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

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2
votes
4answers
67 views

Solving $\int\cos(x)\ln\left(\frac{9}{6-\sin(x)}\right)\,dx $

I have: $$\int\cos(x)\ln\left(\frac{9}{6-\sin(x)}\right)\,dx $$ I've tried by parts but without results, i don't know how to start, any tips?
2
votes
1answer
89 views

If $\lim_{x\to 0^+}f'(x)=-\infty$ and $\lim_{x\to 0^+}f''(x)=\infty$ it is always the case that $\lim_{x\to 0^+}\frac{f(x)}{f'(x)}=0$?

My teacher proposed the following question in our class: Let $f$ be a twice differentiable function in $\mathbb{R}^+$ such that $\lim_{x\to 0^+}f'(x)=-\infty$ and $\lim_{x\to 0^+}f''(x)=\infty$. ...
0
votes
1answer
43 views

Comparing the roots of two increasing functions

For any $0 \leq x \leq y \leq 1$, define $f(y;x):=\frac{y^2}{2}-\frac{2 y^3}{3}+\frac{y^4}{4} - \frac{x^2}{2} + \frac{x^3}{3}$ and $g(y;x):=\frac{y^2}{3}-\frac{2 y^3}{4}+\frac{y^4}{5} - \frac{x^2}{3} +...
0
votes
1answer
23 views

Area included in the graph of various functions

I've some problems recognizing which one is the area between various function. In this case i need to calculate the area between 3 lines and a curve, exactly between: $x+3,x^2-9,x=5 ,x=0$ I can't ...
0
votes
2answers
33 views

determine all (x,y) of the line Normal to an Ellipse

Hi everyone I have a question that requires me to determine the (x,y) coordinates of all points that intersects the x-axis on this ellipse when the normal line has a slope of -4, and I'm curious to ...
2
votes
2answers
59 views

For which point of $x+y+3z+k=10$, the expression $x^2+y^2+9z^2+4k^2$ is minimal? [on hold]

For which $x,y,z,k \in \mathbb R$ of $x+y+3z+k=10$ is the expression $x^2+y^2+9z^2+4k^2$ minimal?
0
votes
0answers
20 views

Confusion of limit when doing in two ways. [duplicate]

https://www.artofproblemsolving.com/community/u43127h1274925p6680268 This limit question is an eye opener for me.where is the problem in attachment .guide
1
vote
2answers
50 views

Symmetry in functions

What's the difference between $f(x)=f(a-x)$ and $f(x)=f(x-a)$ ? It's a pretty simple question maybe, but I'm unable to understand this one.
-3
votes
0answers
26 views

double integrals setting up using bounderies [on hold]

Evaluate $$\iint R(3xy+8)dA$$ where R is the region bounded by $$y=x^2, y=x+2$$ I am really struggling with these sorts of problems where there is a boundary. I have done good setting up and solving ...
0
votes
3answers
37 views

The value of an investment in Canada Savings Bonds is modelled by $A(t) = A_0 e^{0.0255t}$… Rest of question below.

The value of an investment in Canada Savings Bonds is modeled by $$A(t) = A_0 e^{0.0255t}$$, where A is the amount the investment is worth after $t$ years, and $A_0$ is the initial amount invested. At ...
0
votes
2answers
46 views

Integration by substitution starting from a result

I know: $$\int\frac{1}{x^2\sqrt{a^2-x^2}}=-\frac{\sqrt{a^2-x^2}}{a^2x}$$ and I want to calculate: $$\int\frac{1}{x^2\sqrt{3-16x^2}}$$ I think I have to apply a substitution but i don't know how to ...
3
votes
4answers
74 views

Is there a function $y=f(x)$ such that $\frac{d^3y}{dx^3} = f(x)$? [duplicate]

The exponential function $y=e^x$ is its own derivative, the hyperbolic functions $y=\frac{e^x+e^{-x}}{2}, y= \frac{e^x-e^{-x}}{2}$ are equal to their own second derivatives, and the trigonometric ...
0
votes
3answers
61 views

Setting Up a Double Integral Over a Region

Evaluate $$\iint_R(x+y)\ \mathrm dy\ \mathrm dx$$ where the region $R$ is bounded by $y=\frac{1}{9}x, x=6$ and the x−axis. can anyone show me how to set this up? I thought that it would integrated ...
0
votes
1answer
34 views

line integral of 3D vector field

Suppose I have a 3D vector field $\vec v(x,y,z)=(v_1,v_2,v_3)$ and I want to compute $$\int_C \vec{v}\cdot \vec n\, dS$$ where $C$ is the unit circle $C\equiv\{(x,y)\in\mathbb{R}^2\,:\, x^2+y^2=1\}$ ...
0
votes
0answers
6 views

