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

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2answers
67 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 ...
0
votes
2answers
30 views

Directional derivatives exercise from Courant's introduction to calculus and analysis

Show for $z=f(x,y)=\sqrt[3]{xy}$ that $f$ is continuous and that the partial derivatives $\partial z/\partial x$ and $\partial z/\partial y$ exist at the origin but that the directional derivatives in ...
2
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2answers
39 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 ...
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 ...
2
votes
1answer
43 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
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1answer
561 views

proof of the second generalized mean value theorem for integrals

Let $f,g,g´$ be continous on $[a,b]$ and $g$ monotone on $[a,b]$; then there exist $c\in (a,b)$ so that $$\int_{a}^{b}f(x)g(x)dx=g(a)\int_{a}^{c}f(x)dx+g(b)\int_{c}^{b}f(x)dx$$ Ineed to apply the ...
1
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3answers
50 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 ...
-1
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0answers
26 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 ...
5
votes
6answers
117 views

Show that $h(x)=x^5+3x+6$ is one to one

How do I show that $$h(x)=x^5+3x+6$$ is one to one? I set $$f(a)=f(b)$$ and try to isolate for $a$ and $b$ but I get stuck because I have a term of "$a$" that is degree $5$ and a term of "$a$" that is ...
1
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4answers
71 views

Maximum of subtended angle $\theta$

Following Problem, from Jim Fowler's Mooculus class: A painting is mounted on a wall. The bottom of the painting is 5 feet above eye level, and the top of the painting is 14 feet above eye level. If ...
1
vote
2answers
29 views

Spline approximation for $g(t) = \frac{t e^{-t}}{(x+t^2)^2}$

Is there any nice way to do a spline approximation for $$ g(t) = \frac{t e^{-t}}{(x+t^2)^2}\,, $$ where $x$ is some constant? I tried finding nice interpolation points, however this proved very ...
2
votes
4answers
114 views

Finding the function of these numbers $1, 2, 5, 13, 34, 89, 233, 610$

Firstly I used the differences between them but I found the numbers return again. How can I find the function of these numbers
0
votes
1answer
27 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 ...
17
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6answers
185 views

What is the importance of $\sinh(x)$?

I stumbled across $\sinh(x)$. I am only a calculus uno student, but was wondering when this function comes into play, and what is its purpose? Last, does it have world applications, or is it a ...
2
votes
1answer
50 views

Surface of revolution of an ellipse

I have been working on this question, but I end up getting the wrong answer overtime: The ellipse $$\frac{x^2}{a^2}+ \frac{y^2}{b^2} = 1$$ where $a>b$ is rotated about the $x$-axis to form a ...
0
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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
0answers
33 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
vote
1answer
46 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 ...
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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
25 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
28 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
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1answer
29 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, $y=x^2+3$
2
votes
8answers
122 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
49 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 ...
8
votes
1answer
142 views

Alternative ways to evaluate $\displaystyle \int^1_0 \frac{\text{Li}_2(x)^3}{x}\,dx$

In the following link here I found the integral & the evaluation of $$\displaystyle \int^1_0 \frac{\text{Li}_2(x)^3}{x}\,dx$$ I'll also include a simpler version together with the question: is ...
0
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3answers
632 views

Optimize volume of an open cardboard box made from flat square of cardboard…

Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. ...
4
votes
2answers
353 views

If $\,x>1$, then $\displaystyle\lim_{n\rightarrow\infty}\frac{\left\lfloor x^{n+1} \right\rfloor}{\left\lfloor x^n \right\rfloor}=x$.

How can I prove that $$ \lim_{n\rightarrow\infty}\frac{\left\lfloor x^{n+1} \right\rfloor}{\left\lfloor x^n \right\rfloor}=x, $$ whenever $x>1$. Here $\left\lfloor \cdot\right\rfloor$ denotes the ...
0
votes
2answers
73 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 ...
4
votes
2answers
110 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 ...
5
votes
7answers
344 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 ...
1
vote
3answers
49 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.
8
votes
3answers
138 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 $$(x^3)\:\text{dt}+(1-t)[(x^2+1)t-1]\:\text{dx}=0$$ ...
4
votes
1answer
67 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 ...
36
votes
8answers
1k views
-2
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0answers
40 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 ...
2
votes
2answers
62 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 ...
2
votes
1answer
89 views

How prove $\frac{2}{3}<\frac{3x^6+15x^2+2}{2x^6+15x^4+3}\le\frac{3}{2}$

Let $x\in(0,1]$, show that $$\dfrac{2}{3}<\dfrac{3x^6+15x^2+2}{2x^6+15x^4+3}\le\dfrac{3}{2}$$ My try: since $$\begin{align}\dfrac{3x^6+15x^2+2}{2x^6+15x^4+3} ...
3
votes
2answers
52 views

How find this limit $\lim_{x\to 0}\frac{1}{x^4}\left(\frac{1}{x}\left(\frac{1}{\tanh{x}}-\frac{1}{\tan{x}}\right)-\frac{2}{3}\right)=?$

Find this following limit $$\displaystyle \lim_{x\to 0}\dfrac{1}{x^4}\left(\dfrac{1}{x}\left(\dfrac{1}{\tanh{x}}-\dfrac{1}{\tan{x}}\right)-\dfrac{2}{3}\right)=?$$ My try: since ...
1
vote
2answers
120 views

How solve equation:$y^4+4y^2x-11y^2+4xy-8y+8x^2-40x+52=0$

Let $x,y\in \mathbb{R}$, solve this follow equation:$$y^4+4y^2x-11y^2+4xy-8y+8x^2-40x+52=0$$ My try: Since $$8x^2+4x(y^2+y-10)+y^4-11y^2-8y+52=0$$ then I can't. Maybe this problem have other nice ...
2
votes
3answers
255 views

Spivak problem on Schwarz inequality

I have a question regarding problem 19 in the 3rd Ed. of Spivak's Calculus. Specifically, part (a). The question concerns the Schwarz inequality: $$ x_1y_1 + x_2y_2 \leq ...
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 ...
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: ...
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 ...
1
vote
3answers
48 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?
2
votes
0answers
24 views

Are these two option valuation formulas equivalent? Why?

I have been reading a finance paper that claims that the following function, which is a value for a financial derivative (1): $$V(s,t)=E_{Q} \left[\zeta\big(S(T)\big)e^{-\int_t^T r_F(\nu) ...
2
votes
4answers
262 views

Proving there exist an infinite number of real numbers satisfying an equality

Prove there exist infinitely many real numbers $x$ such that $2x-x^2 \gt \frac{999999}{1000000}$. I'm not really sure of the thought process behind this, I know that $(0,1)$ is uncountable but I ...
0
votes
3answers
97 views

Evaluation of the integral $\int 3x \cos x^2 \, dx$

I want to solve this: $$\int 3x \cos x^2 \, dx$$ I get this answer: $$ \frac{\sin 2x}{2}+\frac{\cos 2x}{4}+C $$ but the answer should be: $$ \frac{3 \sin x^2}{2}+C $$ Am I doing anything wrong ...
3
votes
1answer
56 views

Writing a proof of the convergence of a series defined recursively

Define the sequence $a_n$ recursively by $a_1=1$ and $$a_{n+1}=\frac13\left(a_n^2+\frac1n\right)$$ (a) Prove, by induction or otherwise, that $(a_n)$ is decreasing. (b) Prove that the series ...
1
vote
2answers
20 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 ...