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

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0
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0answers
29 views

Having trouble understanding what exactly I am integrating for a hydrostatic force/pressure calculus question

So the question is: a trough has vertical ends that are trapezoids with parallel sides of length 4m (top) and 2m (bottom) and a height of 3m. If the trough is filled with water to a depth of 2m, find ...
0
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1answer
14 views

Significance of derivative in finding square free decomposition

If $gcd(f(x),`f(x))=1$ then f(x) is square free. But what is the reason behind taking derivative of f(x)? How one came to this conclusion?
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0answers
43 views

Hint for solving an indefinite integral

I would like to solve the following integral $$\int \frac{dx}{x^2 \sqrt[4]{(a-x^2)(b+x^2)}},$$ where $a$ and $b$ are the real constants. My attempt: $$\sqrt[4]{(a-x^2)(b+x^2)} = \sqrt[4]{-\bigg[x^4 ...
0
votes
1answer
67 views

For what x the series converges?

I have the following problem: For what values of x the following series: $$ \sum_{n=1}^\infty n! \cdot (x-4)^n$$ a) Converges absolutely. b) Converges conditionally. I started by using the ratio ...
5
votes
1answer
88 views

Is my proof correct? If $f$ has a finite number of discontinuities on $[a, b]$, then it is integrable on $[a, b]$

Question: Suppose a function $f(x)$ over the interval $[a, b]$ is bounded and has only a finite number of discontinuous points on $[a, b]$. I intend to prove that it must be integrable on $[a, b]$. ...
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1answer
24 views

Forms of functions in dynamical systems

I wanted to read some introductory material about dynamical systems since I might need a basic understanding of them in a related task. So, as far as I see, in a continuous time dynamical system, we ...
2
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0answers
57 views

Graphing the surface $z = xy$

Let the surface $S \subset \mathbb{R}^3$ be the graph of the function $f:\mathbb{R}^2 \to \mathbb{R}, f (x, y) = xy$. Let $U$ be the portion of $S$ for which $x^2 + y^2 ≤ 2$ and let $C$ be the ...
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1answer
36 views

Show that there is a continuous function $h$ over $[a,b]$ for which $h(x) \leq f(x)$ and $\int_{a}^b (f(x)-h(x))dx < \epsilon$

Assume $f$ is integrable over $[a,b]$ and $\epsilon > 0$. Show that there is a continuous function $h$ over $[a,b]$ for which $h(x) \leq f(x)$ for all $x \in [a,b]$ and $\displaystyle \int_{a}^b ...
1
vote
1answer
23 views

euler's theorem of homogeneuos function

suppose that $f$ is a mapping from $\mathbb R^n \to \mathbb R$. a positively homogeneous of degree $n$ and suppose $f_1, f_2, .... f_N$ are continuous for $a \neq 0$. then $$\sum_{i=1}^n a_if_i(a)= ...
2
votes
2answers
38 views

Question about Leonard Gillman's proof of the divergence of the Harmonic series.

Leonard Gillman (1917 – 2009) was an American mathematician, emeritus professor at the University of Texas, Austin. His proof of the divergence of the Harmonic series appeared in The College ...
2
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2answers
53 views

$(x-x_0)^0$ in power series [duplicate]

When I first studied power series in high school, the teacher gave the following general definition: \begin{equation} f(x)=\sum_{n=0}^{\infty}a_n (x-x_0)^n \end{equation} He then proceeded to ...
1
vote
3answers
19 views

Uncertainties in angle measurement

I wonder why uncertainties in angle measurement MUST be in radians. For example, I want to calculate the uncertainty in measuring the function $y= \sin (\theta)$ when the angle is measured $\theta = ...
1
vote
2answers
46 views

Finding an integral involving logarithmic functions: $\int_0^\infty\frac{1}{z[\ln(z)]^2}dz$ [closed]

Finding the following integral: $$\int_0^\infty \frac{1}{z[\ln(z)]^2} dz$$
2
votes
2answers
28 views

