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

learn more… | top users | synonyms

1
vote
3answers
100 views

what is $\int_{0}^{\infty}e^{-x^2}dx$ [duplicate]

I have a question: $$\int_{0}^{\infty}e^{-x^2}dx=?$$ Thanks for your help. Thanks ahead.
1
vote
2answers
53 views

Show a double-sided infinite integral of $\sin(x+b)$ exists iff $b=n\pi$

More formally: Show that $$\lim_{a\rightarrow \infty} \int_{-a}^a \sin(x+b)$$ exists if and only if $b=n\pi$ for some $n \in \mathbb{Z}$. I get the intuition fine. The function is just a horizontal ...
1
vote
5answers
78 views

Calculate $\displaystyle \lim_{x \to \frac{3}{2}} \frac{2x^2-3x}{|2x-3|}$

I know that $\displaystyle \lim_{x \to \frac{3}{2}} \frac{2x^2-3x}{|2x-3|}$ does not exist, because the lateral limits are different and I also know that the absolute value on the denominator has ...
1
vote
4answers
126 views

Double integral for $\int_{0}^{1} \int_{-1}^{0} \frac {xy}{x^2 + y^2 + 1}\ dy\ dx$

I'm trying to evaluate this $$\int_{0}^{1} \int_{-1}^{0} \frac {xy}{x^2 + y^2 + 1}\ dy\ dx$$ tried substition $$ u = {(x^2+y^2+1)}^{-1} \ \ du = \ln {(x^2+y^2+1)}$$ but du is not found in the ...
1
vote
2answers
79 views

Convergent or divergent? $\sum_{n=1}^{\infty}{\frac{n^n}{(n+1)^{n+1}}}$

Does the series $$\sum_{n=1}^{\infty}{\frac{n^n}{(n+1)^{n+1}}}$$ converge?
1
vote
2answers
119 views

Show that each of the series converges.Help Please

Show that each of the series converges on their respective domains. a) $$\sum_{n=1}^\infty \frac{1}{(1 + nx)^2}, x \in (0,\infty)$$ b)$$\sum_{n=1}^\infty e^{-nx}, x \in(0,\infty)$$ For the first ...
1
vote
4answers
128 views

Find the limit as $x$ tends towards $\frac{\pi}{4}$

In looking at the corresponding graph and differentiating it after reducing it to a different form, I know the that limit is equal to $2$ but I am unsure as to how I can show this algebraically. Any ...
1
vote
5answers
116 views

Anyone have a good proof for the second part of FTC?

Does anyone have a good proof for the second part of the fundamental theorem of calculus? I haven't been able to find any good videos on it so far so I'd like someone to write it down and I can throw ...
1
vote
3answers
254 views

Derivatives of sine and cosine at $x=0$ give all values of $\frac{d}{dx}\sin x$ and $\frac{d}{dx}\cos x$?

In video 3 of the video lectures by MIT on Single Variable Calculus presented by David Jerison, the latter says: Remarks: $\dfrac{d}{dx}\cos x\left|\right._{x=0}=\lim\limits_{\Delta ...
1
vote
4answers
77 views

Given $(a_{n+1} - a_n) \rightarrow g$ show $\frac{a_n}{n} \rightarrow g$

So yea, basically this is the problem. $(a_{n+1} - a_n) \rightarrow g$ show $\frac{a_n}{n} \rightarrow g$ It looks like Cauchy's sequence, but I'm not sure. Can we say that if $(a_{n+1} - a_n) ...
1
vote
4answers
86 views

Evaluate $\lim_{x\to 0} \frac{(1-x)^{1/3}-(1+x)^{1/2}}{x}$

Evaluate the limit $$\lim_{x\to 0} \frac{(1-x)^{1/3}-(1+x)^{1/2}}{x}$$ I know the limit is $-5\over6$ by looking at the graph of the function, but how can I algebraically show that that is the limit?
1
vote
2answers
203 views

Integral: $I=\int\limits_{0}^{1}\dfrac{x^2dx}{\sqrt{3+2x-x^2}}$

Evaluate: $\displaystyle I=\int\limits_{0}^{1}\dfrac{x^2dx}{\sqrt{3+2x-x^2}}$
1
vote
3answers
61 views

