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

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1
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1answer
22 views

Find convergence domain of integral

I need to find convergence domain of $$\int_1^2 \! \frac{\ln(x-1)}{(4-x^2)^p} \, \mathrm{d}x$$ I've tried to use estimates like $\frac{\ln(x-1)}{(4-x^2)^p} < \frac{1}{(4-x^2)^p}$ and change of ...
1
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1answer
30 views

Prove that if $\lim_{x\to a} f(x) =0$ and $g(x)$ is bounded, then $\lim_{x\to a} f(x)\cdot g(x)=0$ [duplicate]

Question: Prove that if $\lim_{x\to a} f(x) =0$ and $g(x)$ is bounded, then $\lim_{x\to a} f(x)\cdot g(x)=0$ Attempt: I don't really understand the meaning g(x) is bounded. I did this problem in ...
0
votes
1answer
17 views

Zero functions on open interval

Are there non-constant differentiable functions that are zero on an open interval of real line? I've tried using the product integral: $$ f(x) = \exp(\int_0^1 \log(x-u) \mathrm{d}u ) = \frac{x^x ...
0
votes
0answers
9 views

Gradient Vector of Homogeneous Functions

I've been given the definition $f:\mathbb{R}^n \to \mathbb{R}^m$ is homogeneous of degree k if $f(\lambda x)=\lambda^kf(x)$ $\forall x\in\mathbb{R}^n, \lambda>0$ and asked to show $<\nabla ...
1
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0answers
9 views

Problem about find the extreme of a function (Multipliers of Lagrange)

Good morning, i have a problem with this: Find the maximum and minimum distances from the origin to the curve $g\left(x,y\right)=5x^{2}+6xy+5y^{2}$ I make this: Function to optimize: ...
0
votes
1answer
29 views

Value of $f'(3^{1/5})$ from the given differential equation

A function $y=f(x)$ satisfies $$xf'(x)-2f(x)=x^4 f(x)^2$$ and given that $f(1)=-6$ and $x$ belongs to all positive real numbers then prove that $f'(3^{1/5}) =8$ I have tried in this way...... Given ...
-1
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1answer
41 views

Evaluate Double Integral [on hold]

Evaluate the following double integral: $$\int _0^{2\pi }\int _0^1 \left(x - 2x^2 \sin\left(y\right) \cos\left(x^2+1\right)\right) \text dx\,\text dy$$ Please note that the answer is ${\pi}$
1
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0answers
13 views

I have an problem with the function to optimize with lagrange multipliers

I need help with the restriction of the problem, because i cannot find the function to optimize. The problem: Find the maximum and minimum distances from the origin to the curve ...
1
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1answer
30 views

What is the best way to solve $\lim_{n\to \infty}{(e^{i \theta})^n}$?

What is the best way to solve the limit: $\lim_{n\to \infty}{(e^{i \theta})^n}$ $\theta$ is fixed, but you must have a care for cases $\ \theta > 0 , \ \theta = 0 , \ \theta < 0.$ There ...
-1
votes
1answer
52 views

Integral of $\int \frac{xe^x}{\sqrt{1+e^x}} dx$ [on hold]

I need help to solve this: $$\int \frac{xe^x}{\sqrt{1+e^x}} dx$$
2
votes
2answers
98 views

How can $\int_a^b f(x)dx $ exist if either $f(a)$ or $f(b)$ does not exist?

In class, I came across the integral: $$\int_0^1 \frac{dx }{\sqrt{1-x^2}}=\frac{\pi}{2}$$ This is easy enough to prove using a substitution or by recalling the derivative of $\arcsin x$. However, ...
0
votes
0answers
8 views

Solve inequality of composite function

There is a Calculus problem where I got stuck and need a hint to proceed: Let $f(x)=e^x+ \ln(x)-3$ , $x>0$. Solve the inequality $f(\ln(x)-1) <3 $. We can see that $f$ is increasing and that ...
0
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2answers
28 views

$lim_{j->\infty} (j^j)/((j+1)^{j})$ [duplicate]

Can someone please explain this limit: $$lim_{j\rightarrow\infty} \frac{j^{j}}{(j+1)^{j}}=\frac{1}{e}?$$ I got it from this series: $$\sum_1^{\infty}\frac{j!}{j^j}.$$
0
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2answers
16 views

How can I find the limits of this iterated polar integration?

