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

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1answer
43 views

Double Integration word problem

In a certain metropolitan area, the population is approximated by the function: $$P(x,t)=\frac{\ 7274e^{0.5t}}{1+x}$$ Where $x$ is the number of miles from the center of the city, and $t$ is the ...
7
votes
3answers
275 views

What is wrong with this proof of $0=1$?

I am trying to understand what is wrong with the proof posted here that $0=1$ (source): Given any $x$, we have (by using the substitution $u=x^2/y$) ...
0
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1answer
18 views

Finding a power series solution for a given differential equation and identifying the function represented by the power series.

Find a power series for the solution of the differential equation $y'(t)-2y(t)=0 ,\ y(0)=5$, and then identify the function represented by the power series. (I use the following information ...
1
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1answer
42 views

Evaluate Double Integration

Evaluate $\iint−3x^2 dA$ over the region in the first quadrant bounded by the hyperbola xy=16 and the lines $y=x$, $y=0$, and $x=8$. I have drawn a picture, but I am still a little unsure on what to ...
2
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1answer
112 views

How can I prove the integral $ \int_{1}^{x} \frac{1}{t} \, dt $ is $\ln x $ with this approach?

I have been trying to find a proof for the integral of $ \int_1^x \dfrac{1}{t} \,dt $ being equal to $ \ln \left|x \right| $ from an approach similar to that of the squeeze theorem. Is it possible to ...
0
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0answers
36 views

length of continuously differentiable curves

I saw that the length of a continuously differentiable curve $\gamma$ in $\mathbb{R}^n$ with $\gamma(t) \neq 0$ is defined as $\int_a^b |\gamma^{'}(t)|dt$, as can be found here ...
1
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2answers
29 views

function such that the sum of previous f(x) is smaller than f(x)

Just out of curiosity: is there a function $f$, such that $ \forall x, \sum_{x'<x} f(x') < f(x) $ sum or integral...
1
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2answers
24 views

Linear Approximation of a function at point a with change of x

So I know this problem is supposed to be very basic, but I cannot for the life of me get the answer my teacher and book gets. I would very much appreciate if a solution could be posted on how this ...
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3answers
91 views

proving that $g(x)=0$ has one real root

Given $g(x)=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots+\dfrac{x^{2n+1}}{(2n+1)!}$, Need to prove that $g(x)=0$ has one real root. I thought to use the fact that $e^x<T_{2n}(x)$ for all $x<0$, ...
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0answers
28 views

Calculus 2 - Rotating a region about an axis

I am having issues with the disk method and shell method when rotating a region around an axis. Example: The region bounded by $y=x^{\frac{1}{3}}, x=4y$, axis $x=3$ I am thinking that shell method ...
2
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2answers
24 views

creative method to obtain range of newton function ?!

I am searching for more proof that the range of $y=\frac{x}{x^2+1}$ is $ \frac{-1}{2}\leq y \leq \frac{+1}{2}$ these are my tries : domain is $\mathbb{R}$ first : $$y=\frac{x}{x^2+1}\\yx^2+y=x ...
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2answers
34 views

Chain rule for implicit functions

Let $F_1(x_1,x_2,x_3)=f(x_1,f(x_1,x_2,x_3),x_3)$ and $F_2(x_1,x_2,x_3)=f(x_1,x_2, f(x_1,x_2,x_3))$. Find $\displaystyle \frac{\partial F_i}{\partial x_j}$ for all $i=1,2$ and $j=1,2,3$. I know ...
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2answers
125 views

Find $\lim_{n \to \infty} n[(1+\frac{1}{n})^n - e]$

$\lim_{n \to \infty} n[(1+\frac{1}{n})^n - e]$ I let, $x = \frac{1}{n}$, then as $\lim_{x \to 0} \frac{1}{x}[(1+x)^\frac{1}{x} - e] = \infty$ L'hopital's: $\lim_{x \to 0} ...
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0answers
23 views

Newton's method, Hyperbola [closed]

