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

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0
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4answers
44 views

Proof that $\lim_{x \to 0 } \frac{ln(x+1)}{x} = 1$

I've looked around to see a proof for this limit and encountered this: $$ \lim_{x \to 0 } \frac{\ln(x+1)}{x} $$ $$ \lim_{x \to 0 } \frac{1}{x} \ln(x+1) $$ $$ \lim_{x \to 0 } \ln(x+1)^\frac{1}{x} ...
4
votes
2answers
34 views

Proof of expression with inegrals

I have had trouble prooving the following expression. Do you have any hints to help me? Let $f:[a,b]$ be an integrable function for which $$\int_a^bf(x)dx=6$$ Prove that there exist $t_1,t_2\in(a,b)$ ...
2
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2answers
23 views

Multiple choice question about limits and continuity? (Or, $\tan x$ is continuous?!)

I'm doing a test about limits and continuity and got these two wrong. $\mathbf{Q1}$: The function $f(x) = \tan x$: $\hspace{1em}\mathtt{a)}$ is continuous $\hspace{1em}\mathtt{b)}$ is ...
1
vote
2answers
22 views

Generalization of linear approximation?

How is the linear approximation is generalized to the Taylor series? I do not get that concept.
2
votes
0answers
31 views

Density of the rationals in the reals

While studying measure theory I have encountered the following set, $$U_\varepsilon=\bigcup_{n\in \mathbb{N}}(q_n-\varepsilon /2^n,q_n+\varepsilon/2^n),$$ where $(q_n)_{n\in \mathbb{N}}$ is an ...
2
votes
2answers
59 views

Proving a limit using another limit

Let $f(x)$ be a functions that's defined in the area $ x= 0$ $$\lim\limits_{x \to 0} \frac{f(x)}{x} = 3$$ Prove that: $$ \lim_{x \to 0}\frac{f(3x)}{\ln(1+4x)} = 2.25 $$ I really don't know what to ...
1
vote
1answer
15 views

Mass of the body M, Cartesian reference frame.

Oxyz is a Cartesian frame of reference with unit base vectors, $i,j$ and $k$. A rigid body $V$, of uniform density $p$, is bounded by the surfaces $y=(1-x^2)^{(1/2)}, z=0, y=0$ and $z=1-y$ If the mass ...
1
vote
1answer
41 views

Evaluate this infinite product involving $a_k$

Let $a_0 = 5/2$ and $a_k = a_{k-1}^{2} - 2$ for $k \ge 1$ Compute: $$\prod_{k=0}^{\infty} 1 - \frac{1}{a_k}$$ Off the bat, we can seperate $a_0$ $$= -3/2 \cdot \prod_{k=1}^{\infty} 1 - ...
2
votes
4answers
102 views

If we know $x+y+z=1$, $x^2+y^2+z^2=2$, and $x^3+y^3+z^3=3$, how to find $x^4+y^4+z^4$?

Let $x$, $y$, and $z$ be such that $$\begin{align*} x+y+z&=1\\ x^2+y^2+z^2&=2 \\ x^3+y^3+z^3&=3 \end{align*}$$ Then $x^4+y^4+z^4=?$
4
votes
0answers
38 views

Seperating single integral into an double integral.

Please refer to : How to prove that $\int_{0}^{\infty}\sin{x}\arctan{\frac{1}{x}}\,\mathrm dx=\frac{\pi }{2} \big(\frac{e-1}e\big)$ The answer by @Venus. What is the procedure in converting that ...
0
votes
3answers
42 views

Why don't graphing tools represent holes in a graph?

Why don't graphing tools represent holes in the graph of a function? A hole at a point in a graph is point where function is not defined. Suppose there is a function $$\frac{x}{\sqrt{x-1}-1}$$ Its ...
1
vote
0answers
15 views

Show that weak local minimum of a convex function $\mathbb{R}^N\rightarrow \mathbb{R}$ is its weak global minimum.

