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

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1
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2answers
48 views

Do solutions of $\dot{x} = \frac{x}{t^2} + t$ exist satisfying $x(0) =0$

Suppose we have the 1-dimensional ODE \begin{equation} \dot{x} = \frac{x}{t^2} + t \end{equation} Do there exist solution curves with initial condition $x(0)=0$? If you proceed in a standard way ...
0
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1answer
25 views

Indicated composite function?

Please i really need help, I have my problem below and also my solution I've tried
0
votes
3answers
44 views

Proof with Fundamental Theorem of Calculus

If $f'(t)\leq 10$ for $0 \leq t \leq 5$ and $f(0)=3$: How I can explain with $\int_a^b f'(t)\,dt=f(b)-f(a)$ what the maximum of $f(5)$ is?
0
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0answers
29 views

Analysing an integer problem

I am trying to reverse analysis a -hopefully- simple function found in a software. I know the function is extremely fast (so it should be simple operation, no cryptography involved). However I am ...
0
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2answers
36 views

Proof with integral properties

I'm trying to explain/looking for an answer whether a positive function $u=f(t)$ exists, for which $\int_{t=0}^{1}u\,dt = \int_{t=1}^{0}u\,dt$ is true. As we all know, the correct theorem is ...
2
votes
1answer
41 views

Large-z limit of the *other* second derivative of the Laguerre polynomial

I'm trying to find the asymptotic behavior of the second derivative of the Laguerre polynomial (more precisely, the associated analytic function), $\frac{\partial}{\partial n^2}L_{n}(z)$, as $z\to ...
0
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1answer
76 views

Finding Absolute Min/Max with given Domain and Equation. f(x,y)

Question is: Suppose that $f(x,y) = 5x+3y$ at which $-3 \leq x$, $y \leq 3$. Find Absolute minimum and maximum of $f(x,y)$. Since $\frac{\partial}{\partial x} f(x,y) = 0$ or ...
2
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0answers
50 views

norm over differentiable functions computable from derivatives only

I'm running an optimization algorithm, minimizing the norm $||f-\hat f||$ of some objective function $f(x_1,x_2,x_3,y_1,y_2,y_3)$. The function $f$ cannot be computed directly, but its second ...
0
votes
1answer
296 views

Maximizing total tax revenue with function Qs+-8+P and Qd=(80/3)-(1/3P)

The supply and demand equations of a good are given by Qs= -8+P Qd=(80/3) - (1/3)P P is measured in dollars. Suppose the government decides to impose a constant per unit tax of $t on the supplier. ...
0
votes
5answers
52 views

How do I simplify inverse tangent?

How do I simplify $\sec(\arctan(x/5))$? I tried using the formula but it didn't work for me.
1
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0answers
27 views

Transformation for two different boundary functions in Stefan problem

Peace be upon on all of you, I have one-dimensional Stefan problem. Let say we have two boundary conditions of $u(t,s_{1}(t))=g_{1}(t)$ and $u(t,s_{2}(t))=g_{2}(t)$, where $u$ is temperature, $t$ is ...
0
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2answers
44 views

Derivative of a definite integral with two constraints

I am new to this, and I want to see if I have the right answer: $$\frac{d}{dx}\int^{2x}_{x} s^2 ds = \int^{0}_{x} + \int^{2x}_{0}= -\int^{x}_{0}+\int^{2x}_{0} =-x^2+8x^3 $$
2
votes
2answers
56 views

Estimating integral $\int_0^{0.5} \ln(1+\frac{x^2}{4})$

Estimate the definite integral $\int_0^{0.5} \ln(1+\frac{x^2}{4})$ with an error of at most $10^{-4}$, using the Alternating Series Estimation Theorem. My approach is as follows: I found the ...
0
votes
1answer
24 views

Find the horizontal and/or oblique asymptote

Find the horizontal and/or oblique asymptotes of the one system function $$f(x) = \begin{cases} \dfrac{1-2x}{3x^2} & \text{if $x < 0$} \\ \dfrac{x^4+x^3+1}{x+1} & \text{if $x \geq 0$} ...
2
votes
2answers
50 views

Derivative of a definite integral with fraction

I think I may be on the right track but need some help pulling my thoughts together. I have this problem: $$\frac{d}{dt}\int^{\frac{1}{t}}_0 \frac{dx}{1+x^2}$$ So, I believe I want to spread this ...
2
votes
1answer
31 views

Rate of change of rectangle inside triangle

A rectangle is inscribed inside a right angled triangle with hypotenuse 50cm and an angle of 30 degrees. I have supplied a diagram below. The vertical line marked h is moving to the right at 3cm per ...
15
votes
6answers
356 views

Evaluate $\int_0^1 \frac{x^k-1}{\ln x}dx $ using high school techniques

Is there a way to compute this integral, $$\int_0^1 \frac{x^k-1}{\ln x}dx =\ln({k+1})$$ without using the derivation under the integral sign nor transforming it to a double integral and then ...
0
votes
0answers
47 views

For what values $q,r$ does the improper integral $\int_0^1 x^q (1-x^2)^r dx$ converge?