Minimize the inner product of this tensor function

Minimize the following function: $ f(V) = || V \otimes V - U_1 \otimes U_2 ||$ where $U_1, U_2 \in SU(n)$ are fixed and we minimize over all $V \in SU(n)$. The norm is from the trace inner product. ...
0
votes
2answers
33 views

Usage of implicit function theorem for $f(x,y)=x^2+2xy-y^2-a^2$

Find the derivative of the following implicit function with the implicit function theorem: $$F(x,y)=x^2+2xy-y^2-a^2$$ My attempt for this task: $$F(x,y)=0 \Leftrightarrow (x,y)=(a,0)$$ ...
7
votes
2answers
89 views

Can the Substitution Rule be Interpreted as a “Change of Measure”?

I just started learning measure and rigorous integration theory on my own along side my calculus class and I've noticed that with the substitution rule, you have something that looks like this $$ \int^...
0
votes
1answer
22 views

How to know describe the set of levels for functions f(x,y)=c when c varies

Hey im having quite troubles trying to understand how to describes the set of levels in functions. In this problem any ideas? $$f(x,y)=x^2+y^2+1$$
0
votes
1answer
24 views

Double integral for a region

Evaluate the following double integral $$∫∫_R9x^3ydA$$ where R is the rectangle with vertices (0,0), (2,0), (0,6), and (2,6). can some one help show me how to set up this double integral? I have a ...
2
votes
0answers
19 views

For which $\alpha$ the function series $f_{n}(x)=n^\alpha xe^{-nx}$ has uniform convergence to $f(x)=0$ in $[0,1]$?

$$ f_n(x)=\frac{n^\alpha x}{e^{nx}} $$ I'm struggling with the idea of uniform convergence, so i'm not sure this solution is correct. The function is defined between $[0,1]$, positive and continuous. ...
0
votes
1answer
36 views

Laplace Transform

Question: Use Laplace Transform to solve the following differential equation $\ y''+y =sin(t); y(0)=1, y'(0)=-1 $ My try,where F(s) is the transform of f(t)=y(t) $F(s)= \frac{1}{(s^2+1)^2} + \frac{...
0
votes
0answers
50 views

Simplify $F(x) = \exp[-\ln^2x^h]$

I was wondering if the expression $F(x) = \exp[-\ln^2x^h]$ can be simplified even further? As you can see, the $\ln$ which is the natural logarithmic function is raised (and not its argument) to power ...
1
vote
2answers
28 views

Limits, and Continuity - Finding whether a function is continuous or not

$$ f(x) = \lim_{n \to \infty} \frac{\log(2 + x) - x^{2n}\sin x}{1 + x^{2n}} $$ then, check the continuity of the function at $x = 1$. I found this question in a text. After some thinking I ...
4
votes
2answers
98 views

Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$

Problem: If $f(x)$ is continous at $x=0$, and $\lim\limits_{x\to 0} \dfrac{f(ax)-f(x)}{x}=b$, $a, b$ are constants and $|a|>1$, prove that $f'(0)$ exists and $f'(0)=\dfrac{b}{a-1}$. This ...
1
vote
1answer
130 views

Show that $\lim\limits_{x\to\infty} f(x)$ exists if $f'(x) = \frac{1}{x^2+f(x)^2}$ and $f(1)=1$

Let $f(x)$ be a real differentiable function defined for $x\geq 1$ such that $f(1)=1$ and $f'(x)=\dfrac{1}{x^2+f(x)^2}.$ Show that $$\lim_{x\to \infty}f(x)$$ exists and is less than $1+\frac{\pi}{4}$ ...
0
votes
1answer
40 views

Solution for an inequality

I want to solve this inequality for $z$ $$(z+1) \left(1-e^x\right)-e^y\geq 0$$ where $-\infty <x\leq \log \left(\frac{1}{z+1}\right)$ and $-\infty <y\leq 0$. I am struggling because $z$ ...
0
votes
4answers
47 views

