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

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0
votes
1answer
28 views

If $f(x) = x^3 + 4x^2 + ax + 1$ is a monotonically decreasing function of $x$ in $(-2, -\frac{2}{3})$ then find $a$

If $f(x) = x^3 + 4x^2 + ax + 1$ is a monotonically decreasing function of $x$ in the largest possible interval$ (-2, -\frac{2}{3})$ then find $a$. My work: $$f'(x)=3x^2+8x+a$$ For it to be ...
2
votes
4answers
42 views

How to simplify trigonometric functions with having higher multiples of $x$ if the function is complex?

$$ \int \frac{(\cos 9x + \cos6x)}{2 \cos 5x - 1} dx $$ I know that it simplifies to $ \cos x + \cos 4x $ but I have no idea how to do that. I tried expanding $\cos 9x $ and $\cos 6x$ by using the ...
0
votes
0answers
32 views

How can I solve the antiderivative with upper limit as another integral using the fundamental theorem of calculus

How can I approach to solve $d(\int_{0}^{\int_{0}^{x}{1+e^{-t^2}}dt}{cos^2u\over 1+u^2}du)\over dx$ using the fundamental theorem of calculus. Can I substitute U with all the upper limit like this: $$...
0
votes
0answers
5 views

How to give a formula of the perimeter of a $r$-neighborhood of a smooth set in $2D$?

Let $A$ be a simply connected open set in $\mathbb{R}^2$ with smooth boundary. Define $$A^r := \{x \in \mathbb{R}^2: d(x, A) \le r \},$$ where $d$ is the distance function. Let $P$ denote the ...
0
votes
0answers
21 views

An alternative formula for a second order Taylor expansion?

I read in a book that the second order Taylor expansion of a function (around $x^0$) can be written as: $$f(x)=f(x^0)+\sum_{j=1}^n df(x^0)/dx_j*(x_j-x_j^0)+\sum_{j=1}^n\sum_{i=1}^nd^2f(x^1)/dx_idx_j*(...
0
votes
1answer
58 views

How to solve the inverse function of $f(x) = x^a + x^b$ [on hold]

I don't even know where to start. I have tried everything. I even googled it.
-3
votes
0answers
36 views

What will be the value of x? [on hold]

Find $\lim_{x\to \frac 52} \lfloor x\rfloor$ , where $\lfloor\,\,\rfloor$ denotes the greatest integer function , Please do not close this question or downvote .
1
vote
0answers
22 views

Integral of determinant of Jacobian of a $C^{1}$ application.

Let $f \in C^{1}(\mathbb{R}^{n}; \mathbb{R}^{n})$, suppose that exists $r>0$ such that $f(x)=0$ if $|x|>r$. Prove that exists $k>0$ such that $\int_{\overline{B_{k}(0)}} det Jf(x) dx=0$. My ...
0
votes
1answer
34 views

Difficulty setting up an iterated integral

I am trying to integrate the function $\frac{1}{\sqrt{2y-y^2}}$ over the region in the first quadrant bounded by $x^2=4-2y$. Given that this region is between bounded by an convex parabola and in the ...
-1
votes
1answer
18 views

Find the volume of the solid generated by rotating about the x-axis the region bounded by the curves

I'm new to this site, and this will be my last question post. My boyfriend has a few problems that he's unable to complete because of a family emergency and I have decided to try and help him. His ...
0
votes
2answers
29 views

Find the depth of the propane in the tank when it is filled to one-quarter of the tank's volume.

I'm new to this site. My boyfriend has some calculus problems that he's unable to complete due to a family emergency, and so I am trying to help him. His professor hasn't emailed him back about ...
3
votes
1answer
21 views

Finding the volume of a solid generated by revolving around the line y=6

I'm new to this site. My boyfriend is unable to complete these problems due to a family emergency and I was going to try to help him because they are due in a few hours and his professor hasn't ...
1
vote
0answers
20 views

Reading an article that takes the cross product of a vector and time.