Show that weak local minimum of a convex function $\mathbb{R}^N\rightarrow \mathbb{R}$ is its weak global minimum. Does the same happen to strong minimums? I know that when $f$ is convex, then we ...
-3
votes
0answers
29 views

How to find the area within the curve $ a^2 \cdot y^2 = x^3(2a - v) $? [on hold]

Here is the equation of a curve: $ a^2 \cdot y^2 = x^3(2a - v) $. Now I want to find the whole area of the curve . How can I find the whole area?
4
votes
1answer
36 views

Multiple integrals involving product of gamma functions

The following integral was posted a few days back on Integrals and Series forum: $$\int_0^{2\pi} \int_0^{2\pi} \int_0^{2\pi} \frac{dk_1\,dk_2\,dk_3}{1-\frac{1}{3}\left(\cos k_1+\cos k_2+ \cos ...
1
vote
0answers
14 views

Shifting integration variables

I'm not sure how to pose this question precisely, but I'll try. I'm trying to see what happens when you have an integral of the form $\int \mathrm{d}x \,f(x-g(z))$ and you try and write it as $\int ...
3
votes
2answers
64 views

Surface area of a solid of revolution: Why does not $ \int_{b}^{a} 2\pi \,f(x) \,dx $ work?

Why does not $ \int_{b}^{a} 2\pi \,f(x) \,dx $ yield the correct answer when calculating the surface area of a solid of revolution?
1
vote
2answers
63 views

Why does this inequality stand?

I want to ask something about: "Since $i \log_e i$ is concave upwards, it is easy to show that $$\sum_{i=2}^{n-1} i \log_e i \leq \int_2^n x \log_e x \,dx \leq \frac{n^2 \log_e ...
2
votes
1answer
41 views

Determining a radius convergence of a power series

Let $$ \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^{3n+1} $$ Is there an immediate way to determine $R=1$?
0
votes
1answer
27 views

Find the volume $z \geq 3x^2+2y^2, \ \ 3x^2+2y^2+5z^2 \le 1$

Find the volume of solid defined by the following inequalities : $$z \geq 3x^2+2y^2, \ \ 3x^2+2y^2+5z^2 \le 1$$ We have an ellipse, which the semi-axis are $\sqrt{\frac{z}{2}}$ and ...
-1
votes
0answers
25 views

Show existence of a sub-sequence $(f_{n_k})$ which is uniformly convergent to a function in $C[0,1]$

Let $f_n:[0,1]\rightarrow R$ be a sequence of continuously differentiable function, Let $M>0$ be such that for any $0\le x \le 1$ and natural $n$, $|f_n(x)|$, $|f'_n(x)|<M$ Show existence of a ...
1
vote
2answers
32 views

Raising and Lowering Through Differentiation

I'm calculating the Christoffel symbols of the second kind which is of course defined as multiplying the symbol of the first kind multiplied by the contravariant metric. I was thinking of how to make ...
1
vote
3answers
49 views

Fundamental Theorem of Calculus 1 - definite integral

I have two problems, they're not from a book so I can't check the answer for one of them and the other I'm not sure on what to do. $$ {d\over dx}{\int^{1}_{x^{2}}} {\sqrt{t^{2}+1}} {dt} $$ $$=-{d\over ...
0
votes
1answer
26 views

Computing the value of a function whose derivative is another function

Apologies if this is something relatively trivial, my calculus is a bit rusty. Let say I have function $f(t)$ which is increasing at a non-constant rate. This rate is also a function of $t$, lets say ...
0
votes
0answers
91 views

Prove the Schwarz inequality using $ 2xy \leq x^2 + y^2 $

Im really bad at analysis and this problem was recommend to me to help me grasp some basics of $\epsilon $ $\delta $ So im doing a problem ( though its like 12 pieces ) this is i guess the fourth ...
0
votes
0answers
30 views