Problem 1, Ch. 6 in Piskunov's, Differential and Integral calculus

Find the curvature of the curve at indicated points $b^2x^2+a^2y^2=a^2b^2$ at $(0,b)$ and $(a,0)$ My attempt $\displaystyle{\kappa=\frac{|\frac{d^2{y}}{dx^2}|}{\left[1+\left(\frac{dy}{dx}\...
5
votes
2answers
444 views

Find the average value of the function…

Find the average value of the function $$F(x) = \int_x^1 \sin(t^2) \, dt$$ on $[0,1]$. I know the average value of a function $f(x)$ on $[a,b]$ is $f_\text{avg}=\dfrac1{b-a} \int_a^b f(x) \, dx$, ...
8
votes
4answers
364 views

Does the series $1-\frac12+\frac12-\frac1{2^2}+\frac13-\frac1{2^3}+\frac14-\frac1{2^4}+\frac15-\frac1{2^5}+\cdots$ converge or diverge?

$1-\frac{1}{2}+\frac{1}{2}-\frac{1}{2^2}+\frac{1}{3}-\frac{1}{2^3}+\frac{1}{4}-\frac{1}{2^4}+\frac{1}{5}-\frac{1}{2^5}+\cdots$ I've be trying to figure out how to write this series symbolically so I ...
1
vote
2answers
45 views

Which of the following subsets of $\Bbb R^2$ are homeomorphic to the set $\{(x, y) \in \Bbb R^2 \mid xy = 1\}$?

Which of the following subsets of $\Bbb R^2$ are homeomorphic to the set $\{(x, y) \in \Bbb R^2 \mid xy = 1\}$? $a. \{(x, y) ∈ \Bbb R^2 \mid xy − 2x − y + 2 = 0\}.$ $b. \{(x, y) ∈ \Bbb R^2 \...
-1
votes
0answers
27 views

At what point does the curve y=(1/x) have max curvature? What happens to the curvature as x approaches infinity?

I have found kappa and kappa prime however have no clue where to go from here. Thanks Kprime = 6/-x^4(1+(1/x^4)^3/2-12/x^3(1+(1/x^4)^1/2) all over (1+(1/x^4))^3/2
0
votes
1answer
14 views

How to solve $c_1 me^{mL}+ c_2 (-m)e^{-mL}= -(h/k)(c_1 e^{mL} +c_2 e^{-mL})$?

$c_1 me^{mL}+ c_2 (-m)e^{-mL}= -(h/k)(c_1 e^{mL} +c_2 e^{-mL})$ Where L is variable, e is constant of the base of natural logarithm and everything else is constant. Label the name of your method of ...
1
vote
3answers
41 views

Area between four functions

I'm not sure about calculating the area between: $f(x)=x+3$ ,$g(x)=x^2-9$,$k(x)=5$ and $y=0$ My idea , but i'm not sure about the first integral, is: $$$$ $\int_\sqrt{14}^05-(x-3)dx$ + $\int_0^25-(...
2
votes
1answer
51 views

Deriving an Expression for the Coordinates of a Partial Hollow Torus as a Function of the Angle

I'm modeling a shape that is best described as a partial, hollow torus. Here's what it looks like: http://i.imgur.com/3h4H5KQ.png In my application, the angle can vary from 0 to 85 degrees. I'm ...
2
votes
9answers
237 views

Integrate $\int\frac{x+1}{\sqrt{1-x^2}} \; dx$ without using trigonometric substitution

I want to solve: $$\int\frac{x+1}{\sqrt{1-x^2}} \; dx$$ I know how to solve this using trigonometric substitution, but how can I solve the integral in an other way ?
1
vote
2answers
40 views

Verification of rolle's theorm [closed]

If we have to verify rolle's theorm for a question, if we show the existence of the point without showing the applicability of rolle's theorm, Are we done? consider the truth table of p implies q too ...
0
votes
2answers
28 views

Finding the tangent line of a piecewise-defined function

I have $ f(x) = \begin{cases} \frac{e^x-1}{log(x+1)} & \quad \text{if } x>-1 ,&x\not=0 \\ 1 & \quad \text{if } x= 0\\ \end{cases} $ I need the tangent line of ...
0
votes
1answer
52 views

Let $F(x) = \int_{\sin(x)}^{\cos(x)} e^{t^2 + xt} dt.$ Find $F'(0)$.