How can compute the area of the triangle whose corners are at the origin, (1,0) and (1,1). I solved this with r integral first but I could not find the correct limits for theta integral first order. ...
1
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2answers
46 views

Why do I get different results for the same integral?

The variables $ a, b, s, c $ are constants, so: $$ \int \left ( a \cos(s + cx) - b \sin(s + cx) \right ) dx = \frac{a\sin(s + cx) + b\cos(s + cx)}{c} +C $$ But if $c=0$ then: $$ \int \left ( a ...
0
votes
1answer
15 views

Cost Functions and differentiation

A firm uses capital $K$,to generate output, $Q$, according to the production function $Q = K^a$.The input price is $r$ and fixed costs are $C_0 > O$. a) Derive the firms cost function in terms of ...
3
votes
4answers
62 views

$\lim_{n \to \infty} (\frac{(n+1)(n+2)\dots(3n)}{n^{2n}})^{\frac{1}{n}}$ is equal to :

$\lim_{n \to \infty} (\frac{(n+1)(n+2)\dots(3n)}{n^{2n}})^{\frac{1}{n}}$ is equal to : $\frac{9}{e^2}$ $3 \log3−2$ $\frac{18}{e^4}$ $\frac{27}{e^2}$ My attempt : $\lim_{n ...
0
votes
1answer
42 views

What does the symbol “$|_{\epsilon=0}$” mean with a derivative? [duplicate]

What does the notation "$|_{\epsilon=0}$" at the bottom of the derivative mean?
0
votes
1answer
46 views

How do I differentiate an improper integral?

I would like to differentiate a function of the type $\int_x^\infty f(x, t) dt$ with respect to $x$ ($f$ real or complex valued). Does differentiation under the integral sign apply? What are better ...
0
votes
1answer
27 views

How to integrate an equation with multiple non-independent variables

I'm a little lost with this particular equation, I have three variables which need to be integrated and can't quite wrap my mind to get the correct result. I have this: $$ \frac{dH}{dt}=8\pi ...
-2
votes
1answer
28 views

Calculate limit of function 4 [duplicate]

Why \begin{equation} \lim_{x\to-\infty}\sqrt{x^2-x-1}-x=+\infty, \end{equation} Thanks
3
votes
1answer
92 views

Is there a closed form of this integral $ \int_0^\infty \sin(xe^{-x})dx\, $? [on hold]

I have tried by subsititution method and it got more complicate than before. Can anyone help me to evaluate this integral. $$ \int_0^\infty \sin(xe^{-x})dx\,. $$
1
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0answers
52 views

How to derive this identity: $\lim _{x\to \infty} (1 + f(x))^{\frac{1}{g(x)}} = \mathrm e^{\lim_{x \to \infty} \frac{f(x)}{g(x)}}$ [on hold]

How to prove this identity: $\lim _{x\to a} (1 + f(x))^{\frac{1}{g(x)}} = e^{\lim_{x \to a} \frac{f(x)}{g(x)}}$ I've found references to this identity but no derivation EDIT:Here the identity only ...
2
votes
1answer
21 views

Having trouble deriving the symbols used in a quadratric approximation problem.

I'm refreshing my calculus by studying MIT OCW's Single Variable Calculus course online. The problem is 2A-11, part of Unit 2 "Applications of Derivatives". It's a problem dealing with quadratic ...
-1
votes
1answer
64 views

How can I find $dy/dx$? [on hold]

What does $dy/dx$ represent? $y = x^5$ $y = x+5$ $y = b$, $b$ is a constant Am I supposed to divide the $y$ by $x$? So, $\frac{y}{x^5}$ and $\frac{y-5}{x}$? If so, what do I do with the third one? ...
0
votes
1answer
27 views

How can I find left and right limits as $x$ approaches nonremovable discontinuity?