Minimize total cost of one kilometer

The cost of the fuel consumption of a locomotive is proportional to the square of its speed plus 100 pounds per hour without regard to its speed. The cost of the fuel consumption is 25 pounds per hour ...
0
votes
2answers
22 views

Calculus minimum cost for an open box

An open box with a squared base of volume $128 \ m^3$. The cost of the material used for the base of the box is $2$ pounds per $m^2$, and that of the material used for the lateral faces is $0.5$ ...
-1
votes
2answers
28 views

Calculus: Maximum profit for a factory [on hold]

A factory earns 20 pounds for each unit if it produces 800 units per week. If the production will increase, the profit for each unit will decrease 0.02 pounds. Find the number of units to be produced ...
2
votes
1answer
21 views

Finding the PDF from the CDF where the CDF is not differentiable at some point

I got the following problem: Let $X$ be a continuous random variable with $CDF$ denoted $F_X$ defined as follows: $F_X(x)= \begin{cases} 1-x^{-4/3}, & x\in[1,\infty) \\ 0, & x\in ...
2
votes
3answers
25 views

Determine monotone intervals of a function

Let $$ f(x) = \int_1^{x^2} (x^2 - t) e^{-t^2}dt. $$ We need to determine monotone intervals of $f(x)$. I tried to differentiate $f(x)$ as follows. $$ f'(x) = \left(x^2 \int_1^{x^2} e^{-t^2}dt \right)' ...
0
votes
1answer
28 views

Find the area between the two functions--integrals [on hold]

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region $y=5x^2$ and, $y=x^2+3$
5
votes
2answers
55 views

Evaluating sums using residues $(-1)^n/n^2$

I am an alien towards compelx analysis, with very little know I am posing a question, who someone may want to help with. Evaluate: $$\frac{1}{4}\cdot \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$$ In ...
2
votes
8answers
114 views

How to show that $f(x) = 0$ if $\int_a^bf(x)\,\text{d}x=0$ for all $a,b\in\mathbb{R}$?

I found this problem on the web: Let $f(x)$ be a real-valued, continuous function with the property that $$\int_a^bf(x)\,\text{d}x=0$$for all real numbers $a,b$. Prove that $f$ is identically $0$. ...
7
votes
1answer
46 views

Differentiating a constant and switching order

Why does this work? $$\int x^2e^{ax}dx = \int \frac{d^2}{da^2}e^{ax}dx = \frac{d^2}{da^2}\int e^{ax}dx = \frac {d^2}{da^2} \frac {e^{ax}}a = \frac{e^{ax}(a^2x^2-2ax+2)}{a^3}$$ $a$ is a constant, so ...
0
votes
1answer
39 views

Proof of limit of a piecewise function, rational, irrational

Prove that: If $f(x) = 0$ for irrational $x$ and $f(x) = 1$ for rational $x$ then $\lim_{x \to a} f(x)$ does not exist for any $a$. So begin by the opposite assumption: Assume $\lim_{x \to a} f(x) ...
1
vote
1answer
42 views

Proving the existence of a sequence of polynomials convergent to a continuous function $f$.

I need to show that if $f$ is continuous function ($f:\mathbb{R}\rightarrow \mathbb{R}$), then there exists a sequence of polynomials which converges to $f$ on any compact subset of $\mathbb{R}$. I ...
0
votes
2answers
70 views

Does $\displaystyle\lim_{x \to 1}x\ln(x - 1)$ exist? WolframAlpha says yes

The solution to one exercise says that $$\lim_{x \to 1}x\ln(x - 1) = -\infty$$ How can this be, if $\operatorname{dom} \ln(x - 1) = (1, +\infty)$? Only the limit from the right exists, but the other ...
8
votes
3answers
136 views

How to solve the differential equation $(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$?