Difficulty finding a power series representation

I have to find a power series representation and interval of convergence for $$f(x) = \frac{x-x^2}{(1+2x)^3}$$ Noting that $\frac{1}{1+2x}=\frac{1}{1-(-2x)}=\sum_{n=0}^\infty(-2x)^n$, I start taking ...
0
votes
0answers
26 views

Finding mass, moments about x-, y- axes and center of mass [closed]

Find the mass, moments about $x$-, $y$-axes, and the center of mass for the following thin plate of uniform density. Sketch the region and indicate the location of the center of the mass. The region ...
1
vote
1answer
25 views

Show that there is a step function $g$ over $[a,b]$

Assume $f$ is integrable over $[a,b]$ and $\epsilon > 0$. Show that there is a step function $g$ over $[a,b]$ for which $g(x) \leq f(x)$ for all $x \in [a,b]$ and $\displaystyle \int_{a}^b ...
0
votes
1answer
19 views

If $f(x)=\sum_{0}^{\infty}b_n(x-5)^n$ for all $x$, write a formula for $b_8$.

If $f(x)=\sum_{0}^{\infty}b_n(x-5)^n$ for all $x$, write a formula for $b_8$. Now I know that $b_n=\dfrac{f^{(n)}(5)}{n!}$. I have tried various things but I think there is something wrong with my ...
0
votes
1answer
23 views

Mean Value Theorem question, numbers that satisfy the theorem [closed]

Verify the function satisfies the hypothesis of the Mean value theorem on the given interval. Then find all numbers $c$ that satisfy the conclusion of the Mean Value Theorem $$f(t) = 3t^3 - 9t + 6 ...
1
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2answers
62 views

Integrating functions I've never dealth with

If $a>0$ show that $$\lim_{n \to \infty} \int_a^\pi \frac{\sin(nx)}{nx} dx = 0.$$ I've never dealt with non-elementary integral functions before and I'm not sure why this would show up on a ...
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0answers
21 views

Necessity of $C^{1}$ hypothesis in fundamental theorem for line integrals

The statement for the fundamental theorem for line integrals I have in my (unpublished) textbook is: Let U ⊆ Rn be an open set, let φ : [a,b] → U be a piecewise smooth curve, and let $Ω = C_{φ}$. Let ...
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0answers
36 views

Do irrational derivative orders exist?

There are many notations for a derivative of $y$ with respect to $x$. Two, most popular are $y'(x)$ or just $y'$ and $\frac{dy}{dx}$. For higher order derivatives, the more consistent notation is ...
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3answers
47 views

How to write the following polynomial in $(1-\frac{x}{a}) (1-\frac{x}{b}) (1-\frac{x}{c}) (1-\frac{x}{d})$?

I was given the following problem: Write the polynomial $f(x) = \frac{1}{24} \displaystyle \prod_{i \mathop = 1}^4 (x-i)$ in the form $(1-\frac{x}{a}) (1-\frac{x}{b}) (1-\frac{x}{c}) (1-\frac{x}{d})$ ...
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3answers
62 views

How do I compute the following complex number? [on hold]

This was the problem I was given: Compute the complex number for $\frac{(18-7i)}{(12-5i)}$. I was told to write this in the form of $a+bi$. So please give me a hint of how to do this. :)
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votes
5answers
92 views

Evaluate $\int_0^5 e^{-2t}\sin(t)\, \mathrm{d}t$.

$$\int_0^5 e^{-2t}\sin(t) \,\mathrm{d}t$$ I know I should be able to integrate this by parts but I can't seem to get the parts I choose to make the result any easier to integrate.
8
votes
1answer
59 views

Prove that $\exists x_0, x_1\in (0,1)$, such that $\frac{f'(x_0)}{x_0}+\frac{f'(x_1)}{x_1^2}=5$

Let $f:[0,1]\to\mathbb{R}$ be a differentiable function, such that $f(0)=0$ and $f(1)=1$. Prove that there exist different $x_0, x_1\in (0,1)$, such that ...
0
votes
1answer
29 views