Question: For what values $q,r$ does the improper integral $\int_0^1 x^q (1-x^2)^r dx$ converge? I had some ideas from a previous thread but now I have a different attempt that I would like to ...
1
vote
0answers
67 views

Problem in understanding the process of finding antiderivative.

Antiderivative or indefinite integral is the family of functions the derivative of which gives the original function. Now, let's elaborate the process. Suppose $F(x)$ is the derivative of the ...
0
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2answers
28 views

Evaluate and prove the limit at negative infinity

Find $$\lim_{x\to -\infty} f(x)$$ where $f(x)=\frac{x^2+1}{x-2}$ and prove your result. Not sure how to work with negative infinity, please help
0
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1answer
182 views

Height of liquid in half-filled cylinder-cone on its side

This is a question in my test review. I've been trying to figure it out for a while now, to no avail. Please help. There's a container comprised of a cylinder with a cone stacked on top with the same ...
2
votes
1answer
96 views

Using Intermediate Value Theorem and Rolle's Theorem to solve for roots.

I was having trouble with this question. You are supposed to use the Intermediate Value Theorem and Rolle's Theorem to prove that $x^{4}+4x-8$ has exactly two real roots. How would I do this? Thank ...
1
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1answer
31 views

Evaluate $ \lim_{x \to +\infty}\frac{2x}{(x-3)}$

Evaluate the limit as $ \lim_{x \to +\infty}$ and prove the result: $f(x) = \frac{2x}{(x-3)}$ I know that the limit is $2$ but having a hard time to find the right $\epsilon$. Please help
-1
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1answer
26 views

partial sum question

find a formula for the $n$th partial sum of the series and use it to find the series' sum if the series converges. $$2+\frac 23+\frac 29+\frac2{27}+\dots+\frac{2}{3^{n-1}}$$ The answer is ...
0
votes
1answer
112 views

Logistic differential equation to model population

Problem Description: The population of the world was about 5.3 billion in 1990. Birth rate in the 1990s ranged from 35 to 40 million per year and death rates ranged from 15 to 20 million per year. ...
0
votes
1answer
21 views

Logistic growth

The rate at which a population grows dP/dt is $0.3P/((5)(5-P/500))$. The initial population is 2 at time = 0 I solved the problem to get P(t) = $2500/(1+1248e^-.3t). by integrating the integral and ...
5
votes
1answer
63 views

One Question about the Fubini's Theorem

The Fubini's Theorem says: If function $f:X \times Y \rightarrow R$ is integrable over $X \times Y$, then $$ \int_{X \times Y}f(x,y)dxdy = \int_{X}dx\int_{Y}f(x,y)dy = \int_{Y}dy\int_{X}f(x,y)dx. $$ ...
0
votes
2answers
83 views

An odd function $f$ is differentiable at zero. Prove $f'(0)=0$?

I know that $f'$ of an even function is odd function, thus I have $f(x)=f(-x)$. However I'd no idea how to prove that $f'(0)=0$? Please answer my question...
0
votes
2answers
37 views

Chain rule using the expression F=150W^1/3

Suppose the attendence of a baseball game was denoted by W alone in the format F=(150W)^1/3. Is this function (strictly) concave or convex. Explain. To which I answered that it would be strictly ...
0
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0answers
55 views

Fun Lagrange multiplier problem?

Do any of you have a fun or interesting Lagrange multiplier problem that would be suitable for undergraduate calculus students? I'm planning on working through a standard Lagrange multiplier problem ...
-2
votes
2answers
60 views

Consider the integral.

So I missed class today and decided to take a look at the homework assigned. This notation is unfamiliar to me. Up until this point, we have just been finding over and underestimates based on ...
0
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0answers
60 views

Optimization to minimize cost function

I have the function $C=Tq^{\frac 1a }+F$. Where $C$ is total cost, $q$ is output, $a$ is a positive parametric constant, $F$ is fixed cost, and $T$ measures the technology available to the firm ...
2
votes
2answers
40 views

Can I use l'Hôpital's rule on this?

Question: $\displaystyle\lim\limits_{x\to\pi/2}\frac{\ln(\sin x)}{\cos x}$ This was in a set of questions about l'Hôpital's rule. However, the numerator is undefined and the denominator equals ...
0
votes
1answer
55 views

Find the range of a complicated function

I need to find the range of the following function : $$f(x,y) = \sqrt[4]{\frac{4x - 3y + 5}{3y-4x + 13}}$$ So my thoughts about it are first the bottom part $( 3y - 4x + 13 )$ must be greater than ...
2
votes
2answers
61 views

Ellipse with center in origin

The purpose is to fit data to a ellipse which center is the origin $(x_0=0,y_0=0)$. I found the general quadratic curve: $$ax^2+2bxy+cy^2+2dx+2fy+g=0$$ Reference: ...
1
vote
2answers
26 views