I have this equation in the form: $$V(t)=ϕV(t−1)+r×t+c+\epsilon $$ Why is $r$ crossed with $t$. The variable $t$, in the article (shown below) is time which I believe is a scalar. I suppose time ...
3
votes
1answer
86 views

How do I prove this $\int_{0}^{\infty}{e^{-x^n}-e^{-x^m}\over x\ln{x}}dx={\ln{\left(m\over n\right)}}?$

How do I prove this $$\int_{0}^{\infty}{e^{-x^n}-e^{-x^m}\over x\ln{x}}dx=\color{blue}{\ln{\left(m\over n\right)}}.\tag1$$ I know of the standard integral $$\int_{0}^{1}{x^m-x^n\over \ln{x}}dx=\...
0
votes
0answers
24 views

$\nabla u \in L²(B_1(0)) \Rightarrow u \in H^1(B_1(0))$ [on hold]

how do I show the following: $\nabla u \in L²(B_1(0)) \Rightarrow u \in H^1(B_1(0))$. Thanks for your answers.
0
votes
2answers
32 views

Can I have a net equation for distance traveled?

So I was working on a problem from my Calculus textbook and I wanted to know why the approach I took to solve the problem did not work. The problem is as follows: A subway train travels 400 feet ...
0
votes
1answer
18 views

What to write when computing double integrals in order to minimize errors

This is a bit of an odd question, but I am learning how to compute lots of calc things from books so I don't have the benefit of watching how a teacher does it. I tend to be very error prone when ...
1
vote
1answer
19 views

rotating a linear equation about the $x$ axis

I am really having a hard time doing this problem its taking a lot of work to try to figure this out. Find the volume of the solid generated by rotating about the x-axis the region bounded by the ...
1
vote
1answer
27 views

Evaluating a Gaussian integral on $\mathbb{R}^{n}$.

For $t>0$ I want to show that $$\frac{1}{(4\pi t)^{n/2}}\int_{\mathbb{R}^{n}}e^{\frac{-\|x\|^{2}}{4t}}dx=1$$ So far, I have $$\begin{aligned} \frac{1}{(4\pi t)^{n/2}}\int_{\mathbb{R}^{n}}e^{\...
0
votes
1answer
17 views

volume of solid around y axis rotation

I am working on this problem and need some help figuring it out. I will give the problem and then give the ways i tried to solve it: Find the volume of the solid generated by revolving the region ...
1
vote
2answers
61 views

Computation of the definite integral $ \int _{\log _e\left(8\right)}^{\infty }\sqrt{\frac{e^x+1}{e^x-3}}\: $

Good evening to everyone! I have the following integral: $$ \int _{\log _e\left(8\right)}^{\infty }\sqrt{\frac{e^x+1}{e^x-3}}\: $$ and I don't know how to study its convergence. Here's what I tried: $$...
0
votes
0answers
25 views

More elegant way of expressing Lagrange polynomial.

On wikipedia the Lagrange polynomial looks messy, I think I found a elegant way to express the Lagrange polynomial: Like this (where $\Delta L(x)$ represents $L(x+1)-L(x)$: $$L(x)=\sum_{i=0}^{\infty}...
0
votes
1answer
13 views

How is the class related to derivability?

Good evening to everyone. I have a question where they require me to find the derivability. After I read the answer sheet I saw that the function has the class $ C^1 $. How is the class related to ...
-1
votes
0answers
15 views

How long is 1 turn of conical helix? [on hold]

The cone has diameter 120mm and angle of slant is 31°. What is the lengh of 1 turn of conical helix (angle 8°)?
1
vote
1answer
46 views

Is L a linear transformation?

I have to prove is L is a linear transformation on the field $P_3(R)$, if it is then I'd have to find the matrix of the linear transformation from the standard base vectors $p(1),p(x),p(x^2),p(x^3)$. ...
1
vote
2answers
27 views

Comparison of two slopes to get the best value

I'm trying to get something in excel that may be solvable by coming up with the proper formula. I believe it's a comparison of two slopes. The theory comes an economies of scale: How can I spend the ...
2
votes
0answers
33 views

A sufficient condition on a real smooth function

Let $f : [0, \infty) \to \mathbb{R}$ be a smooth function. I would like to find a sufficient condition on $f$ in order to have that $$ \liminf_{t\rightarrow \infty} \int_0^t \Big(\frac{t - s}{s} \Big)...
1
vote
3answers
78 views

Why is dividing by $dx$ or some other differentitator in an integral considered taboo?

