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

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2
votes
1answer
188 views

Area bounded by three functions

How would I do the following question. Sketch the region bounded by the graphs of $f(x)=cos(x)$ $g(x)=x-\frac{\pi}{2}$ and $x=0$ I know that $\cos(x)=0$ has an intercept at $\frac{\pi}{2}$ and ...
2
votes
1answer
79 views

Find radius and interval of convergence for $\sum_{n=1}^\infty$ $\frac{5^n}{n+2^n}x^n$

Find radius and interval of convergence for $\sum_{n=1}^\infty$ $\frac{5^n}{n+2^n}x^n$ So if i apply ratio test, I get $\lim_{x\to \infty} 5|x||\frac{n+2^n}{n+1+2^{n+1}}| $ Now need help to to check ...
10
votes
2answers
133 views

How to find the minimum of $f(x)=(\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x))^2$?

I need to find the minimum of $f(x)$ with $$f(x)=(\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x))^2$$ Could you help me with some clues?
2
votes
2answers
1k views

Finding area in a bounded region

How would I solve the following problem? Sketch the region bounded by the graphs of $f(x)=x^2-4x$ and $-3x^2$ and then find the area of the region. I have made a sketch for this question but ...
1
vote
2answers
967 views

Linear approximation to ln(x) at x = 1, then estimate ln(1.08)

I know that the derivative of $\ln(x)$, or log of whatever base (x) = $(1/x)$ *the original function. If x is a more complicated expression, then the derivative would be $(x'/x)*f(x)$. If I knew the ...
0
votes
3answers
1k views

Linear approximation to $y = \sqrt{1-x}$ at $x=0$, then approximate $\sqrt{0.9}$ and $\sqrt{0.99}$

How do I find this? I know that the derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$. Here, I would plug in $(1-x)$ instead of $x$. When $x = 0$, the slope would evaluate to $\dfrac{1}{2}$. I got ...
0
votes
1answer
74 views

Maximum and minimum of $y = 4x-8*(\cos(x))$ between $-\pi$ and $\pi$

I have found that the maximum of this function is at $\pi$, where the function will equal $$4\pi+8,$$ which is approximately $20$. However, I tried to get the minimum value, and it was incorrect. The ...
2
votes
3answers
113 views

Linear approximation to 1/0.254

The question says: Use linear approximation to approximate $1/0.254$. I know that $1/0.25 = 4$. Where do I proceed from next. Do I subtract $0.004$ from the answer, or what else could I do?
3
votes
2answers
54 views

Graph Concavity Test

I'm studying for my final, and I'm having a problem with one of the questions. Everything before hand has been going fine and is correct, but I'm not understanding this part of the concavity test. ...
1
vote
0answers
97 views

A seemingly complicated proof.

Let $$ f(x) = (ax)^{\big(b\sin x\big)} $$ $$ g(x) = (x - \pi/2)(\sin x)^{\cfrac 1 {(x-\pi/2)}} $$ Also, let $ h(x) = g(x) - f(x) $ be defined in $(\pi/2, \pi) $ If $h(x)$ is an increasing function ...
2
votes
6answers
1k views

Prove that $\cos(x)$ doesn't have a limit as $x$ approaches infinity.

I've been working on this one for quite a long time now. I have to prove that $\cos(x)$ has no limit as $x$ approaches infinity. Let $\epsilon>o$ and M be any number greater than 0, so that for ...
4
votes
2answers
194 views

About the Beta function : $\text{B}\left(\frac{4}{3},\frac{2}{3}\right)$.

Find the value of : $\text{B}\left(\frac{4}{3},\frac{2}{3}\right)$, where $\text{B}(x,y)$ is the Beta function. Why do I need this ? Because I want to calculate : $$ \int\limits_{ - \infty }^\infty ...
2
votes
0answers
142 views

show that $\lim f '(x)=0$ as $x\to \infty$; and deduce that $\lim f(x)$ exist

If $f:[0,\infty)\rightarrow\mathbb R$ be a continuously differentiable function s.t. $$ \ f'(x)=\dfrac{1}{(x^2+\sin^2(x)+f(x))} ,\forall x\geq 1$$ Show that $ \lim f'(x)=0 $ as $x\to \infty$; and ...
2
votes
2answers
106 views

What is $ \lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$?

How to solve the following limit question? $$\lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$$ Thanks a lot.
2
votes
1answer
350 views

Differentiation under the Integral sign for the Lebesgue integral

I want to prove the following version of Liebniz's Rule: Let $f:[a,b]\times [c,d]\to \mathbb{R}$ be integrable with respect to the first variable, $\phi,\psi:[c,d]\to [a,b]$ be differentiable and let ...
3
votes
1answer
73 views

When does $\lim\limits_{n\to\infty}\int_{b}^{a_n}f_n(x)dx=\lim\limits_{n\to\infty}\int_b^\infty f_n(x)dx$ hold?

