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

learn more… | top users | synonyms

1
vote
1answer
34 views

Hyperbolic trigonometric functions identity

$$\cosh(\sinh^{-1}x) = \sqrt{x^{2}+1}$$ I used the fact that $\cosh(x) = \frac{1}{2}(e^{x}+e^{-x})$ and that $\sinh^{-1}(x) = \ln(x + \sqrt{x^{2}+1})$ Eventually I simplified to ...
0
votes
1answer
31 views

Prove or disprove irrational numbers [duplicate]

If x^(1/3) is an irrational number, then x is also irrational. I tried using contrapositive, but it's not the right way.
0
votes
1answer
26 views

$f''(x)=12x^2+6ax+3=3(2x+a/2)^2+3(1-a^2/4)$ Where can be this derivative negative? Can it be negative?

$f''(x)=12x^2+6ax+3=3(2x+a/2)^2+3(1-a^2/4)$ I still don't how to calculate for which a parameter can this expression be negative.
1
vote
2answers
71 views

What are the exceptions to the 2nd Derivative Test?

Today in class my professor went over the 2nd derivative rule and was mentioning how there are some exceptions. I wanted to know if anyone I had a more in depth analysis of it. An explanation of what ...
2
votes
3answers
62 views

Find norm of operator $L(x,y)=(x+3y,y-x)$

I'm trying to tackle the following question, but with no success... Let $L: \ \mathbb{R^2}\to\mathbb{R^2}$ be an operator such that $L(x,y)=(x+3y,y-x)$. Find $\|L\|$. So, I know that I need ...
0
votes
1answer
30 views

Taylor series and integration

Let $f$ be a twice continuously differentiable function on $\mathbb{R}$. Let $0=t_0<t_1<\ldots < t_n=t$ be a partition of $[0,t]$. Then: $$f(t)-f(0)=\sum_{i=0}^{n-1}f(t_{i+1})-f(t_i) ...
2
votes
1answer
50 views

How to calculate this complicated intergral

While solving this differential equation system I wanted to express a function in another way. I came up with the following integral: $$\int \frac{dx}{\cos^2 x \cdot \left(\frac{1 + \sin x}{\cos ...
1
vote
2answers
34 views

Existence of a function with finite $\lim\limits_{n\to \infty} f(n)$

Does there exist a continuous and bounded on $\mathbb{R}$ function $f(x)$ such that $\lim\limits_{x\to +\infty} f(x)$ does not exist but there exists the limit of the sequence $\lim\limits_{n\to ...
5
votes
1answer
97 views

Integral of $\int x^{-x} dx$

Question: $\int x^{-x} dx =$ ? Hint: $$ e^{x\ln \frac{1}{x}} = \sum_{n=0}^\infty \frac{x^n}{n!} \left(\ln\left(\frac{1}{x}\right)\right)^n$$ I figure since $\int x^{-x} dx = \int e^{x\ln ...
-2
votes
1answer
412 views

Calculus word problems

Need help with this word problem, not sure how to complete this question. A cop is trying to catch drivers who speed on the highway. She finds a long stretch of the highway. She parks her car behind ...
0
votes
1answer
68 views

Evaluating $\int _0^{\pi }\int _0^x\sqrt{1-x^2}\:dydx$

Evaluate: $\int _0^{\pi }\int _0^x\sqrt{1-x^2}\:dydx$ I've gotten it done to: $\int _0^{\pi }x\sqrt{1-x^2}dx$ Should I now change to polar coordinates because of the pi, or how should I proceed?
0
votes
1answer
23 views

Probability Density Function (Integration)

A probability density function is given by $$f(x)=\left\{\begin{matrix} ke^{-2x} &x\geq0 \\ 0&otherwise \end{matrix}\right.$$ Find $k$ My attempt, $k\int_{0}^{\infty }e^{-2x}dx=1$ But ...
1
vote
1answer
21 views

Under Uniform Convergence Can you Bring an Integral Inside a Double Sum

Let's say that $\sum_{n=0}^{\infty} a_n(x)$ and $\sum_{n=0}^{\infty} b_n(x)$ are both uniformly convergent and that $\int_{\mathbb{R}}a_n(x)dx<\infty$, $\int_{\mathbb{R}}b_n(x)dx<\infty \forall ...
0
votes
2answers
51 views

Why is a limit of an integer function $f(x)$ also integer?