Forgive my ignorance but I remember when I was learning calculus, I remember that when we integrate, we always multiply the differentiator to $F(x)$. However, it was never explained to me why we ...
0
votes
2answers
28 views

Differential calculus by Piskunov - Chapter 4, Problem 50

I can't seem to solve this limit question correctly. It would nice, if someone could help with a neat solution. $\begin{align}L &=\lim_{x\rightarrow{\pi/2}}{(\cos{x})^{(\pi/2-x)}} \end{align}$ ...
0
votes
2answers
31 views

Given a parabola $y= x^2 - 2x + 2.$ The tangent $r$, which is not horizontal, goes through the point $(2,1).$ What is the slope of $r$?

Given a parabola $y= x^2 - 2x + 2$. The tangent r, which is not horizontal, goes trough the point (2,1). What is the slope of r? I've already used the formula $\lim_{x\to 2}$$\frac{f(x)-f(2)}{x-2}$....
-2
votes
0answers
17 views

Price Elasticity of demand [on hold]

Vik and Fleet produce trainers in the sports-shoe market. For one of their main products they have the following demand curves: Vik PV ¼ 175 1:2QV Fleet Pf ¼ 125 0:8Q f where P is in £ and Q is in ...
1
vote
1answer
72 views

Closed form of the series.

Write the following series in closed form $$\frac{1}{a-1}+\frac{2}{a-2}+\frac{2}{a-3}+\frac{1}{a-4}+\frac{1}{a-5}+\frac{2}{a-6}+\frac{2}{a-7}+\frac{1}{a-8}+\frac{1}{a-9}+\frac{2}{a-10}+\frac{2}{a-11}...
0
votes
0answers
17 views

Determining the spherical coordinate parametrization of an area in $\mathbb{R}^3$.

Let $\epsilon_0,\epsilon_1,\epsilon_2$ be three iid random variables with a symmetric distribution and let $\lambda>1$. I want to calculate $$ P(\epsilon_0>0 \quad;\quad\epsilon_1>-\lambda\...
1
vote
1answer
25 views

Compute expected received balls from boxes

I have 6 boxes: $A,B,A',B',C \text{ and } D$. The box $A$ has $n_1$ red balls that are numbered from $1, \cdots, n_1$. The box $B$ has $n_2$ green balls that are numbered from $1, \cdots, n_2$. Make a ...
0
votes
0answers
18 views

Any simplifications possible for the following integral?

Here is an integral $$\int_0^a\mathop{\mathrm{d}z}f(z)\mathrm{e}^{\int_0^zf(z')\mathrm{d}z'+\int_0^zg(z')\mathrm{d}z'},$$ where all functions are real and positive and $a>0$. Is there any ...
1
vote
0answers
41 views

Calculating Riemann sum limit.

What is the minimal or the most basic sufficient condition for $f$ that this limit exists: $$\lim_{n \to \infty} \sum_{i = 1}^n \frac {f\left( \frac{i^3 - i + n^3}{n^3} \right) - f \left( \frac{i^3 ...
0
votes
4answers
33 views

How to know if N is normal to the plane or not?

I have N = <1 , -2 , 3 > and the plane 2x - 4y + 6z = 5 How do i know if N is normal to the plane or not? please be detailed with the answer as i'm very lost.
2
votes
1answer
101 views

Proof from Calculus 1

Last days, from going into a website of the university of Pisa, I found an exercise given in the previous exams, in 1999. The problem was like: Given a continuous function $f$ in $\mathbb R$, and ...
0
votes
1answer
37 views

How this integral is evaluated $\frac{\partial }{\partial x}\left(\int _y^x\cos \left(-5t^2-2t-4\right)\:dt\right)$?