A cable is freely suspended between two poles 100m apart on a flat ground and the height of each pole is 10m. The overhang required is 6.5m to ensure safety of traffic passing below the cable. The ...
0
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3answers
104 views

A simple looking integration : $\left(\frac{x^3}{1+x^5}\right)$

One of my friends gave me this problem about a week back and since then, I have been toiling to get a solution to this problem, but I just get stuck at some step. Can someone please tell me the steps ...
0
votes
1answer
34 views

Fastest way to write a number in the format of $8\times A\times B$

example split $5216$ into two numbers. Then write both of these numbers in the format of $8\times A\times B$. One parameter of the second number should not be bigger than $10$, while one parameter in ...
2
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5answers
98 views

Mistake in evaluating $\int\dfrac{dx}{\ln(x)}$

Evaluate: $$I=\int\dfrac{dx}{\ln(x)}$$ My attempt: $$$$ $$I=\int \dfrac{x'}{\ln(x)} dx$$Integrating by Parts,$$\dfrac{x}{\ln(x)}-\int\dfrac{x}{(\ln(x))'}dx$$$$=\dfrac{x}{\ln(x)}-\int ...
0
votes
2answers
41 views

Euler Cauchy equations, change of variables

To convert an euler cauchy: $x^{2}y''+pxy'+qy=0$ equation into a linear one we perfom the substitution $x = e^z$ from which we get: $$z=\log x$$ $$\frac{\mathrm{d} x}{\mathrm{d} z} = e^z =x $$ ...
0
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1answer
71 views

How to integrate $\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$?

How to integrate: $$\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$$ I don't really have a clue? Do I need to simplify it first somehow?
0
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1answer
36 views

Number of triangles having particular area

If $g:R\to \ N\cup\big\{0\big\}$ and $g(x)=n$,where $x$ represents the area of triangle joining the two fixed points and a variable point $R(p,q)$such that $\angle PRQ=\frac{\pi}{2}$ and $n$ ...
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0answers
37 views

why at saddle point is second derivative 0? [closed]

what exactly is SECOND DERIVATIVE and why does at saddle point it becomes 0
0
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1answer
22 views

A Geometrical Interpretation of the line integral of $f$ along $C$ with respect to $x$ and $y$.

I'm studying the line integral of a function along a curve $C$ with respect to $x$. Is the assertion as the following figure indicated true or false? I have read the questions Interpreting Line ...
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0answers
21 views

Find a Maclaurin series representation for $f(x)=3e^{-x^2/2}$ and approximate $R_n < \frac{1}{10000}$

I am tasked with the following: Find a Maclaurin series representation for $f(x)=3e^{-x^2/2}$ and use the power series to approximate $\displaystyle \int_{0}^{0.5}3e^{-x^2/2}$ with error ...
4
votes
2answers
63 views

Minimum value of $a+b$

If the graph of $f(x)=2x^3+ax^2+bx$ intersects the $x$-axis at three distinct points, then what is minimum value of $a+b$? Here $a$ and $b$ are natural numbers. My attempt: As the graph intersects ...
0
votes
2answers
22 views

Confusion regarding interval on which a function is increasing

The question is as follows: If the function $f(x)=\cos x$ is strictly increasing on the open interval $(0,\pi)$, where will it be increasing ? The answer to this question is $[0,\pi]$. I am a ...
6
votes
1answer
43 views

$|g(x)| \leq K \int_a^x|g| \ \ \forall x \in I$ [duplicate]

Let $I:=[a,b]$ and let $g: I \to \Bbb R$ be continuous on $I$. Suppose that there exists $K > 0$ such that $$|g(x)| \leq K \int_a^x|g| \ \ \forall x \in I.$$ Then $g(x) = 0\ \ \forall x \in I $. ...
2
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0answers
54 views

Upperbound for $\sum_{i=1}^n\frac{1}{x_i^2}$?