Finding $y^{\prime}$ of $y=\log_7 e^{8x}$

Find $y^{\prime}$ of $y=\log_7 e^{8x}$ I know that $\dfrac{d}{dx}(e^{8x})=8e^{8x}$, but I am confused on how to work the rest of the problem. Is this correct: $\log_ex=\ln x$ and that ...
1
vote
4answers
139 views

Help with derivative of $y=x^2\sin^5x+x\cos^{-5}x$

Find $y^{\prime}$ of $y=x^2\sin^5x+x\cos^{-5}x$ My try: $\dfrac{d}{dx}(x^2\sin^5x)=x^2(-5\sin^4x)+(2x\sin^5x)$ $\dfrac{d}{dx}(x\cos^{-5}x)=x(-5\cos^{-6}x)+1(\cos^{-5}x)$ This doesn't seem ...
1
vote
3answers
107 views

Nspire CAS spitting out a wrong answer?

Consider the integral: $\int \frac{8x+11}{(2x+3)(x+1)}$ My Nspire CAS tells me that the answer to this is $ln\left((x+1)^3 \cdot (2x+3)\right)$ (replace parentheses with absolute value signs, I ...
1
vote
2answers
115 views

How to check if the series $\sum_{n=0}^{\infty} \sqrt{n+1}-\sqrt{n}$ is convergent

or divergent?? I tried few tests, but I didn't success to discover if the series is convergent or is divergent... $$\sum_{n=0}^{\infty} \sqrt{n+1}-\sqrt{n}$$ Thank you!
1
vote
6answers
550 views

How do you derive this trig identity from the common ones? $\cos^2x=\frac{1+\cos2x}{2}$

$$\cos^2x=\frac{1+\cos2x}{2}$$ Just came across this identity one today. Where does this come from? Is this an easy derivation from the more popular identities, or is this one you just take it at ...
1
vote
3answers
164 views

Find $\lim_{ n \to \infty} (\frac{n!}{n^n})^{\frac{1}{n}}$

Find the limit : $\lim_{ n \to \infty} (\frac{n!}{n^n})^{\frac{1}{n}}$ My working : Let $$t = \lim_{ n \to \infty} (\frac{n!}{n^n})^{\frac{1}{n}}$$ Now taking log on both sides : $$\log t = ...
1
vote
3answers
116 views

Find $\lim_{n\to\infty} \frac{n^2}{2^n}$

$$\lim_{n\to\infty}\frac{n^2}{2^n}$$ Do you have some tips so I could solve this problem, without the use of L'Hôpital's rule? Indeed, we didn't see formally L'Hôpital's rule, nor Taylor series so ...
1
vote
2answers
329 views

maximize $\int \limits ^b_a (24 - 2x - x^2)^{1/3}$

I need to find $a$ and $b$ in $\int \limits ^b_a $(24 - 2$x$ - $x^2$)$^{1/3}dx$ such that the value of the integral is maximized. I know I need to solve the integral, plug in $a$ and $b$, and then ...
1
vote
2answers
151 views

Necessary conditions for a continuous function $f(x)$

In every textbook the condition for a function $f$ to be continuous at a point $x_0$ are: $f(x_0)$ exists $ \lim\limits_{x \to x_0}f(x)$ exists $f(x_0)=\lim\limits_{x \to x_0}f(x)$ Are conditions ...
1
vote
4answers
121 views

Integration of $\int_{}^{}\frac{1}{x(1+x)^3}dx $

we got this integration problem $$\int_{}^{}\frac{1}{x(1+x)^3}dx $$ it seems a fairly simple problem but what i am struggling with it is doing its partital fractions $\int_{}^{}\frac{1}{x(1+x)^3}dx ...
1
vote
3answers
520 views

How do I solve this Partial Fractions question?

$$\frac{x^4}{(x^2-3)(x^2+3)}\;$$ How would I do this? My attempt started with this: $$\frac{A+Bx}{(x^2-3)}\; + \frac{C+Dx}{(x^2+3)}$$ But when I start working it all out, A, B, C and D all ...
1
vote
3answers
197 views

Can you take out the exponent from an integral?