Posivite Values P For Which a Series Converges

Consider the following series $$ \sum_{n=1}^{\infty}\frac{(n!)^2}{(Pn)!} $$ For which positive values of $P$ does the series converges? Not sure if the Ratio Comparison Test would be useful here: ...
0
votes
2answers
49 views

Power Series: Derivative

Given a Banach space $E$. Consider a series: $$|t|\leq R:\quad\sum_{k=0}^\infty A_k t^k\quad(A_k\in E)$$ Is there an elegant proof of: $$\left(\sum_{k=0}^\infty A_k t^k\right)'=\sum_{k=0}^\infty A_k ...
2
votes
1answer
41 views

Closed-form of prime zeta values

The prime zeta function is defined as $$P(s)=\sum_{p\,\in\mathrm{\mathcal P}} \frac{1}{p^s},$$ where $\mathcal P$ is the set of prime numbers. It converges for all $\Re(s)>1$. There is a related ...
2
votes
1answer
32 views

How does this pattern work?

I know that $$ \sum_{k=0}^{\infty}\frac{1}{k\,!}=e=\lim_{n \to \infty}{(1+\frac{1}{n})^n} $$ but why $$ \sum_{k=0}^{\infty}\frac{1}{(2k)\,!} = \cosh(1) $$ and $$ \sum_{k=1}^{\infty}\frac{1}{(2k+1)\,!} ...
1
vote
1answer
32 views

How to compute the following taylor series expansion

I'm supposed to find the Taylor series expansion of $(\arcsin(x))^2$, but I can't think of a proper solution .The derivative doesn't show much promise since it still contains the $\arcsin(x)$ ...
1
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4answers
110 views

Proof that $f(x)=\frac{(\sin x)^3}{x}$ gets maximum in $(0,\infty)$.

I have this problem, and I got stuck in my proof Prove $f(x)=\dfrac{(\sin x)^3}{x}$ gets maximum in $(0,\infty)$. My Proof $$(1)\lim_{x \to 0+} \frac{(\sin x)^3}{x}= 0$$ $$(2)\lim_{x \to \infty} ...
-1
votes
2answers
49 views

Evaluation of $\lim_{n\to\infty} \frac{1}{\ln n}\sum_{k=1}^n \frac{k}{k^2+1}$

I want to evaluate $$ \lim_{n\to\infty} \frac{1}{\ln n}\sum_{k=1}^n \frac{k}{k^2+1} $$ I can already see that $$\lim_{n\to\infty}\frac{1}{\ln n} = 0$$ so how do we go about solving this?
1
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0answers
25 views

Classification of discontinuities of multidimensional functions/maps

I know that for functions $f: \mathbb{R} \rightarrow \mathbb{R}$ there exist plenty of references which describe various discontinuities that such functions can exhibit (e.g. jump, asymptotic, etc). ...
0
votes
1answer
34 views

Integration of logarithms

Integral of $$1\over x(log_4^x)^2$$. I changed $$(log_4^x)^2$$ to $$2(ln(4)/(ln(x))$$ Then I integrated and got $$.5ln(4)(ln(ln(x))$$ The answer to the problem is $$ -ln(4)/(log_4^x) + C...$$ ...
0
votes
1answer
17 views

Calculating line integral over this vector field?

The path is the straight line from (1,1,1) to (2,-3,3). The force field is x^2yi +zj +(2x-y)k. I had the path as x(t)= (t+1,-4t+1,2t+1) from 0 to 1. The final answer I got was 20/3, but I'm not sure I ...
0
votes
2answers
40 views

Unbounded operator

Assume you have an operator $T : \operatorname{dom(T)}\rightarrow H$. Now we also know that $ran(T)$ is finite-dimensional. Does this imply that $T$ is bounded?( So is $T$ a bounded map $T \in ...
1
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0answers
24 views

Functional derivative of a repeated integral

For a given function $f$, the functional derivative of the functional $\mathcal{F}[\rho]=\int f(x,\rho(x))\,dx$ is well-known to be $\frac{\delta}{\delta \rho(x)}\mathcal{F}[\rho]=\frac{\partial ...
0
votes
2answers
50 views

What y=0 is also a solution for: $y' = y^{2}$?

By using integration, I can know that: $y=\frac{1}{-x+C}$ is a solution. But why $y' = y^{2}$ is also a solution?
0
votes
1answer
117 views

Find the values of $a$ and $b$ such that $b = \lim_{n \to + \infty } n^a \sum_{k = 1}^n {\frac{1}{{\sqrt k }}}$

Evaluate $a ;b$ values which satisfy this equality : $$b = \lim_{n \to + \infty } n^a \sum_{k = 1}^n {\frac{1}{{\sqrt k }}}$$ My solution $$ \sqrt n \le \sum\limits_{k = 1}^n ...
0
votes
1answer
30 views

Linear aproximation using only variables

yet again another school assignment. I have a hard time with conceptual math that only involves variables. I much prefer to have real values, but I need to learn! I am pretty stuck, so baby steps ...