Suppose that $x_i>0$, $i=1,\ldots,n$. I'm looking for an upperbound (doesn't have to be particularly tight) of $\sum_{i=1}^n\frac{1}{x_i^2}$ in terms of some symmetric function of ...
2
votes
5answers
77 views

if we have $(f(x))^2 = 2 \int_0^xf, \ \forall x>0,$ then $f(x) =x \ \forall x\geq0$.

Let $f: [0, \infty) \to \Bbb R$ be continuous and $f(x) \neq 0 \forall x>0$. If we have $$(f(x))^2 = 2 \int_0^xf, \ \forall x>0,$$ then $f(x) =x \ \forall x\geq0$. We have $(f(x))^2 = 2 ...
1
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2answers
46 views

$n$-th term of an infinite sequence

Determine the $n$-th term of the sequence $1/2,1/12,1/30,1/56,1/90,\ldots$. I have not been able to find the explicit formula for the $n$-th term of this infinite sequence. Can some one solve this ...
0
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1answer
11 views

$F(x) := (n- 1)x-\frac{ (n- 1)n}{2}$ for $x \in [n- 1, n), n \in \Bbb N$ using this result to evaluate $\int_a^b[x]dx.$

Let $F(x)$ be defined for $x \geq 0$ by $F(x) := (n- 1)x- (n- 1)n/2$ for $x \in [n- 1, n), n \in \Bbb N$. Show that $F$ is continuous and evaluate $F'(x)$ at points where this derivative exists and ...
2
votes
2answers
50 views

Is there enough information given to solve this related rates problem?

This is the question from a practice exam: Suppose a pyramid has 4 lateral faces that are all equilateral triangles. Find the rate at which the volume of the pyramid is changing if each side of each ...
1
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1answer
64 views

(Putnam) Let $f:[1,3] \rightarrow \mathbb{R}$ such that $-1 \leq f(x) \leq 1 $ for all x and

The following is a Putnam math competition problem: Let $f:[1,3] \rightarrow \mathbb{R}$ such that $-1 \leq f(x) \leq 1 $ for all x and $ \int_{1}^{3}f(x)dx = 0 $. What is the max value of ...
3
votes
2answers
221 views

Solving Limits with L'Hospital's Rules

I am having difficulties solving this limit. I was given the question and equation: Try using L’Hospital’s Rules to evaluate the follwing limit: $$\lim\limits_{u \to \infty } \frac{u}{\sqrt{u^2 ...
0
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1answer
24 views

Derivatives of Implicit Functions (Abstract Case)

I have never been good at differentiation of implicit functions in cases when in a function is given, much less in abstract cases with composite functions. Hopefully someone can help me get started on ...
6
votes
0answers
24 views

limit of a region of integration in $\mathbb{R}^2$ approaches a line

I am trying to follow the derivation of derivatives in a paper published in some japanese journal but there seems to be a mistake in the proof. I will present the problem in 2D and in 2 variables so ...
2
votes
0answers
26 views

How does one prove that $ C_{\mathbb P} (\mathbb{I_r})$ is closed where…

How does one prove that $ C_{\mathbb P} (\mathbb{I_r})$ is closed where $$\mathbb{I_r}=[x_0-r,x_0+r]$$ and $$\mathbb{P}=\{(x,y): |y-y_0|\leq a, |x-x_0|\leq b\}\subset \mathbb G $$ where $\mathbb G-$ ...
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0answers
21 views

On utilizing the Leibniz rule of integration on a non compact interval.

I am following some slides that you can find here. At slide $\approx$ 24 a problem arises, to find $$\DeclareMathOperator*{\argmin}{\arg\!\min} \argmin_{\hat{y} } -\int_{-\infty}^{\hat{y}} (y ...
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2answers
29 views

Using Lagrange's Method in Finding Extreme Values (New to This Method)

Did I do this hw question correctly (at least in theory, I do not expect anyone to check my algebra work)? In particular, did I solve for lambda and plug lambda back into my equations for x,y, and z ...
2
votes
2answers
88 views

Is $\lim S_{n,m}=\sum_{k=1}^n({-1})^k{n\choose k}k^{-m}<\infty $ for $ n \to \infty$ and $m$ large?