Let $\{a_n\}\subset \mathbb{R}$ be sequence and $$f_n:[b,\infty)\longrightarrow \mathbb{R}, \qquad n=1,2,\dots .$$ Assume that $$\lim_{n\longrightarrow\infty}a_n=+\infty.$$ Obviously, from the ...
6
votes
1answer
116 views

Find the value of a function with definite integrals

I am trying to understand a paper of Maynard Smith (1974), that connects biology with game theory. I don't want to overwhelm you with useless stuff, but I have this definite integrals: ...
3
votes
2answers
101 views

Calculate $\sum_{n=2}^\infty ({n^4+2n^3-3n^2-8n-3\over(n+2)!})$

Calculate $\sum_{n=2}^\infty ({n^4+2n^3-3n^2-8n-3\over(n+2)!})$ I thought about maybe breaking the polynomial in two different fractions in order to make the sum more manageable and reduce it to ...
2
votes
1answer
36 views

parametric integration

Is there a mistake in the bottom of page 5 of this document? INTEGRATION: THE FEYNMAN WAY $$\frac{\partial}{\partial b} e^{be^{ix}}= e^{ix}e^{be^{ix}}$$ instead of $ib e^{be^{ix}}e^{ix}$? thank you ...
1
vote
1answer
147 views

Solve the integral $\int_{x=0}^{\infty}\frac{1}{x}\int_{y=0}^{x}\frac{\cos{(x-y)}-\cos{x}}{y}dydx$

Find the value of: $$I=\int_{x=0}^{\infty}\dfrac{1}{x}\int_{y=0}^{x}\dfrac{\cos{(x-y)}-\cos{x}}{y}dy \ dx$$ I think we could take: ...
1
vote
1answer
162 views

Enigmatic optimization problem

My problem, which I proposed to myself months ago is based on the simple optimization problem in which you find the best path for a lifeguard to rescue a drowning victim. Obviously the shortest ...
3
votes
2answers
146 views

If the iterated limits are equal does the double sequence converge?

If $(a_{m,n})$ is a double sequence (in $\mathbb{R}$ or $\mathbb{C}$) and $\lim_m\lim_n a_{m,n}=\lim_n\lim_m a_{m,n}=a$ then can we deduce $\lim_{m,n}a_{m,n}=a$? More specifically can we deduce ...
1
vote
1answer
20 views

Why are these calculus expressions equivalent?

Why are these two expressions equivalent? $e^{-t^2}[\frac{dy}{dt}-2ty] = e^{-t^2}bt$ $\frac{d}{dt}e^{-t^2}y=e^{-t^2}bt $ I realize that $ \frac{d}{dt}e^{-t^2}=-2te^{-t^2} $
0
votes
1answer
156 views

Prove that D (the differential operator) maps V (a vector space) into V.

I'm quite confused about what "into" means here and, more importantly, how I am supposed to prove that something maps a vector space into (not onto) another vector space. Here's some of the ...
2
votes
5answers
81 views

Radius convergence of power series

How would I go about finding the radius of convergence using a limit ratio test? Can I get a hint for this one? $\displaystyle\sum_{j=1}^\infty\frac{(jx)^j}{j!}$
3
votes
1answer
255 views

About calculating a Green's function

For a positive integer $d>=3$ and a real number $h_0$ I have the differential equation for a function $f$ of $x$, $$x^2 h^2 f'' + \frac{6-d}{2}xh^2 f' = \frac{d(d-2)}{2}h^2f$$ where $h = h_0 ...
2
votes
1answer
235 views

Finding interval of convergence for $\sum\limits_{k=1}^\infty k!(x-4)^k$

I'm trying to find the interval of convergence for the series... $$\sum_{k=1}^\infty k!(x-4)^k$$ I found that $$\lim_{k\to\infty}\left|\frac{(k+1)!(x-4)^{k+1}}{k!(x-4)^{k}}\right|=\infty$$ which as ...
4
votes
1answer
147 views

How to find a partial derivative of an implicitly defined function at a point

Suppose that the relation $\frac{x^2}{2} + \frac{y^2}{2} + \frac{z^2}{2} + xy + xz =\frac{7}{2}$ defines $z$ as a function of $x, y$ around the point $(1, 1, 1)$. Find $\frac{dz}{dy}$ at $(1, 1, ...
0
votes
1answer
61 views

Approximation of sum of square

I would like to know if there is a way to compute/approximate this formula: $$\sum_{i=0}^n (x_i-y_i)^2$$ when we only know: $$\sum_{i=0}^n x_i$$ and $$\sum_{i=0}^n y_i$$ Thanks for any help!
2
votes
2answers
170 views

Prove the next integral converges for any $x>0$: $\int_0^{\infty}t^{x-1}e^{-t}dt$

Prove the next integral converges for any $x>0$: $$\int_0^{\infty}t^{x-1}e^{-t}dt$$ I can't find a proper way to prove that, But what i did so far was: integration by parts: ...
1
vote
3answers
133 views

How to differentiate integrals with variable limits?

