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

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2answers
164 views

Evaluate $\int_0^1 \sqrt{2x-1} - \sqrt{x}$ $dx$

I'm trying to calculate the area between the curves $y = \sqrt{x}$ and $y= \sqrt{2x-1}$ Here's the graph: I've already tried calculating the area with respect to $y$, i.e. $\int_0^1 ...
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3answers
113 views

Evaluate $\int \frac{1}{(2x+1)\sqrt {x^2+7}}dx$

How to do this indefinite integral (anti-derivative)? $$I=\displaystyle\int \dfrac{1}{(2x+1)\sqrt {x^2+7}}dx$$ I tried doing some substitutions ($x^2+7=t^2$, $2x+1=t$, etc.) but it didn't work out.
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58 views

Differentiate $y = |x|(5 − x^2)^.5$

The curve: $$y = |x|(5 − x^2)^.5$$ is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point $(2, 2)$. How would I differentiate the absolute value?
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2answers
138 views

What is the probability of going bankrupt in roulette?

Imagine that the bank has the money $M_1$ and the player has the money $M_2$. The rules are the following: You win with a chance of $\frac{17}{36}$ and lose with $\frac{19}{36}$ each round. Now you ...
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1answer
56 views

Convergence of a subsequence .

If every subsequence of $x_n$ has a further subsequence which converges , is it true that the sequence is convergent? NOTE : This is not a duplicate ofthis . In this problem it is not given that the ...
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2answers
128 views

Jordan measure zero discontinuities a necessary condition for integrability

The following theorem is well known: Theorem: A function $f: [a,b] \to \mathbb R$ is Riemann integrable if and only if its set of discontinuities has Lebesgue measure zero. Now if we change ...
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1answer
67 views

A definite integral involving $\exp(-a\cosh x)$

Wolfram alpha can not integrate the following integral: $$\int_0^\infty x^n \exp(-a\cosh x)dx$$ Thanks- Mike
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2answers
55 views

solution of a quadratic equation

If I have an equation of a form: $$x^2+\alpha x + 10 =0$$ my book says that both roots have the same sign because "10" is positive. I'm trying to understand why the book makes this claim. Is there ...
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2answers
80 views

I need to solve $\dfrac{dx}{dt}= 2x(1-0.0001x)-0.01xy, \dfrac{dy}{dt} = -0.5y+0.0001xy$

I need to solve $$ \begin{align} \frac{dx}{dt} &= 2\,x\,(1-0.0001\,x)-0.01\,x\,y \\ \frac{dy}{dt} &= -0.5\,y+0.0001\,x\,y \end{align} $$ Can anyone tell how do we solve such problems, if ...
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1answer
53 views

Find slope of [(2/x) - f(-3) / x + 3 ]

I'm trying to find slope of the following equation, it will then be used to find the equation of the tangent line. lim x-> -3 [(2/x) - f(3) / x + 3] f(x) = (2/x) P(-3, (-2/3)) Use the definition ...
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4answers
71 views

I need help figuring this error percentage homework problem.

Question: The period of a simple pendulum is given by $T=2\pi\sqrt{\frac{L}{g}}$ where $L$ is the length of the pendulum in feet, $g$ is the constant of acceleration due to gravity, and $T$ is ...
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3answers
103 views

Evaluating the integral of an exact differential

What is wrong with evaluating the closed path integral as the following? $$ \oint_\gamma \frac{x\,dy-y\,dx}{x^2+y^2}= 2\pi\ne\oint_\gamma d\left(\arctan\left(\frac{y}{x}\right)\right)=0 $$ where the ...
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1answer
74 views

Proving $\int^\infty_0 x^n e^{-x} \, dx = n!$

I was motivated by this question on the various applications of integration by parts to prove the following integral: $$\int^\infty_0 x^n e^{-x} \, dx = n!$$ Here's what I have done, I feel I am ...
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3answers
36 views

Optimization in Calculus

As you can see I found the equation but I don't know how to find the points. As far as I tried was $(7, 49)$ but it was wrong.
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2answers
79 views

Creating a mathematical formula to price a taxi booking

I've been asked to create a mathematical formula that will be used to price taxi bookings at a local taxi company. Current system used: A table is used as a reference Variables: x is the ...
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3answers
81 views

Calculation of Trigonometric Limit with Summation.

If $\displaystyle f(x)=\lim_{n\rightarrow \infty}\sum_{r=0}^{n}\left\{\frac{\tan \left(\frac{x}{2^{r+1}}\right)+\tan^3\left(\frac{x}{2^{r+1}}\right)}{1-\tan^2\left(\frac{x}{2^{r+1}}\right)}\right\}.$ ...
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2answers
57 views

How do I go about solving this derivative of inverse tangent?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$8\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=8\frac{1}{1+x^2}$$would ...
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3answers
165 views

A counter-example to differential function but not twice differential

Find a function $f$ that is differentiable, but not twice differentiable and which does not belong to the following type: $$f(x) = \begin{cases} x^\alpha \sin(x^{\beta}) & x \neq 0 \\ 0 & ...
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1answer
34 views

How do I solve this trig derivative in respect to $x$?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=\frac{1}{1+x^2}$$would ...
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2answers
421 views

When are there no critical points?