I'm trying to do a simple integration and I was wondering if this is possible or if it breaks a law: $$\int { \frac { 1 }{ \cos^{ 2 }4x } } \, dx\\ ={ \left[ \int { \frac { 1 }{ \cos 4x } } \,dx ...
1
vote
5answers
1k views

Taking the limit of $n(e^{-1/n}-1)$ as $n$ approaches infinity

The form is infinity times zero and that is indeterminate which means I need to use L'Hospital's rule, but I have tried to do that but every time I would find another indeterminate form. How can I ...
1
vote
2answers
116 views

Absolute convergence, conditional or divergence? $\sum_1^\infty \frac{(-1)^{n+1}}{n^\frac{1}{3}}$

$$\sum_1^\infty \frac{(-1)^{n-1}}{\sqrt[3]{n}}$$ This seems fairly simple to me, I know I can look at the absolute value and if that diverges than I am done. $$\sum_1^\infty ...
1
vote
4answers
417 views

Show that $f$ is integrable and $\int_{I} f=0$

Let $I$ b a generalized rectangle in $\Bbb R^n$ Suppose the bounded function $f:I\to \Bbb R$ assumes the value $0$ except at a single point $x \in I$ Show that $f$ is integrable and $\int_{I} f=0$ ...
1
vote
2answers
153 views

Calculating $\lim_{x\to 0}\left(\frac{1}{\sqrt x}-\frac{1}{\sqrt{\log(x+1)}}\right)$

Find the limit $$\lim_{x\to 0}\left(\frac{1}{\sqrt x}-\frac{1}{\sqrt{\log(x+1)}}\right)$$
1
vote
3answers
104 views

is the limit continuous or not?

is $(x^2 + y^2) / (x^2 - y^2)$ continuous or not at $(0,0)$? I think it is not continuous at $(0,0)$. To check I just plugged in the points and got $0$. Did I do that right? and also is there another ...
1
vote
2answers
231 views

True or false: if $f'(c)<0$, then $f$ is concave down at $x=c$?

How can I determine the following statement is true or false? If $f'(c)<0$, then $f$ is concave down at $x=c$ ?
1
vote
3answers
108 views

Critical number $y = \frac{1}{x^2 + 2}$

Seems pretty straight forward but my book seems to be giving an incorrect answer without any explanation to their magic. $$y = \frac{1}{x^2 + 2}$$ I know that this has no 0 so that rule of finding a ...
1
vote
2answers
106 views

What is the derivative of $\frac{x}{2-\tan x}$?

I am too stupid to figure this out so I won't even try anymore $$y = \frac{x}{2 - \tan x}$$ I am sure this will take someone about four seconds to solve, but I spent about ten minutes looking at it ...
1
vote
2answers
190 views

Determine whether this integral converges: $\int_1^\infty\frac{(x+1)\arctan x}{(2x+5)\sqrt x}$

Determine whether the next integral converges: $$\int_1^\infty\frac{(x+1)\arctan x}{(2x+5)\sqrt x}$$ I has this one on a test and lost all my points on this one. Since we were given no answers to the ...
1
vote
3answers
603 views

The population of a certain bacteria can multiply threefold in 24 hours. If there are 500 bacteria now, how many will there be in 96 hours?

The population of a certain bacteria can multiply threefold in 24 hours. If there are 500 bacteria now, how many will there be in 96 hours? I figured out this bacteria $=500(3)^{96/24}$ but then my ...
1
vote
3answers
230 views

Composition of Continous function is continuous

I have an example, prove that the function y = |cosx| is continuous. We can make two function viz. let g(x) = |x| f(x) = cosx As we know that |x| is continuous function and cosx is also continuous ...
1
vote
3answers
1k views

Prove that $S = \{(x,y)\in \Bbb R^2 : x>0, y>0\}$ is open.

Prove that $S = \{(x,y)\in \Bbb R^2 : x>0, y>0\}$ is open. Can anyone help me, I need to prove this statement only by the use of a ball.
1
vote
5answers
746 views

Limit question as $x$ and $y$ approach infinity?