How this integral is evaluated? $$\frac{\partial }{\partial y}\left(\int _y^x\cos \left(-5t^2-2t-4\right)\:dt\right)$$ And in general, are there general methods for partial differentiation ...
0
votes
0answers
24 views

Determine whether the function $ \arcsin \left(\sqrt{1-2\log _e^2\left(x\right)}\right) $ is continuous or not

I don't know how to study the continuity of this function$$ \arcsin \left(\sqrt{1-2\log _e^2\left(x\right)}\right) $$ Do I have to take the first and last value of its domain and pass them to limits ...
0
votes
3answers
47 views

Angle Between a Parabola and a Vertical Line

I am trying to find the angle between a parabola $y=-0.000314x^2+0.3716x$ and a vertical line $x=738$. I found that I have to use this formula: $$\tan \theta=\frac{m_2-m_1}{1+m_1.m_2}$$but I'm not ...
10
votes
2answers
626 views

A slightly problematic integral $\int{1/(x^4+1)^{1/4}} \, \mathrm{d}x$

Question. To find the integral of- $$\int{1/(x^4+1)^{1/4}} \, \mathrm{d}x$$ I have tried substituting $x^4+1$ as $t$, and as $t^4$, but it gives me an even more complex integral. Any help?
1
vote
0answers
10 views

4 step area of a solid of revolution

I have been working on this problem all night and its driving me crazy no matter which way I try to tackle it I just cant seem to nail down the right answers even though my work seems correct. would ...
0
votes
0answers
22 views

Limit of a generalized convex combination

I would like to find the following limit: $\lim_{x\to \infty}[\lambda p^x + (1-\lambda)q^x]^{1/x}$, where $0<p\le q<1, 0<\lambda<1$ My attempt is the following. $[\lambda p^x + (1-\...
1
vote
2answers
76 views

Can $\sin(x)$ be written as the product of its infinite roots $(0,\pi,-\pi..)$ as - $\sin(x)=(x)(x+\pi)(x-\pi)(x+2\pi)(x-2\pi)..$

My Question is that can $\sin(x)$ be written as the product of its infinite roots as - $$\sin(x)=(x)(x+\pi)(x-\pi)(x+2\pi)(x-2\pi)...$$ I have already seen in many places $ \sin(x)$ being written as-...
1
vote
1answer
55 views

What functions satisfy the product of their derivatives being equal to their product

Suppose $$\frac{\frac{d}{dt}x(t)}{x(t)} =\frac{y(t)}{\frac{d}{dt}y(t)}$$ How would we go about proving solutions other than $x(t)=y(t)=e^{t}$ exist, and furthermore if we could prove other solutions ...
0
votes
2answers
73 views

Evaluate $\iint { \sqrt{\left| y-{ x }^{ 2 } \right|}\, dx\,dy } $ over a rectangle

Question: I want to evaluate $\iint_R {\sqrt{ \left| y-{ x }^{ 2 } \right|}\, dx\,dy }, $ where $R=[-1,1]\times[0,2]$. Indeed $x\in[-1,1]$ and $y\in[0,2]$. My approach: Since, $|y-x^2|$ is positive ...
2
votes
0answers
83 views

Is this notation common in Calculus?

Okay this is going to be quite a stupid question, but to me this seems... wrong, or at the very least not completely correct. In the material I'm reading there's a part that states that $y$ evolves ...
1
vote
1answer
30 views

functional equation

Let $$f(x) = \frac{1}{2}\left[f(xy)+f\left(\frac{x}{y}\right)\right]$$ for all $x\in \mathbb{R^{+}}$ such that $f(1)=0$ and $f'(1) = 2.$ Then $f(x)$ $\bf{My\; Try::}$ Using Inspection above $f(x)=2\...
1
vote
0answers
18 views

Maximum value of $(1-F(t))t$ for probability distribution

Consider a continuous distribution on $(0,1)$ with cumulative distribution function $F$. For the value of $t\in(0,1)$ that maximizes $$P(t)=(1-F(t))t,$$ what is the lower bound of $P(t)$? For example, ...