Is there ever a time when there are no critical points of a function? For example, I am trying to find the critical points and the extrema of $\displaystyle f(x)= \frac{x}{x-3}$ in $[4,7]$ I am not ...
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2answers
95 views

Can all functions be expressed in terms of elementary functions?

After being introduced to the non-elementary function through an attempt to evaluate $\int x \tan (x)$, an interesting question occurred to me: Can the non-elementary functions be decomposed to ...
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1answer
30 views

turn $D=\{(x,y) | x \leq y \leq 2x, a\sqrt{x} \leq y \leq b \sqrt{x}, x\geq 0, 0 <a<b\}$ into a rectangle

Im trying to transform the region $D=\{(x,y) | x \leq y \leq 2x, a\sqrt{x} \leq y \leq b \sqrt{x}, x\geq 0, 0 <a<b\}$ to a rectangle by using a variable change. I'm doing this in order to ...
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45 views

Calculus, local linearization

(a)Given that$$ f(7)=13$$ and$$ f′(7)=−0.38$$, estimate f(7.1). My answer was$$ f(6.1)= 13+ -0.38(x-7)$$ = 13.342. (b)Suppose also $$ f′′(x)<0 $$for all $x$. Does this make your answer to part ...
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2answers
36 views

Question about a particular limit (at infinity)

I have a question about limits, a problem specifically. I have been asked to solve the following limit in any way I see fit: $$\lim_{x\to 2\pi^-}x\csc x$$ I know that the domain of $\csc$ is all ...
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2answers
65 views

Calculus/ zero or undefined ? with explanation

How can I know if the it suppose to be zero or undefined, because when I try to substitute in the original equation with 5/2 I will end up with a number and y. Same goes to (c) I will end up with x. ...
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1answer
48 views

Can somebody explain the resolution of the following limit? [closed]

$$\lim_{x\to-\infty}\frac{x-1}{|1-x|}$$ I checked at wolfram alpha and the result is $-1$, but i can't understand the resolution yet.
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1answer
1k views

find polar area (inner loop): $r=1+2sin(\theta)$

find polar area (inner loop): $r=1+2sin(\theta)$ I get that the zeros occur at $\frac {7\pi}{6} and \frac {11\pi}{6}$ and in turn this should be where the upper and lower bounds are (I'm actually not ...
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1answer
57 views

Maclaurin series of (1+x)^(1/x)

how can i find the Maclaurin series of $f(x)=(1+x)^{1 \over x}$? $f(0)$ is not even defined, or should I define it as $f(0)=e$? I stopped at the first derivative as it gets terribly messy. thank ...
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1answer
72 views

Why is $\ln(-x)$ the antiderivative of $x^{-1}$ if $x < 0$?

In relation to this question, ln|x| vs. ln(x)? When is the ln antiderivative marked as an absolute value? "On the other hand, if $x$ takes negative values then the derivative of $\ln(−x)$ ...
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1answer
43 views

Convergence of improper integral with logarithm

I would like to determine the nature of $A$ without calculating it. $$ A= \int_0^1 \ln(1-t^{a}) dt . $$ In $t=1$ we have a problem, so how should I proceed?
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4answers
85 views

what does b represent in the straight line equation?

I need your help, we know that the line equation is y= mx+b but my question is what does b represent in the straight line equation and how it effect the line ? Clear example with images would be ...
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2answers
2k views

curl of (cross product of two vectors), i know the formula, but not sure how to prove it

$$\text{curl } \left(\textbf{F}\times \textbf{G}\right) = \textbf{F}\text{ div}\textbf{ G}- \textbf{G}\text{ div}\textbf{ F}+ \left(\textbf{G}\cdot \nabla \right)\textbf{F}- \left(\textbf{F}\cdot ...
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1answer
105 views

Determination of $1^\infty$ indeterminate forms

Recently I have been learning some of the basic concepts of limits and in my academics. There I have been taught some methods to evaluate indeterminant forms like $1^\infty$, $0^0$ and $\infty^0$. ...
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2answers
88 views

Limit of an integral $\int_0^{\pi/2} \frac{x\sin^nx}{\sqrt{1+\sin^2 x}}\ dx$

What's the limit of $$\lim_{n\longrightarrow \infty} \int_0^{\pi/2} \frac{x\sin^nx}{\sqrt{1+\sin^2 x}}\ dx\ ?$$ Lebesgue's theorem is useless here I think, since $\sin x$ has no limit for ...
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2answers
36 views

How to find the convergence interval and radius of $\sum_{n=1}^{\infty}\sqrt[n]{n}(2x+5)^n$?