Solve $ \int_{0}^{x}ty(t)dt=x^2y(x)$ [closed]

Solve $ \int_{0}^{x}ty(t)dt=x^2y(x)$ it is difficult for me, i can't sovle it
0
votes
1answer
38 views

Solve $(x^2+y^2+1)dy+xydx=0$ [closed]

Solve $(x^2+y^2+1)dy+xydx=0$ I tried many times but I can't solve it
1
vote
1answer
38 views

Why does $\frac{\frac12 x+\frac18x^2+O(x^3)}{\frac12x-\frac18x^2+O(x^3)}=1+\frac12x+O(x^2)$?

I was reading the solution to a limit through Taylor expansion but did not understand this passage: $$g(x)=\frac{1-\sqrt{1-x}}{\sqrt{1+x}-1}=\frac{\frac12 ...
0
votes
2answers
19 views

How would I calculate an derivative with two unknown variables?

I'm learning calculus II. I recently wondered what if I had two unknown variables in an function, and wanted to take an derivative. Let's say there is a function $f(x,y)=2x^3+7y^2$ How would I ...
2
votes
1answer
42 views

Convergence of $\int_{0}^{1} \frac{\sqrt {e^2+x^2} - e^{\cos x}}{\tan^ax}dx$

The problem I'm facing is as it follow: For which values of $a$ the integral converges: $$\int_{0}^{1} \frac{\sqrt {e^2+x^2} - e^{\cos x}}{\tan^ax}dx$$ So far I figured that if $a< 1$, the ...
0
votes
1answer
25 views

linear ODE problem

A substance evaporates at a rate proportional to the exposed surface. If a spherical mothball of radius $\frac{1}{2}$ cm has radius $0.4$ cm after $6$ months, how long will it take: For ...
0
votes
1answer
58 views

Suppose $f$ is analytic and $f(a) = f(b) = 0$. Show that $|f(z)| ≤ |{z − a \over 1 − z\bar{a}}| · |{z − b \over 1 − z\bar{b}}|$.

Suppose $f$ is analytic from $D(0, 1)$ to $D(0, 1)$ and $f(a) = f(b) = 0$ for two different numbers $a, b$ in $D(0, 1)$. Show that $\left\vert f(z) \right\vert ≤ \left\vert{z − a \over 1 − z\bar{a} ...
3
votes
0answers
71 views

Sketching the surface $x^2+y^2+4z^2 = 1$

Let the surface $S \subset \mathbb{R}^3$ be the solutions of the equation $g(x, y, z)$ $ = 1$ where $g(x,y,z)=x^2 +y^2 +4z^2$. Let $U$ be the finite region of S satisfying $z > 0$ and let $C$ be ...
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1answer
18 views

Differentiating a squared quantity

I was reading through my electromagnetism book where i came across this statement where when we differentiate wrt a squared quantity rather than a single quantity we multiply it by $\frac{1}{2}$. ...
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0answers
49 views

Theorem proving help required for finals [closed]

Please help me with this question, I have been trying to solve it but I cannot derive the answer. Thank you so much in advance
1
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1answer
14 views

Fourier Transform point force on a half-space

I have to calculate the following Inverse Fourier-Transform, which describe the potential function for a point force on a half-space: ...
0
votes
1answer
24 views

Local extrema of $x^3+y^2+6y$

I have to find local extrema of $x^3+y^2+6y$. I found out that the stationary points are $(0,-3)$. I also found the Hess matrix for this function and computed the determinant, which is $12x$. But now ...
0
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1answer
38 views

Limit help required for finals preparation [closed]

please help me solve this following limits. I have been try but I cannot seem to solve them. Thank you in advance!
2
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1answer
32 views

Calculating the flux of $\langle x,y,x\rangle$ over $z=1-x-y$

Find the flux of $\bar F = \langle x, y, x\rangle$ over $z = 1 - x - y$ in the first octant use the upward unit normal ($\bar n$) flux = $\int\int_S \bar F \cdot \bar n dA$ $dS = \sqrt 3 dA$ $\bar ...
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0answers
20 views

How do solve the integral $\int_{-\infty}^{\infty} exp(-a|t|)exp(-a|t-\tau|)dt$?