Let $F(x) = \int_{\sin(x)}^{\cos(x)} e^{t^2 + xt} dt.$ Find $F'(0)$. I am reviewing for the mGRE and I came across this problem. My first thought was to simple apply the fundamental theorem of ...
2
votes
1answer
36 views

Does it follow that two finite positive measures are the same?

Suppose $\mu$ and $\nu$ are finite positive measures on the Borel $\sigma$-algebra on $[0, 1]$ such that $\int f\,d\mu = \int f\,d\nu$ whenever $f$ is real-valued and continuous on $[0, 1]$. Does it ...
3
votes
3answers
83 views

Evaluation of $\lim_{x\to 0} \frac{(1+x)^{1/x}-e+\frac{ex}{2}}{x^2}$

Evaluate the following limit: $$L=\lim_{x\to 0} \frac{(1+x)^{1/x}-e+\frac{ex}{2}}{x^2}$$ Using $\ln(1+x)=x-x^2/2+x^3/3-\cdots$ I got $(1+x)^{1/x}=e^{1-x/2+x^2/3-\cdots}$ Could some tell me how to ...
1
vote
1answer
32 views

Meaning of limits, $\int_{\max(0, t-5)}^{\max(0,t-3)} e^{-3s} \, ds $?

What does it mean to have $\max(0,t-3)$ and $\max(0,t-5)$ in the limits? Is it a abbreviation? $$ \int_{t-5}^{t-3} e^{-3s}u(s) \, ds = \int_{\max(0, t-5)}^{\max(0,t-3)} e^{-3s} \, ds $$ Source of ...
-1
votes
3answers
50 views

Rats and snakes populations using calculus

On a certain island, at any given time, there are R hundred rats and S hundred snakes. Their populations are related by the equation: $$(R−13)^2+16(S−20)^2=68$$ What is the maximum combined number of ...
18
votes
3answers
432 views

The entry-level PhD integral: $\int_0^\infty\frac{\sin 3x\sin 4x\sin5x\cos6x}{x\sin^2 x\cosh x}\ dx$

I hope you find this integral interesting. Evaluate $$\int_0^\infty\frac{\sin\left(\,3x\,\right)\sin\left(\,4x\,\right) \sin\left(\,5x\,\right)\cos\left(\,6x\,\right)}{x\,\sin^{2}\left(\,x\,\...
1
vote
1answer
17 views

If $f'(x)\geq c$, show that $f(x)\geq f(0)+cx$ if $x\geq0$ and $f(x)\leq f(0)+cx$ if $x\leq0$

Suppose that $f$ is differentiable on Reals and that there is a positive number c such that... If $f'(x)\geq c$ for all $x$, show that $f(x)\geq f(0)+cx$ if $x\geq0$ and $f(x)\leq f(0)+cx$ if $x\leq0$...
21
votes
2answers
2k views

If $f(x)$ has a vertical asymptote, does $f'(x)$ have one too?

So here is what I understand: If $f(x)$ is increasing/decreasing, then its derivative $f'(x)$ is positive/negative and... If $f(x)$ is increasing/decreasing, then the derivative of $f'(x)$ (...
0
votes
1answer
20 views

Value of $V/(250\pi)$

A cylindrical container is to be made from certain solid material with the following constraints: It has fixed inner volume $V$ mm${}^3$ ,has a $2$ mm thick solid wall and is open at the top. The ...
1
vote
1answer
44 views

Finding $\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $

As the question says, $$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $$ where a is a constant, $a>0$.
0
votes
4answers
51 views

Area of circle (double integral and cartesian coordinates)?

I know that the area of a circle, $x^2+y^2=a^2$, in cylindrical coordinates is $$ \int\limits_{0}^{2\pi} \int\limits_{0}^{a} r \, dr \, d\theta = \pi a^2 $$ But how can find the same result with a ...
1
vote
1answer
30 views

Building volume using lagrange multipliers

A rectangular building with a square front is to be constructed of materials that costs 20 dollars per square foot for the flat roof, 20 dollars per square foot for the sides and the back, and 14 ...
26
votes
2answers
361 views

If $f$ is a smooth real valued function on real line such that $f'(0)=1$ and $|f^{(n)} (x)|$ is uniformly bounded by $1$ , then $f(x)=\sin x$?

Let $f : \mathbb R \to \mathbb R$ be a smooth ( infinitely differentiable everywhere ) function such that $f '(0)=1$ and $|f^{(n)} (x)| \le 1 , \forall x \in \mathbb R , \forall n \ge 0$ ( as usual ...