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

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2
votes
2answers
38 views

Determine equation of tangent plane?

Determine equation of tangent plane in points $(\frac{1}{2},1,f(\frac{1}{2},1) )$ $f(x,y)=x^{4}-x^{2}+y^{2}$ I know usually how these examples work, but I am confused with these $3$ points. I have ...
3
votes
2answers
75 views

Show that $\int\limits_a^b |f(t)|dt \leq (b-a)\int\limits_a^b|f'(t)|dt$

Let $f:[a,b]\to\mathbb{R}$ be continuously differentiable. Suppose $f(a) = 0$. Show that $$ \int\limits_a^b|f(t)|dt \leq (b-a)\int\limits_a^b|f'(t)|dt $$ By the mean value theorem, for every $t\in[...
0
votes
0answers
48 views

Unusual integration of 1/cx [duplicate]

Consider an integral: $$\int_2^3 \frac{1}{cx} dx$$ where $c$ is a constant So we can take that out of the integral, so $$\int_2^3 \frac{1}{cx} dx = \frac{1}{c} \int_2^3 \frac{1}{x} dx $$ all is ...
1
vote
0answers
37 views

Continuous positive function such that $\int\limits_1^{\infty}f(x)dx$ converges, while $\int\limits_1^{\infty}f^2(x)dx$ diverges [duplicate]

Does there exist a continuous positive function such that $\int\limits_1^{\infty}f(x)dx$ converges, while $\int\limits_1^{\infty}f^2(x)dx$ diverges? I have proved that if $f$ is decreasing ...
-1
votes
1answer
85 views

Can you solve this integral [on hold]

Iam tried to solve this integral, but i couldn't; i dont know by what substitution or by what method. Can you help me to find its exact solution? $$\int\frac {1}{3^x+x}dx$$
-4
votes
2answers
42 views

separation of variables to find brine level [on hold]

I need help with this calculus problem. I dont know how to do it at all. A 800-gallon tank initially contains 700 gallons of brine containing 75 pounds of dissolved salt. Brine containing 4 pounds of ...
1
vote
6answers
178 views

Solve definite integral: $\int_{-1}^{1}\arctan(\sqrt{x+2})\ dx$

I need to solve: $$\int_{-1}^{1}\arctan(\sqrt{x+2})\ dx$$ Here is my steps, first of all consider just the indefinite integral: $$\int \arctan(\sqrt{x+2})dx = \int \arctan(\sqrt{x+2}) \cdot 1\ dx$$ ...
-2
votes
1answer
53 views

finding the initial height of the tree [on hold]

I need help with this calculus problem A tree has been transplanted and after xx years is growing at rate $1+\frac{1}{(x+9)^2}$ meters per year. After 3 years, it has reached a height of 10 meters. ...
3
votes
1answer
65 views

sum of series using mean value theorem

Let $f(x)$ be a function which is differentiable on $[0,1]$ with $f(0)=0$ and $f(1)=1$. Show that for every $n\in \Bbb N$ there exists numbers $x_1,x_2,\ldots,x_n\in [0,1]$ such as $$ \sum_{k = 1}^n \...
2
votes
1answer
64 views

Minimum of a function in $(0,1) \times (0,+\infty)$

I would like to minimize the function $$ (\alpha,\theta) \mapsto F(\alpha,\theta) := -\theta x^\alpha + \sum_{k=1}^N \ln(1+p_k(e^{\theta \ell_k^\alpha}-1)) $$ where $\theta \in (0,+\infty)$, $\alpha \...
-2
votes
1answer
61 views

about integration of expression [on hold]

I am a class $11$ student from India. I thought that Integration is nothing but addition of minute quantities . Then why is $\int x^2 = 2x$. In class $5$ , we were taught that $x^2+x^2+x^2 = 3x^2$. ...
-8
votes
1answer
55 views

bubble blowing using differential equation [on hold]