I have to prove that the limit of the function $\frac{x^2}{x^2+y^2}$ as $x$ approaches infinity and as y approaches infinity does not exist. I thought about finding the side limits, and if they are ...
1
vote
5answers
613 views

Calculus graph questions?

How would I solve the following questions? Sketch the graph of a differentiable function f that satisfies the given conditions if that possible or explain why its not possible: $f(0)=-3$ $f(3)=0$ ...
1
vote
3answers
59 views

Demonstrate that $f$ is constant.

Let $f:\mathbb{R^2}\rightarrow \mathbb{R}$ a function differentiable that: a) $f(x,0)=0$ for all $x \in \mathbb{R}$ b) $\dfrac{df}{dy}(x,y)=0$ in all points. Demonstrate that $f$ is constant.
1
vote
2answers
139 views

Evaluate $\lim_{n\to\infty}\frac{n}{2^n}\sum_{k=1}^{n}\dbinom{n-1}{k-1}\{\sqrt{k^2+2k+2}\}$

Evaluate $$\lim_{n\to\infty}\frac{n}{2^n}\sum_{k=1}^{n}\dbinom{n-1}{k-1}\{\sqrt{k^2+2k+2}\}$$ where $\{x\}$ is the fractional part of $x$. Some suggestions here? Thanks!
1
vote
3answers
512 views

If $f'(x)$ is bounded in $[a,b]$ then $f(x)$ is uniformly continuous in $[a,b]$

Help me please with this question. Let $f(x)$ is a differentiable function in$ [a,b]$. How using Mean value theorem to show that if $f'(x)$ is bounded in $[a,b]$ then $f(x)$ is uniformly continuous ...
1
vote
3answers
83 views

Question about limit of a product

Is it possible that both $\displaystyle\lim_{x\to a}f(x)$ and $\displaystyle\lim_{x\to a}g(x)$ do not exist but $\displaystyle\lim_{x\to a}f(x)g(x)$ does exist? The reason I ask is that I was ...
1
vote
2answers
100 views

integrating not explicitly?

Let $$f(x)=\int_1^x\frac{\log(t)}{t+1}dt$$ if $x>0$. Compute $f(x)+f(1/x)$ I tried to calculate integrals explicitly but obviously failed. How should I approach this problem? Hints are ...
1
vote
3answers
1k views

Length of the curve defined by $y=6 x^{3/2} - 7$ from $x=1$ to $x = 9\;$?

Find the length of the curve defined by $y=6 x^{3/2} - 7$ from $x=1$ to $x = 9$. I need help with this section. I would really appreciate it. Thank you!
1
vote
4answers
131 views

Calculate $\int_0^2 x^2 e^x dx$

How do I calculate $$\int_0^2 x^2 e^x dx$$ Is there a product rule for integration ?
1
vote
3answers
232 views

Area Between Curves

The problem I am working on is, "In Exercises 17 and 18, find the area of the region by integrating (a) with respect to and x (b) with respect to y." The two functions: $g(y)=4-y^2$, and $f(y)=y-2$ ...
1
vote
2answers
96 views

Derivative of $\frac{e^x−e^{−x}}{e^x+e^{−x}}$

I am curious as how to solve this I have been trying here and I know the answer is $\dfrac{4}{{(e^x+e^{-x}})^2}$ I have this derivative $y=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}$. So here is how we do it. ...
1
vote
2answers
146 views

How to integrate $\frac{x^2}{1-x^2}$

I want to integrate \[\frac{x^2}{1-x^2},\] what I have try is trigonometric substitution and partition function and integration by part but still cannot solve it Thx for your reading!
1
vote
2answers
875 views

How to find constants a and b in this function?

$\displaystyle \lim_{x\to0} \frac{\sqrt[3]{ax+b}-2}x = \frac 5{12}$ How do you solve for constants $a$ and $b$?
1
vote
3answers
101 views

Give an example about limits: $\lim\limits_{x\to0}f(x^{2})$ exists but $\lim\limits_{x\to0}f(x)$ does not.

Can someone help me find an example where $\lim\limits_{x\to0}f(x^{2})$ exists but $\lim\limits_{x\to0}f(x)$ does not. Thanks.