Why is a limit of an integer function $f(x)$ also integer? For example, a function that's defined on interval $[a, \infty)$ and the limit is $L$
0
votes
1answer
20 views

Unable to prove an integral inequality involving

Let $f(x)=1+x+\frac{1}{2} x^2+\frac{1}{6}x^3$. Let $g(x)=1+\int_0^xf(t)dt$ $h(x)=1+\int_0^xh(t)dt $. Show that $$h(x)-f(x)=g(x)-f(x)+\int_0^x(h(t)-f(t))dt.$$ This was quite easy: Since ...
1
vote
7answers
85 views

Prove the inequalities $1-\frac{x^2}{2}\le \cos(x)\le1-\frac{x^2}{2}+\frac{x^4}{24}$

$1-\frac{x^2}{2}\le \cos(x)\le1-\frac{x^2}{2}+\frac{x^4}{24}$ I think I should check the two sides. but how can I show that $1-\frac{x^2}{2}- \cos(x)\le 0$? And on the other side $0\le-\cos ...
0
votes
2answers
80 views

Prove that $\int _ {a}^{b}x^n f(x)dx=0$ implies $f=0$ by using the Taylor theorem

Let $f$ be a smooth function on a closed interval $[a,b]$. Assume that there is $M>0$ such that $\left|f^{(n)}\right|<M$ for all $n\geq1$ and all $x\in[a,b]$. Prove that if $$\int _ {a}^{b}x^n ...
1
vote
2answers
51 views

How to evaluate $\lim_{x\to\infty}\arctan (4/x)/ |\arcsin (-3/x)|$?

I don't know how to start evaluate this limit, I cannot use L Hopital's rule. Thank you very much for all responses. $$\mathop {\lim }\limits_{x \to + \infty } \left({{\arctan \left({4 \over ...
0
votes
2answers
67 views

prove $(1+\ldots+o(x^{n-1}))^4=(1+\ldots+o(x^{n-1}))$

I would like to prove that : $$(1+\ldots+o(x^{n-1}))^4=(1+\ldots+o(x^{n-1}))$$ i took that from the picture below My Proof: note that $$(1+x)^a = 1 + ax + \frac{a(a-1)}{2!}x^2 + ...
-2
votes
1answer
36 views

Does the series diverge? [closed]

If $\sum\limits_{k=1}^\infty a_k$ diverges and each $a_k>0$. Is it true that $\sum\limits_{k=1}^\infty \frac{a_k}{k}$ also diverges?
4
votes
3answers
56 views

$\lim_{x\to 0+}=(1/x)^{\sin x}$? [duplicate]

$\lim_{x\to 0+}=(1/x)^{\sin x}$ I think I should rewrite it into a from $e^{\ln}$ , but I can't continue the calculation after this step.
1
vote
2answers
41 views

Show $F(z)=\int_{0}^{1}{g(t)\over t-z}dt$ is holomorphic in $\Bbb{C}\setminus[0,1]$. Limit problem.

Show $\int_{0}^{1}{g(t)\over t-z}dt$, $g(t):[0,1]\to \Bbb{R}$ a continuous function, is holomorphic in $\Bbb{C}\setminus[0,1]$. Trying to simplify ${F(z+h)-F(z)\over h}$ I arrived at ...
0
votes
0answers
14 views

How can I show that these 2 Taylor series expansions are equivalent (in 2 dimensions)?

I've been given the following questions: For part $(i)$ I've found that the expansion is given by $f(x+h, y+k)= 1+2h+k+3h^{2}+2hk+\frac{k^{2}}{2}$. However, for part $(ii)$ I've found that ...
1
vote
1answer
27 views

Why must the outside limits of an iterated be constant?

My book claims that in an iterated integral $$\int_a^b \int_{g(x)}^{h(x)} f(x, y) \, dy \, dx$$ $h$ and $j$ are allowed to be any functions of $x$ not containing $y$, but $a$ and $b$ must be constant ...
1
vote
0answers
14 views

Limits of integration of a multivariable function?