How do you determine the autocorrelation of $g(t) = exp (-a|t|)?$ Plugging it into the equation $ \int_{-\infty}^{\infty} g(t)g(t-\tau) dt$ would result into something like $ \int_{-\infty}^{\infty} ...
2
votes
1answer
40 views

A problem about tangents and areas.

Let $f(x) = x^3 - 4x^2 + 4x$, its graph denoted $C$. If for any $x_1 \not = \frac 43$, the tangent of $f(x)$ at the point $P_1(x_1, f(x_1))$ intersects $C$ at another point $P_2(x_2, f(x_2))$, and ...
2
votes
3answers
96 views

Solve $(x^3+e^y)y'=3x^2$. Help me please I can't solve it. [closed]

Solve $(x^3+e^y)y'=3x^2$. Help me please I can't solve it.
-1
votes
0answers
16 views

minimizing interval of integral [closed]

I'm locking for a solution (or at least a hint) for the following problem: given a function $f(x)$ and a constant $c$ solve $ \int_{a}^{b} f(x) dx \geq c$ and $ |a-b|$ should be minimal thanks in ...
3
votes
1answer
79 views

Why does $z^{-1}$ not have an anti derivative?

I had been given the question as shown in the following image, with the answer also given. Surely however the anti derivative of $z^{-1}$ would be $\log(z)$ ? I have seen a similar question asked, ...
3
votes
3answers
281 views

Evaluating $\lim\limits_{x \to -\infty} \sqrt{x^2 + 3x} - \sqrt{x^2 + x}$. Is Wolfram wrong or is it me?

What am I doing wrong? My attempt $$\begin{align} \lim_{x \to -\infty} \sqrt{x^2 + 3x} - \sqrt{x^2 + x} &= \lim_{x \to -\infty} \sqrt{x^2 + 3x} - \sqrt{x^2 + x} \cdot \frac{\sqrt{x^2 + 3x} + ...
4
votes
3answers
78 views

Why $\displaystyle\sum_{n=1}^\infty {\ln(n^2) \over (n^2) }$ converges?

I have to prove that the series $\displaystyle\sum_{n=1}^\infty {\ln(n^2) \over n^2 }$ converges. The ratio test is inconclusive, so I should use the comparison test, but which series should I compare ...
3
votes
2answers
34 views

Number of real roots of $f ' ( x )$

Let $$f(x)=(x-a)(x-b)^3(x-c)^5(x-d)^7 $$ where $a,b,c,d$ are real numbers with $a < b < c < d$ . Thus $ f ( x )$ has $16$ real roots counting multiplicities and among them $4$ are ...
2
votes
4answers
49 views

ODE $y(1+2xy)dx+x(1-xy)dy=0$

$$y(1+2xy)dx+x(1-xy)dy=0$$ I have tried to isolate $\frac{dy}{dx}$ and got the following: $$\frac{dy}{dx}=-\frac{y(1+2xy)}{x(1-xy)}$$ but I understand that the terms have to be in the same ...
0
votes
1answer
23 views

Derivative of $\arctan y $

I'm given $f(x)=\tan x$ and $g(y)=\arctan y.$ I am also told that $g'(y)=\frac{1}{f'(g(y))}.$ I'm then asked to use this to find the derivative of $\arctan x.$ Here is what I have so far: ...
0
votes
0answers
7 views

Question conserning the existence and continuity of derivatives of function's shperical mean

I heard a rumor that the claim beneath is true and I'm trying to prove it (or find a counterexample). Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$, $f\in C^k(\mathbb{R}^n)$. Fix $\varepsilon > 0$ ...