The book says to "simplify, find crucial numbers, determine sign in intervals then determine limits." $f(x) = \frac{(x-2)(x-1)}{(x-3)(x-2)}$ I don't understand what the directions mean. I know that ...
1
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0answers
25 views

Question regarding terminlogy and wording of the derivative

When doing calculus, we typically say that we "take the derivative of a function f(x)." However, rigorously, f(x) is not a function but rather the value of the function f evaluated at x. Thus, in ...
1
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1answer
61 views

Solving $\int_0^2\int_{y/2}^1 ye^{-x^3}\,dx\,dy$ [on hold]

So I have to solve the integral above but I wasn't really sure how to start? I know it is integrable since it is an integral over a nice boundary and that I can solve it using iterated integrals but ...
0
votes
0answers
25 views

A knotch in a tree : do you prefer geometry or integral calculus of volumes to solve? [on hold]

A notch is cut in a cylindrical vertical tree trunk. The notch penetrates to the axis of the cylinder and is bounded by two half-planes. Each half-plane is bounded by a horizontal line passing through ...
2
votes
1answer
29 views

Order of remainder term in Taylor series approximation

I'm having trouble verifying a bound on the remainder term of a Taylor series approximation. I have a $C^\infty$ function $f$ of compact support. Using the two-term Taylor series for $f$ centered at ...
1
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1answer
18 views

Equation of a Parabola given starting coordinates, starting angle, and vertex height

I would like to find the equation of a downward-facing parabola given a starting point (x,y) and angle L, and the height of the vertex (k). I started with three equations: $y = ax^2 + bx + c$ ...
-3
votes
1answer
56 views

How to find a general solution using substitution for $\frac{dy}{dx}$ + $e^{2x−3y}$ = −$e^x$ [on hold]

Using the substitution $y=\frac13 \log f$ to find general solution for $\frac{dy}{dx}$ + $e^{2x−3y}$ = −$e^x$
1
vote
1answer
39 views

Dude with taylor polynomial

Good night, i'm working with an problem of polynomial taylor, but i have a problem with the residue. Get a quadratic approximation $f\left(x,y\right)=\sin\left(x\right)\sin\left(y\right)$ near the ...
2
votes
1answer
34 views

Integrating variation of error function: $\int_1^2e^{-nx^2} dx$

Show that $$\lim_{n\to\infty} \int_1^2e^{-nx^2} dx = 0.$$ After much googling, I learned that I am working with a variation of the error function! Yay. I've never heard of it in my life and I ...
2
votes
2answers
55 views

Does an analytical form exist for the following integral

I have an integral $$f(n,a)=\int_0^{2\pi}\mathop{\mathrm{d}x}\frac{\cos(nx)\cos^2x}{1-a\cos^2x},$$ where $n$ is an even integer and $0<a<1$ is a real number. Does an analytical form exist for ...
1
vote
1answer
23 views

Find the curl of the vector field G

Find the curl of the vector field: $\underline{G}=(8r^7x-5r^3y)\underline{i}+(-8r^7y+5r^3x)\underline{j}$ where $r=(x^2+y^2)^\frac{1}{2}$ Since r is in the vector field, does it require calculation ...
1
vote
4answers
113 views

Show that if $f(x) > 0$ for all $x \in [a,b]$, then $\int_{a}^b f(x) dx > 0$

Assume $f$ is Riemann integrable and nonnegative over $[a,b]$. Show that if $f(x) > 0$ for all $x \in [a,b]$, then $\int_{a}^b f(x) dx > 0$. This seems very obvious to me. One thing I would ...
0
votes
1answer
26 views

Use Algebraic Means to Determine Limit $\lim_{x \to -2} \frac{2 - |x|}{2 + x}$

I'm attempting to understand a particular step taken on our answer sheet for the following problem. Find the Following Limit using Algebraic Means: $$\lim_{x \to -2} \frac{2-|x|}{2+x}$$ The answer ...
0
votes
3answers
22 views

How to determine intervals where $f$ is greater than $g$?