Let $m$ be a fixed positive integer ($m>1)$ and let $$S_{n,m}=\sum_{k=1}^n({-1})^k{n\choose k}k^{-m}$$ be a partial sum of real series. My question here is : Is $\lim S_{n,m} <\infty $ as $ n ...
2
votes
1answer
48 views

Green's function for Helmholtz equation for the plane with a hole

That is find $G$ which satisfies \begin{align} (\nabla^2+k^2)G(\mathbf{x}, \mathbf{y},\omega) = \delta(\mathbf{x}- \mathbf{y}) \end{align} subject to $$\frac{\partial G}{\partial y_n} = 0 ...
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1answer
41 views

Decide if the following functions are not continuous on $(-\infty, \infty)$

Suppose $g(x)$ is continuous on $(-\infty, \infty)$. Determine if the following functions are or are not cont. on $(-\infty, \infty)$ and explain. a) $k(x) = \frac{x^2}{4 - (g(x))^2}$ b) $j(x) = ...
3
votes
2answers
122 views

Confused about calculating the area under the curve

What is the area under the curve of the following function? $f(x) = x² + 2x -3$ $x=-4$ $x=2$ Please, I'd like to see an image. Here is the graphic: https://www.desmos.com/calculator/oe0ja17spg
3
votes
3answers
68 views

Using the definition of derivative to find $\tan^2x$

The instructions: Use the definition of derivative to find $f'(x)$ if $f(x)=\tan^2(x)$. I've been working on this problem, trying every way I can think of. At first I tried this method: $$\lim_{h\to ...
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votes
2answers
72 views

Evaluate the integral: $\int\ x\ 2^{x^2}\ dx$ [closed]

$\int\ x\ 2^{x^2}\ dx$ using this formula: $\int a^x\ dx =\frac{a^x}{ln(a)}$ I have calculated the answer is $\frac{x^2\ 2^{x^2}}{x^2\ ln\ 2}+C$
0
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2answers
24 views

Find the rate of change of the frequency when D, L, σ and T are varied singly.

I'm reading Calculus made easy to learn the notation (I know derivatives with the limit/prime style) and also some integral calculus which I haven't seen at school yet. You can check it here: ...
1
vote
4answers
69 views

Evaluate the Integral: $\int \frac{\log_{10}\ x}{x}\ dx$

$\int \frac{\log_{10}\ x}{x}\ dx$ $du=x\ln10\ dx$ $\log_{10}x\ \ln10+ C$ Is this answer correct? If not what step should I take to convert the log into a term I can manipulate?
2
votes
2answers
59 views

Spivak's 'Calculus', 5-21(b): Is there an easier/shorter way?

Some personal background: I'll be going into my second year as a maths undergraduate in September of this year, and I'm currently working my way through Spivak's Calculus. While $\epsilon$-$\delta$ ...
2
votes
4answers
186 views

Can you help me make sense of this notation?

I am reading through my calculus textbook, and came across an algebra technique that I can't decipher. The author sets up $$e^x \sin(x) = (1 + x/1! + x^2/2! + x^3/3! + ...)(x -x^3/3! + ...)$$ Which ...
1
vote
6answers
86 views

Evaluate the Integral: $\int(x^5+5^x)\ dx$

$\int(x^5+5^x)\ dx$ I made the the terms within the parenthesis u $u=x^5+5^x$ $du=5x^4+5^xln\ 5$ $du=5x^4+5^x\ ln\ 5 dx$ $\frac{u}{5x^4+5^xln\ 5}\ du$ I am stuck at this point. Is there a ...
0
votes
2answers
37 views

Irrational Conjugate

I have irrational number: $\sqrt{3}-\sqrt{2}$ It's has 3 conjugate numbers: $\sqrt{3}+\sqrt{2}$ $-\sqrt{3}-\sqrt{2}$ $-\sqrt{3}+\sqrt{2}$ First variant - it's a standrart form for me. It's ...