I have the following homework problem that I am stuck on and cant get to work A teen chewing bubble gum blows a huge bubble, the volume of which satisfies the differential equation: $$\frac{dV}{dt}=...
7
votes
2answers
290 views

real values of $x$ which satisfy the equation $\sqrt{1+\sqrt{1+\sqrt{1+x}}}=x$

All real values of $x$ which satisfy the equation $\sqrt{1+\sqrt{1+\sqrt{1+x}}}=x$ $\bf{My\; Try::}$ Here $\sqrt{1+\sqrt{1+\sqrt{1+x}}} = x>0$ Now Let $f(x)=\sqrt{1+x}\;,$ Then equation convert ...
0
votes
2answers
73 views

Integration by parts proof 1 = 0

Let's integrate $\int\frac{f^\prime(x)}{f(x)} dx$ by parts $$ \\ \mbox{ Let } dv= f^\prime(x)dx,u=\frac{1}{f(x)} \\ \mbox{ Then }v=f(x), du=-\frac{f^\prime(x)}{[f(x)]^2}dx \\ \mbox{ This implies }\int\...
4
votes
2answers
71 views

the uniform convergence of the sequence of functions

Let $f_1:[a,b]\rightarrow \mathbb{R}$ be a Riemann integrable function. Define the sequence of functions $f_n:[a,b] \rightarrow \mathbb{R}$ by $f_{n+1}(x)=\int_a^x f_n(t)dt,$ for each $n\ge 1$ and ...
4
votes
1answer
164 views

New series formula for $\arctan(x)$?

I discovered this equation, but have no idea if it has been previously discovered. Please help determine if it has been previously developed. Or please prove that the equation is not correct. $$\...
3
votes
1answer
111 views

Old books on calculus

I'd like to know if there are other old books of the same level of the classic and well-known books like Apostol, Courant, Spivak and Hardy.
1
vote
1answer
30 views

Volume by integration - Disk Method only/Non-coordinate axis

PROBLEM: Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 5. (Use disk method) $$ xy = 3, y = 1, y = 4, x = 5 $$ So first I ...
2
votes
4answers
60 views

Find positive values such that $xy = 32$ and the sum $4x+y$ is as small as possible.

Find positive values such that $xy = 32$ and the sum $4x+y$ is as small as possible. How can I solve this? I know that the answer is $2\sqrt{2} = x$ and $8\sqrt{2} = y$, but I can't seem to figure ...
2
votes
3answers
53 views

prove that $\sum_{n=1}^\infty \frac{2^n+n^2+n}{2^{n+1}n(n+1)}$ is convergent and find the limit when $n \to \infty$

does the following sumatory converges? if yes find the limt when $n \to \infty$ $$\sum_{n=1}^\infty \frac{2^n+n^2+n}{2^{n+1}n(n+1)}$$ ideas? i have tried by the comparison test.
0
votes
2answers
33 views

Confusion about the different ways of writing Taylor Polynomials

For the sake of using a simple example, let's say I want to approximate $y=x^3$ with a second degree polynomial, and let's say I want to construct my polynomial around the point $x=4$. One way I ...
1
vote
2answers
50 views

How do I solve $\lim(1+1/x)^{x^2y/(x+y))}$

How do I solve this limit: it looks like euler can be used here, any ideas? The answer is 1.
0
votes
3answers
29 views

Derivatives: Combining Product & Chain Rules

So I'm working through the material on Khan Academy, and the question is: "Consider the function $f(x) = x^n\ln x$, defined for $x > 0$. Determine, in terms of $n$, the value of $x$ for which $f'...
0
votes
3answers
80 views

A basic question about limits [closed]

How does one compute $\lim\limits_{(x,y)\to (0,0)}\frac{2x^2 y}{x^4+y^2}$?
3
votes
5answers
221 views

properties of distributions

If $$\int_{-\infty}^\infty f dx = 1$$, with $f > 0 \forall x$, then prove or disprove: $$\int_{-\infty}^\infty \frac{1}{1 + f} dx $$ diverges. The hint I got is to consider the measure of the set$(...
0
votes
0answers
27 views