Solve $$(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$$ I tried the substitution $y^2=t$ ; $2y\:\text{dy}=\text{dt}$ to get $$(x^3)\:\text{dt}+(1-t)[(x^2+1)t-1]\:\text{dx}=0$$ ...
-2
votes
0answers
39 views

perfect competition [on hold]

A (perfectly) competitive firm has total cost given by $$TC(Q) = 5,000,000 + 5Q +\frac{Q^2}{10,000}$$ Regarding its fixed cost of \$5 million, \$4 million can be avoided if the firm produces $0$, but ...
1
vote
3answers
48 views

Greatest value of $f(x)= (x+1)^{1/3}-(x-1)^{1/3}$ on $(0,1)$

Greatest value of $f(x)= (x+1)^{1/3}-(x-1)^{1/3}$ on $(0,1)$ Please guide me to solve this problem. I have differentiated it with respect to $x$ and make equal to zero, but couldn't get any point.
4
votes
2answers
98 views

How to integrate a fraction of the type $\frac{1}{(ax+b)^c(dx+e)^f}$?

I'm working on obtaining chemical reactions' speed, and this is one of the problems I met with. $$ \int \frac{1}{(ax+b)^c(dx+e)^f}dx $$ Can this equation could be solved? If possible, please show ...
0
votes
1answer
28 views

Is this enough to demonstrate divergence of an improper integral?

The integral in question is $$\int_0^\infty (f(x)-a)^2dx$$ Where f(x) is some continuous function and a is some constant. When we expand the integrand,we end up with an $a^2$ term. We can then ...
0
votes
1answer
17 views

Optimization, minimizing volume of an open top box given the volume

The question is: An Anacleto box is a square open box: the bottom is a square, the four sides are equal rectangles, and there isn’t anything on the top. The box should have a volume of 1000 ...
4
votes
1answer
65 views

The equality case of the Schwartz inequality

Question: The fact that $a^2 \geq 0$ $ \forall a \in \mathbb{R}$; elementary as it may seem, is nevertheless the fundamental idea upon which most important inequalities are ultimately based. The ...
2
votes
2answers
54 views

Calculus of Variations. Lagrangian Hamiltonian Mechanics Mathpages.

Over at http://www.mathpages.com/home/kmath523/kmath523.htm is an article about Lagrangian and Hamiltonian Mechanics with a derivation of the Euler-Lagrange equations of motion. Mid-way through is ...
0
votes
2answers
35 views

How do I find this distance?

Find the minimum and maximum distances between the ellipse $x^2+xy+2y^2 = 1$ and the origin. This is what I've attempted so far: Maximize $x^2+y^2+z^2$ with respect to $x^2+xy+2y^2 = 1$. Using ...
5
votes
7answers
326 views

If $x$ and $y$ are not both $0$ then $ x^2 +xy +y^2> 0$

Can't quite finish this proof: Prove that if $x$ and $y$ are not both $0$ then $ x^2 +xy +y^2> 0$ $ x^2 +xy +y^2 +xy -xy> 0$ $ (x +y)^2 -xy> 0$ Without loss of generality define $x\geq ...
2
votes
1answer
29 views

Using Lagrange multipliers to find the extrema of $f(x,y) = e^{2xy}$ subject to $x^2+y^2 = 16$

Find the maximum and minimum values of $f = e^{2xy}$ with respect to $x^2+y^2 = 16$. Using Lagrange multipliers, $\nabla f = \lambda\nabla g$. Therefore, the constraints are the following: ...
1
vote
3answers
47 views

Evaluate $\int {x+3\over x^2+6x+10}dx$ [on hold]

$$\int {x+3\over x^2+6x+10}dx$$ Could anyone help me with this substitution problem?
1
vote
2answers
18 views

Intervals and Signs

In the first and second derivative tests, I find whether the derivative is positive or negative by picking a random number within that open interval. The number I pick is arbitrary; however, what ...
0
votes
1answer
35 views

Minimizing a function in Mathematica

Edit: I simplified the function using $\textbf{Simplify[...]}$ How can I minimize this function of $x$, where $l$ is a positive constant? $$\frac{1}{2} \sqrt{\frac{x}{l}+\frac{l}{x}+4 x^2-2}$$ ...