Find the convergence interval and convergence radius of the sequence $$\sum_{n=1}^{\infty}\sqrt[n]{n}(2x+5)^n.$$ I tried ratio test: $$\lim_{n\to \infty}\left|\frac ...
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2answers
100 views

Oops, $y^2=2x^2+C$ is not the same as $y=\sqrt{2x^2}+C$

Oops, $y^2=2x^2+C$ is not the same as $y=\sqrt{2x^2}+C$ I almost slipped and just assumed $\sqrt{C}=C$ but when you take the square root of both sides, you are really ending up with ...
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7answers
55 views

An example of Inequality

Could someone please show me the step by step solution to the following problem? $$ \frac{x^2+1}{x-1} \leq x $$ the anwer should be $-1 \leq x <1$. I'd like to know how to do this without the use ...
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1answer
30 views

Find the equation of the normal at a point

Find the equation of the normal at the point $(2, 1)$ for the function $x^2 + y^3 - 2y = 3$ I'm still struggling a bit with the application of derivatives, but from what I understand I use the ...
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1answer
99 views

Does $\int_0^{\infty } \cos \left(e^x\right) \, dx$ converge?

So, there he is $$\int_0^{\infty } \cos \left(e^x\right) \, dx$$ Mathematica says it is a convergent integral, but I need some sort of a proof. How do we know that it is actually convergent? I've ...
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2answers
76 views

Integrable function on $[0,2]$ and its antiderivative

I got this question: Let $f$ be the integrable function defined on the interval $[0,2]$ by the rule: $f(x)= \begin{cases} 4x^3 & \text{if $0 \leq x \leq 1$} \\ x^2+2 & \text{if $1<x \leq ...
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2answers
78 views

Derivative of $\frac{\sin \coth x}{\csc \sqrt{e^{\log x}}}$

Derivative Problem: Hello, Ciao tutti, Buenos dias! I am trying find derivative with respect to x of function: $$ G(x)=\frac{\sin \coth x}{\csc \sqrt{e^{\log x}}}. $$ Derivativative rule for general ...
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1answer
65 views

How to find the following limit? $\lim\limits_{x\to 1}(2-x)^{\tan({x\pi}/2)}$

Find the limit $$\lim_{x\to 1}(2-x)^{\tan({x\pi}/2)}$$ I'm not sure of the first step, do I take the log? if so can I use L'H rule? I've never come across this yet and I can't find examples online. ...
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3answers
91 views

How to calculate $\int\sqrt{\frac{1-\sqrt x}{1+\sqrt x}} dx$?

How to calculate? $$\int\sqrt{\frac{1-\sqrt x}{1+\sqrt x}}\, \mathrm dx$$ I try to let $x=\cos^2 t$, then $$\sqrt{\frac{1-\sqrt x}{1+\sqrt x}}=\tan\frac t2,\; dx=-2\sin t\cos t\,\mathrm dt $$ so ...
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1answer
67 views

Using Poisson's integral formula

The problem asks to prove the following equality using Poisson's integral formula (or Poisson kernel, if I understood correctly from Wikipedia): $$\int_0^{2\pi} \frac{e^{\cos ...
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1answer
148 views

Double Integral $\iint_D\ (x+2y)\ dxdy$

$$\iint_D (x+2y)\ dxdy $$ If the area is range by $x=2,\ x=3,\ y=x,\ y=2x$, how to include the lines? How limits for integral will looks like? You mean something like this? ( I made mess) $$\iint_D ...
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3answers
39 views

Finding an exact solution to a difference equation

How would I solve an equation of the form: $u(n+1)=1/2u(n)+(1/3)^n$ when $u(0)=1$? I got an answer of the form $u(n)= c + \sum(1/3)^j*2^{j-1}$ but believe this is incorrect?
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4answers
99 views

Finding the sum of series $\displaystyle\sum \limits_{n=1}^{\infty} (-1)^{n}\frac{n^2}{2^n}$

I have some problems in finding the values of series that follow this pattern: $$\sum \limits_{n=0}^{\infty} (-1)^{n}*..$$ For example: I have to find the value of this series $$\sum ...
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3answers
63 views

Calculating limits using l'Hôpital's rule.

After a long page of solving limits using l'Hôpital's rule only those 2 left that i cant manage to solve $$\lim\limits_{x\to0}{\sqrt {\cos x} - \sqrt[3]{\cos x}\over \sin^2 x }$$ ...
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4answers
82 views

Integrating a term again and again

So if you have $f''(x) = 24x$ you know you want to integrate it, because it would look much better integrated, so now we have $f'(x) = 12x^2$, but it could still look better, so we integrate it to ...
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2answers
142 views

Using integral test on $\sum_{n=2}^{\infty}\frac{1}{n^2 \ln n}$

As stated in the title, I have to use the integral test on $$\sum_{n=2}^{\infty}\frac{1}{n^2 \ln n}$$ to prove that it is convergent but I am having trouble doing that $$\lim_{b\to\infty} ...