My book is filled with problems such as $\int_{1}^{x} f(x, y) dy$ . My question is if there is a good reason why the top limit of integration is an $x$. When we integrate the function with respect to ...
0
votes
3answers
132 views

Evaluating trigonometric integrals of the form $\int \frac{C \; d \theta}{ \sin \theta \sqrt{\sin^2 \theta - C }}$

Can you help me evaluating this integral? (no symbolic software please). \begin{equation} \int \frac{C \; d \theta}{ \sin \theta \sqrt{\sin^2 \theta - C }} \end{equation} It can be reduced to ...
1
vote
1answer
18 views

Two dimensional taylor expansion of arbitrary function

Consider the function dependent on the variables $N_t$ and $N_{t-1}$. Call the function $f$ so $f = f(N_t, N_{t-1})$. Now suppose we could write $N_t = N^*+n_t$ where $N^*$ is constant, and $n_t$ ...
1
vote
4answers
68 views

If $P(x)-P'(x) = x^n\;,$ Where $n$ is a positive integer. Then $P(0)$

$P(x)$ be a polynomial such that $P(x)-P'(x) = x^n\;,$ Where $n$ is a positive integer. Then $P(0)$ $\bf{My\; Try::}$ Let $P(x)=y\;,$ Then equation convert into $\displaystyle ...
0
votes
2answers
33 views

How do I know this critical point is the local maximum?

Let $f(x) = 4ln(10x)-4x$, $x>0$ I found the derivative and the critical point was 1, however, how do I know whether or not this is the local maxima or minima ? The next part of the question also ...
1
vote
0answers
36 views

Smoothness of $x\mapsto \frac{1}{1-x}$

I'm reading lecture about taylor expansion but i wonder in following example why he took $x\longmapsto \dfrac{1}{1-x}$ as function in $]-\infty,1[$ of class $\mathcal{C}^{n+1}$ and not in ...
5
votes
0answers
96 views

Recurrence for $\int \left(\frac{\sin x}{x}\right)^n \, \mathrm{d}x$ [duplicate]

I was playing around and wanted to consider the integral of $$I_n = \int_0^{\infty} \left(\frac{\sin x}{x}\right)^n \, \mathrm{d}x$$ using parts with $u = \sin^n x \implies \mathrm{d}u = n\cos x ...
6
votes
2answers
223 views

Why can we substitute the exponential function while deriving the characteristic equation

I'm studying the way characteristic equation works. The derivation, according to Wikipedia, follows: We have a function $y(x)$ and an equation $$a_n y^{(n)} + \cdots + a_1 y' + a_0 y = 0.$$ Then we ...
0
votes
2answers
101 views

Is $\tan x$ continuous at $0$? If so, how can I prove using $\epsilon$ and $\delta$?

I am new to $\epsilon - \delta$ proof. It seems like that $\tan x$ is continuous at 0, since $\tan x=0$ and right and left limits of $\tan x$ is $0$. However, how can I show that this is continuous ...
2
votes
3answers
57 views

What is the critical point of this function?

The problem reads : $$f(x)=7\frac{e^{2x}}{x} + 4.$$ I am unsure of how to approach this problem to find the derivative. If someone could break down the steps that would be greatly appreciated. ...
2
votes
0answers
20 views

How to write $ u(x,t) $ in Forward Time Central Space scheme?

If I want to apply the FTCS scheme to the following equation: $$ \frac{\partial u(x, t)}{\partial t} = f(x)u(x, t) +A\frac{\partial u(x, t)}{\partial x}+B\frac{\partial^{2}u(x, t)}{\partial x^{2}} $$ ...
1
vote
4answers
42 views

$\int\cosh^3(x)dx$

My attempt: $\int\cosh^3(x)dx = \int\cosh(x)(1+sinh^2(x))^2dx$ $let\ u=sinh(x)$ $du=cosh(x)dx$ $\int u^4 + 2u^2 + 1\ du = \frac 15u^5+\frac23u^3+u = \frac 15sinh^5x + \frac 23 sinh^3(x)+sinh(x)+C$ ...
1
vote
2answers
47 views

Evaluating an integral $\int_{-2}^12(x-4)^2dx$

$\int_{-2}^12(x-4)^2dx$ is the one I'm doing. I got 126 as an answer but am being told it's not right. Spent an hour messing with it, then put in into an online calculator that told me I was right, so ...
1
vote
4answers
53 views

