diverges, Must the series $\sum_{k=1}^{\infty} a_k4^k$ diverge too?

I got this question: Prove or disprove the following: If the series $\sum_{k=1}^{\infty} a_k3^k$ diverges, Must the series $\sum_{k=1}^{\infty} a_k4^k$ diverge too? I tried to find a couple of ...
1
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1answer
23 views

holomorphic functions with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
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votes
1answer
36 views

How to find the integral of $\int \frac{GMm}{r^2}\,dr$ [on hold]

I want to find the integral of: $$\int_R^\infty \frac{GMm}{r^2}\,dr$$
2
votes
3answers
79 views

$\lim_{\theta\to0}\frac{\sin\theta}{\theta}=1$, $\theta$ must be in radians. But $x$ can be in degree for $\lim_{x\to\pm\infty}\frac{\sin x}{x}=0$?

We know that $\lim_{\theta\to0}\frac{\sin\theta}{\theta}=1$ but $\theta$ must be in radians. My first question is what happen when $\theta$ is not in radian? Is it only because in the proof we use ...
0
votes
2answers
35 views

How to use trigonometric substitution to compute this definite integral?

I have searched for a similar question on stack exchange but could not find one. The definite integral: $\large\int_0^1 \frac{x^4}{\sqrt{25-x^2}}$ I realize that I need to use $x = 5\sinθ$ in the ...
2
votes
4answers
49 views

How to apply the alternating series test to the series $\sum (-1)^{n+1} n/2^n$?

So, I need to test the following series for convergence or divergence: $$\sum_{n=1}^\infty (-1)^{n+1}{n\over {2^n}}$$ I know that when you use the Alternating Series Test, the series must satisfy ...
2
votes
0answers
45 views

$|a+b|+|b+c|+|c+a| \leq |a|+|b|+|c|+|a+b+c| \ $ [on hold]

Show that for every arbitrary complex number a,b and c we have $$|a+b|+|b+c|+|c+a| \leq |a|+|b|+|c|+|a+b+c| \ $$ Thanks.
0
votes
2answers
39 views

What is happening to the '2' in this integral?

It is the indefinite integral: $\int \frac{1}{2x-6}$ I am trying to understand it and looking the last step goes from $\frac12 \log(2(x-3))$ to $\frac12 \log(x-3)$ Can someone explain to me why the ...
1
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0answers
22 views

Find the area (to three decimal places) bound by 2 equations

Find the area (to three decimal places) bounded by $f(x)=x^2e^x$ and $q(x)=4-x^2$ So far I've gotten $x^2(e^x+1)-4=0$ and the two $x$ values that make the equation $0$ are $1.027$ and $-1.86$ next I ...
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votes
1answer
18 views

Finding area on a graph using integrals [on hold]

How do I find the area under the curve $f(x)=e^x$ above the $x$-axis, between the lines $x=0$ and $x=1$? I do not know where to start.
0
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3answers
46 views

$\sum_{(p,q) \in {\mathbb{N}^*}^2 and p \land q =1} \frac{1}{p^2 q^2} = \frac{5}{2}$ proof? [on hold]

Can you give me a very precise demonstration of this result please because it's very difficult for me to understand the demonstration on the pic :( $$ \sum_{(p,q) \in {\mathbb{N}^*}^2 \text{, } p ...
7
votes
1answer
50 views

Prove $_2F_1\!\left(\frac76,\frac12;\,\frac13;\,-\phi^2\right)=0$

Please help me to prove the identity $$_2F_1\!\left(\frac76,\frac12;\,\frac13;\,-\phi^2\right)=0,$$ where $\phi=\frac{1+\sqrt5}2$ is the golden ratio.
0
votes
1answer
18 views

Find equation of the straight line tangent to the curve at the point indicated

Find equation of the straight line tangent to the curve at the point indicated: $y=2x^2 -5$ at $(2,3)$ I think I have to use $y=m(x-x_o)+y_0$ etc but I'm not sure how to find the $m$! Thanks for ...
0
votes
2answers
22 views

Having a bit of trouble with min/max distance from sphere to point

The sphere is $x^2 + y^2 + z^2 = 81$ and the point is $(5,6,9)$ I am using Langrane multipliers , but the answers I am getting are so far off. I will post my system of equations soon. I found ...
0
votes
1answer
26 views

Need help getting started with a complicated limits problem

So I've got this thing $\lim_{x \rightarrow 3} \frac{\sqrt[3]{x^2+6x}-3}{x-3}$ I tried rationalizing but then it ended up in a huge mess and I didn't get the correct answer. I don't know if there's ...
1
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4answers
68 views