I have two functions, $f(x) = 2x$ and $g(x) = \frac{x^3}{3}$. I solved for $x$ where $f = g$, finding $x = \pm 6^{1/2}$, then solved for $x$ where $f > g$, $x > \pm 6^{1/2}$, and where $f < ...
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votes
2answers
54 views

Periodic functions $f(x)=\sin{x}+\cos{x}$ [on hold]

Find period of functions $$f(x)=\sin{x}+\cos{x}$$and $$f(x)=|\sin{x}|$$ I need some hints and directions.
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votes
4answers
64 views

how to solve this integral $2\pi\int_1^a {1\over x} {\sqrt {1+{1\over x^4}}} dx $ [on hold]

Can you help me to solve this integral? $$2\pi\int_1^a {1\over x} {\sqrt {1+{1\over x^4}}} dx $$
4
votes
4answers
108 views

Finding out a limit using Taylor series.

So the limit is the following: $$\lim_{x \to 0}{\frac{x^2-\frac{x^6}{2}-x^2 \cos (x^2)}{\sin (x^{10})}}$$ Expansions for $\sin(x)$ and $\cos(x)$ are given: $$\sin x = x-\frac{x^3}{3!} + ...
1
vote
1answer
39 views

The so-called error function defined as: $erf(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^{2}}dt$

The so-called error function is defined as: $$erf(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^{2}}dt$$ show that the function $y(x) = e^{x^2}erf(x)$ satisfies the differential equation: ...
0
votes
2answers
19 views

Find fundamental set of solutions for 2nd order ODE?

I am asked to find the fundamental set of solutions ${y_1,y_2}$ for the equation $$y''-25ty'+25y=0$$ I am told that $y_1=t$ is a solution, how would I go about finding $y_2$, and if $y_1$ was not ...
0
votes
1answer
29 views

Critical Point when not in domain of $f(x)$

I am given the function $f(x)= x/(x^2-1)$. I find that the derivative is: \begin{equation} \frac{x^2+1}{(x^2-1)^2} \end{equation} So, I note that $f'(x)$ is undefined at $x = \pm 1$. However, ...
-1
votes
0answers
55 views

Riemann sum over a partition equence [on hold]

For each positive integer $n$, consider the following integral: $$\int\limits_0^{\infty}x^ne^{-x}\,dx=n!$$ Use a Riemann sum over the partition sequence ...
2
votes
2answers
41 views

Is $f(x)=\sum_{n\geq 1}\frac{(-x)^n}{n^2+1}$ convex at $x=0$?

Let $\sum_{n=1}^{\infty}\frac{(−1)^n}{ n^2+1} x^n$ be the Taylor series of $f(x)$ about $0$. Then, is it that, $f(x)$ is concave up at $x = 0$?
0
votes
1answer
29 views

Using calculus to derive a rate of change given two related rates of change

The surface area of a closed cylinder is given by $$A = 2\pi r^2 + 2\pi rh$$ where h is height r is the radius of the base. At the time where the surface are s increase at the rate of $20\pi\space ...
2
votes
2answers
59 views

Why is this integral not correct?

The book says, "why is this not correct?" $$\int_{-2}^1 x^{-4}\,dx=\frac{x^{-3}}{-3}\Big|_{-2}^1=\frac{-3}{8}.$$ It looks correct to me. It's just the fundamental theorem of calculus, right? What is ...
0
votes
4answers
52 views

Does there exist a function with following properties?

Does there exist a differentiable, everywhere concave down function $f(x)$, defined on the whole real line, such that $lim_{x→−∞}f(x) = 1$ and $lim_{x→∞}f(x) = −1$? Give an example justifying that it ...