Prove $|h(x)|\leq (Mx^2)/2$

I repeated this question because I would really appreciate a hint. Let $h:[0,a]\rightarrow \mathbb{R}$ be twice differentiable, $h'(0)=h(0)=0$, and $|h''(x)|\leq M$ for all $x\in [0,a]$. I proceed ...
3
votes
2answers
80 views

Integral of $\int_0^\infty \frac{\sin^4(u)}{u^{k}}du$ where $k\in(1,3)$

My task is to Evaluate $$\int_0^\infty \frac{\sin^4(u)}{u^{k}}\,du$$ where $k\in(1,3).$ I've tried a few things, but nothing seems to be working. Any help?
0
votes
3answers
81 views

Epsilon and Delta proof for limit

I have a function $f(x)=\frac{x+5}{2x+3}$ and want to show that the limit $$\lim_{x\to -1} f(x) = 4$$ is true using $\epsilon$-$\delta$. I have a trouble finding $\delta$ so that $f(x)-4$ is less ...
-1
votes
1answer
49 views

Prove that $f(x)=x^{\frac{1}{n}}$ is continuous.

I want to show that $f(x)=x^{\frac{1}{n}}$ is continuous everywhere using $\varepsilon$ and $\delta$. In case at point $x=0$, it is easily verified. But for some other points except $x=0$, How can ...
0
votes
2answers
59 views

Limit as $x \to 64$ of $\frac{\sqrt[6] x - 2}{\sqrt x - 8}$ [closed]

The question below is a question my teacher suggested for contest worthy of Grade 12 Calculus students. I don't need this answered right away but it would be nice if someone gave it a try $$\lim_{x ...
2
votes
1answer
58 views

How to show an infinite series diverges [closed]

If $\sum_{k=1}^\infty a_k$ and $\sum_{k=1}^\infty b_k$ converge, formal multiplication would suggest that $\left(\sum_{k=1}^\infty a_k \right)\left(\sum_{k=1}^\infty b_k \right)=\sum_{k=2}^\infty ...
2
votes
1answer
24 views

Measuring 'consistency' of a slope

If you have a series of points (x,y). Is there a mathematical measurement that represents how consistent the slope of the line is. Like, a straight line with a constant slope would be the highest ...
0
votes
2answers
39 views

Calculate the derivative of the product of three functions $e^x\cdot \ln(x) \cdot \cot x$

I am trying to compute the derivative of $$e^x\cdot \ln(x) \cdot \cot x$$ It's a product of three functions. I imagine I should first calculate the derivative of the first pair: ...
0
votes
2answers
56 views

growth rate of n! versus $r^n$

How do you show that $\lim_{n\to \infty} {n!\over {r^n}} $ approaches $\infty$? the growth rate of $r^n$ is slower than $n!$, so the latter one is increasing faster, but how do you show the above ...
0
votes
0answers
34 views

Understand first step of Laplace transform of integral

I am new to Laplace transform, and have some hard time understanding the very first step of the "preparation" before taking the laplace transform. $${f(t) =\int_0^t u \cosh(3u)\,\mathrm{d}u } ...
1
vote
2answers
31 views

The derivative of $-e^{\sqrt{2}\cdot x}\cdot 5x^3$

Calculate the derivative of $$-e^{\sqrt{2}\cdot x}\cdot 5x^3$$ Well, we use the product rule. Which is like "the derivative of the first by the second, plus the derivative of the second by ...
0
votes
1answer
30 views

How do you actually write out the terms in a Cauchy sequence?

For example for $\epsilon>0$ there exist $N$ such $n,m>N$ implies $|s_m-s_n|<\epsilon$. I understand that intuitively, we don't need to know a certain limit and thus this definition for ...
0
votes
4answers
185 views

How to compute $\int_0^{+\infty} \frac{dt}{1+t^4} = \frac{\pi}{2\sqrt 2}.$

How to compute $$\int_0^{+\infty} \frac{dt}{1+t^4} = \frac{\pi}{2\sqrt 2}.$$ I'm interested in more ways of computing this integral. There is always the straight forward method to ...