$N$-dimensional volume (of revolution)

Consider the system of coordinates $\{x_{1},x_{2},...,x_{n}\}$ and an n-dimensional shape such that, in $\{x_{1},x_{n}\}$ (and $x_{2}=x_{3}=...=x_{n-1}=0$) it is inside the lines $x_{n}=ax_{1}+b$ and $...
0
votes
1answer
60 views

curve between two points having maximum area [on hold]

Find the curve(i.e. the segment of a standard curve like circle, ellipse etc.) amongst all curves(segments) that have fixed total length, passes through $ (a,b)$ and $(c,d)$ and has maximum area ...
0
votes
3answers
46 views

Determine if the following vectors are coplanar.

I have no idea to start with this question, I know how to find if vectors are coplanar when the values of the vectors are given to me, but I do not know how to manipulate coplanarity properties well ...
1
vote
2answers
55 views

Find $y(2) $ given $y(x)$ given a separable differential equation

Find what $y(2)$ equals if $y$ is a function of $x$ which satisfies: $x y^5\cdot y'=1$ given $y=6$ when $x=1$ I got $y(2)=\sqrt{6\ln(2)-46656}$ but this answer is wrong can anyone help me figure ...
0
votes
1answer
32 views

marginal analysis and differentials

I don't understand this question hope somebody can help me. Suppose that the cost to produce an LCD computer monitor is $\$75$. Furthermore, suppose that when the selling price is p dollars, the ...
0
votes
4answers
64 views

differential equation particular solution

I need help with this calculus problem: Find the particular solution of the differential equation $e^y\frac{dy}{dx}=e^{−9x}$, such that $y=7$ when $x=0$ I got $-\ln(\frac{e^{-9x}}{9}-\frac{1}{9}-e^...
1
vote
3answers
57 views

newtons law of cooling [on hold]

I need help with this calculus problem i just cant get After 10 minutes in Jean-Luc's room, his tea has cooled to 43∘ Celsius from 100∘ Celsius. The room temperature is 23∘ Celsius. How much longer ...
4
votes
4answers
200 views

How to prove that the series $\sum\limits_{n=1}^\infty {\sin^2n} $ diverges

I want to use a divergence test to prove that $\lim_{n\to \infty} \sin^2n$ does not converge. So $\sum_{i=1}^\infty \sin^2 n $ diverge. But because $\pi$ is an irrational number. So I cannot use ...
4
votes
3answers
112 views

Find all continuous functions $f:[0,1]\rightarrow \mathbb{R}$ that satisfy: $\int_0^1 f(x)dx=1/3 + \int_0^1 f^2(x^2)dx$

(Note that $f^2(x)=f(x)\cdot f(x)$ and not composition.) Since both integrals are defined, derivation is out of the question. I tried integrating the second integral by parts but reached something ...
1
vote
1answer
17 views

Determine is $U$ and $V$ subspace and find basis and dimension?

$V=R^{3}$. $U=\left\{ (x_{1},1,x_{3} )^{T}\in \mathbb R ^{3}: x_{1},x_{3}\in \mathbb R \right\} $ $W=\left\{ (x_{1},x_{2},x_{3} )^{T}\in \mathbb R ^{3}: x_{1}+2x_{2}+x_{3}=0 \right\} $ Determine ...
0
votes
3answers
30 views

bounds on a sequence

It may look that this question is trivial, but: Let $(a_n)_{n=1}^\infty$ a sequence s.t. $\forall n\in \mathbb{N} \ \ a_n<\frac {1}{n}$. Prove/Disprove : there is $c > 1$ s.t. $\forall n\in \...
-1
votes
0answers
14 views

Is there any equality for the integral of the product of normal derivative?