Integration of $x/(x^2+1)$ from $-\infty$ to $\infty$

I am trying to find the area of this graph $\int_{-\infty}^\infty\frac{x}{x^2 + 1}$ The question first asks to use the u-substitution method to calculate the integral incorrectly by evaluating ...
0
votes
0answers
15 views

Generalization of the concept of real valued function of class $k$ where $k \in \mathbb{N}$

Let $f:\text{I} \to \mathbb{R}$ a real valued function where I is an interval contained in $\mathbb{R}$. We say that f is of class 0 if $f$ is a continuous function on I. We say that $f$ is of class 1 ...
0
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1answer
21 views

Simplify $S=\sum_{i=0}^{k}a_i (2n)^{2i+1}$

Can someone simplify this expression (or compute its supremum)? Thanks so much. $$S=\sum_{i=0}^{k}a_i (2n)^{2i+1}$$ where $a_i>0$ and $k>1$, and $\sum_{i=0}^{k}a_i=1$.
0
votes
0answers
37 views

A counterexample 2

Can we find a function $f:\mathbb{R}\to(0,\infty)$ which satisfies $$\limsup_{|x|\to + \infty}\frac{f(x+c)}{f(x)}<+\infty, \ \ \forall c\in \mathbb{R},(\text{limit in }+\infty\text{ and ...
3
votes
3answers
41 views

Differential Equation $\frac{dP}{dt} = kP(1-P)$

I have a question about solving this differential equation. So, the question is to solve it given that $P(0)=\frac23$ So this is what I've done so far $$\frac{dP}{dt} = kP(1-P)$$ $$ k\,dt = ...
0
votes
1answer
38 views

Function whose power series coefficients contain logarithms

Is there a function that can be expressed as a power series $$f(x)=\sum_{n=0}^\infty a_n x^n$$ whose coefficients $a_n$ are expressions containing $\log n$ or something similar?
1
vote
1answer
40 views

Why can we make this integral change of limits? Is it obvious?

When deriving the equation for the impulse-momentum theorem, the following occurs: $$\cdots=\int\limits_{t_1}^{t_2}\frac{d\vec p}{dt}dt = \int\limits_{\vec p_1}^{\vec p_2}d\vec p=\cdots$$ I know the ...
1
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2answers
43 views

Exponential Growth Differential Equation

A population of buffalo grows exponentially (the rate of growth is determined by the population itself) but has a carrying capacity. Its population (in tens of thousands) at a time t ( in years ) is ...
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votes
2answers
25 views

Grade 12 Calculus Lines Intersections Question [on hold]

Give the equations of two lines that meet at the point (-1, 5, 2) and which meet at right angles, but do not use that point in either of the equations. Explain your reasoning
2
votes
0answers
84 views

Opinions on Lax's “Calculus with Applications, 2e”

There's a new calculus book titled Calculus with Applications by Peter Lax (2nd edition of an old one). I really liked his linear algebra and functional analysis books, and I was wondering if this ...
0
votes
1answer
39 views

L'Hopital's Rule with $\lim \limits_{x \to \infty}\frac{2^x}{e^\left(x^2\right)}$

(a) Show that $$\lim \limits_{x \to \infty}\frac{2^x}{e^\left(x^2\right)}$$ is a standard indeterminate form, but that L'Hopital's Rule does not give you any information about the limit. (b) Show ...
1
vote
3answers
68 views

Prove that the limit of $2^{\frac{-1}{\sqrt{n}}}=1$

Prove that the limit of $2^{\frac{-1}{\sqrt{n}}}=1$. I need to show that for each $\epsilon$ there exists an $n_0 \in \mathbb{N}$ such that $ \forall n \geq n_0: |2^{\frac{-1}{\sqrt{n}}}-1|\lt ...
2
votes
1answer
74 views

Differential Equation $\frac{dy}{dt}$ = $y - t$

Given the differential equation $\dfrac{dy}{dt}$ = $y - t$ Is this equation separable? -> No it is impossible to separate this equation because we can't get $y$ alone with $dy$ and $-t$ alone with ...
1
vote
1answer
25 views

Need Help Understanding How To Integrate With An Implicit Variable

My calculus is really rusty (damn Mathematica/Matlab) and I was wondering if anyone could help me with an equation I am having trouble integrating. I have attached a snapshot of the paper I am trying ...
3
votes
0answers
52 views

Integral substitution paradox

Assume $f \in L^+(\mathbb{R})$ and $x>0$. Consider the integral $$ \int_0^\infty \frac{f\left(\frac{x}{y}\right)}{y} \: dy. $$ I am trying to make the substitution $u=x/y.$ I seem to get $$ ...