I am trying to get the proof of $\int\int_DD_uf(x) D_ug(x) dx$. For example in Green Theorem, in integral we use the product of $ \nabla$, when it comes to normal derivative, how can I organize the ...
0
votes
2answers
92 views

Minimum value of $4a+b$

Let $ax^2+bx+8=0$ be an equation which has no distinct real roots then what is the least value of $4a+b$ where $a,b\in \Bbb R$. My Try: I differentiated the given function to get $f'(x)=2ax+b$ now ...
1
vote
0answers
54 views

Cauchy Sequence of Differentials and Point-Wise Limits

Let $D\subseteq R^2$ be an open and connected subset, and $\{f_{n} | D\to R^2\}$ a sequence of differentiable functions. Suppose that $\{(Df_{n}) | D\to Hom(R^2,R^2)\}$, the sequence of Jacobians, is ...
-2
votes
1answer
47 views

Find the rate of change of area, perimeter and the lengths of the diagonals of the rectangle?

The length $l$ of a rectangle is decreasing at the rate of $2 \space cm/sec$ while the width $w$ is increasing at the rate of $2\space cm/sec$. When $L=12cm$ and $W=5cm$. Find the rate of change of ...
-3
votes
0answers
37 views

A differentiable function with discontinuous derivative [duplicate]

I need to give an example of a differentiable function with discontinuous derivative. I thought about f(x) = 1/x, of course it is differentiable, and it's differentiable is not continous at x = 0. ...
4
votes
1answer
47 views

Estimate the value of f at a given point

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be a differentiable everywhere. Assume $f(-\sqrt2,-\sqrt2)=0$, and also that $|\dfrac{\partial f}{\partial x}(x,y)|\le |\sin(x^2+y^2)|$ and $|\dfrac{\...
1
vote
3answers
35 views

finding $k$ and $y(t)$

I am looking for help with this homework problem I am really stuck on. A function $y(t)$ is a solution of $$y′+ky=0.$$ Suppose that $y(0)=100$ and $y(2)=4$. Find $k$ and find $y(t)$. I worked it ...
7
votes
3answers
132 views

$\int_0^4\frac{\log x}{\sqrt{4x-x^2}} dx=0$ [duplicate]

I am having trouble proving that it is equal to zero analytically. I have tried plotting and know that for $0<x<1$ the integrand is negative and positive otherwise. I have tried substitution $u\...
0
votes
1answer
50 views

particular solution of the given differential equation

I need help with this calculus problem I am very confused about how to go through with this problem! Find the particular solution of the given differential equation $$\frac{\text{d}y}{\text{d}x}=−6xe^...
3
votes
6answers
132 views

If $f(x) = \frac{\cos x + 5\cos 3x + \cos 5x}{\cos 6x + 6\cos4x + 15\cos2x + 10}$then..

If $f(x) = \frac{\cos x + 5\cos 3x + \cos 5x}{\cos 6x + 6\cos4x + 15\cos2x + 10}$ then find the value of $f(0) + f'(0) + f''(0)$. I tried differentiating the given. But it is getting too long and ...
0
votes
4answers
72 views

How to approximate

I was reading a book and saw this approximation $(1 - 10^{-3})^{1023} \approx 2^{-1.476}$ I am wondering how it is calculated.
2
votes
2answers
117 views

Evaluation of Irrational Integral

Evaluation of $$\int\frac{x^4}{(1-x^4)^{\frac{3}{2}}}dx$$ $\bf{My\; Try::}$ Let $$I = \int\frac{x^4}{(1-x^4)^{\frac{3}{2}}}dx = -\frac{1}{4}\int x\cdot \frac{-4x^3}{(1-x^{4})^{\frac{3}{2}}}dx$$ ...
6
votes
4answers
825 views

Why doesn't derivative difference quotient violate the epsilon-delta definition of a limit?

So the difference quotient is defined as: $$\lim \limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ So if we take a function such as $f(x)=x^2$ and go through the simplification, we get $$\lim \limits_{h \...
2
votes
2answers
69 views

Exponential fractional limit

I am dealing with the following fraction. I really would need that it still bigger than $0$ as $n\to\infty$ but I think that it is not the case. \begin{equation} \frac{\sum_{i=0}^{n/2-1}{\frac{(an^2)^{...