I'd like to evaluate the following two derivatives. $$ (1) \frac{d}{d\theta^*} \int_{\theta^{*}}^1x^{\theta}\theta^\alpha g(\theta)d\theta $$ $$ (2) \frac{d}{dx} ...
1
vote
0answers
264 views

Hessian after coordinate changing

Let $f\colon \Bbb R^n\to\Bbb R$. Let $z=Px$ coordinate changing. $P$ is $n\times n$ constant matrix, $x$ and $z$ are the variables in $\Bbb R^n$. Does anyone know a formula which express how the ...
2
votes
3answers
88 views

The inverse function of $e^{x^2}$

Is there a way to find the inverse of $e^{x^2}$? Or, if this is not possible,are there some functions that approximate very well the inverse of $e^{x^2}$, especially when $x$ is very small and large?
2
votes
3answers
54 views

Comparison test integral convergence

$$\int_0^{\infty} \frac{e^x}{x^x} \,\mathrm dx$$ How can I tell if this integral converges or not? I was thinking of using the comparison test, but I can't think of anything to compare it to. Could ...
1
vote
2answers
156 views

Prove the following equation: $\int_0^{\infty} \frac{\cos{(x)}}{1+x} \,\mathrm dx=\int_0^{\infty} \frac{\sin{(x)}}{(1+x)^2} \,\mathrm dx$

Prove the following equation: $$\int_0^{\infty} \frac{\cos{(x)}}{1+x} \,\mathrm dx=\int_0^{\infty} \frac{\sin{(x)}}{(1+x)^2} \,\mathrm dx$$
2
votes
1answer
86 views

The rate of increase of the Gamma Function over real numbers

If $$ x_1 > x_2 > 0$$ and $$\Delta{x}>0$$ does it follow that: $$\ln\Gamma(x_1 + \Delta{x}) - \ln\Gamma(x_1) \ge \ln\Gamma(x_2 + \Delta{x}) - \ln\Gamma(x_2)$$ Would it be enough to show ...
2
votes
1answer
59 views

A question regarding a proof about limit and continuity

I am trying to understand a part of the following proof. Prove that $(1)$ and $(2)$ are equivalent: $(1)$ $\lim_{x \to c}f(x)=f(c)$ $(2)$ $f$ is continuous at $c$. I understood the proof of $(1) ...
4
votes
3answers
224 views

Radical integral question calculus

I have a question in calculus. Let $F(x)=\int_1^\sqrt{x} t^2\cos( \pi t)dt$ Find $F'(4)$ I know $F'(X$) $=\int_1^\sqrt{x} x^2\cos( \pi x)dx$ So I made $u=x^{\frac{1}{2}}$ and I got ...
1
vote
2answers
149 views

The right way to calculate the volume obtained by rotating the area between 2 graphs around the x axis

If i have 2 graphs: $f(x)=x\cdot \frac{\sqrt{1-x^{2}}}{2},\:g(x)=\frac{\sqrt{1-x^{2}}}{2}$ And need to calculate the volume obtained by rotating the area between $f(x)$ and $g(x)$ Around the ...
0
votes
2answers
307 views

Finding the centroid of a polar curve

The curve is $r = e^{-b\theta}$ where $b > 0$ and $θ \in [0, \infty)$. I got that the arc length is $\frac{\sqrt{b^2 + 1}}{b}$ (is this correct?), but computing the centroid $(x, y)$ looks awful. ...
0
votes
2answers
69 views

Convergence of $\sum_{n=1}^{\infty }\left ( 1+\frac{1}{n} \right )^{n^{2}}\cdot \frac{(-1)^n}{ne^{n}}$

Why the $\sum_{n=1}^{\infty }\left ( 1+\frac{1}{n} \right )^{n^{2}}\cdot \frac{(-1)^n}{ne^{n}}$ is converges and $\sum_{n=1}^{\infty }\left ( 1+\frac{1}{n} \right )^{n^{2}}\cdot \frac{1}{ne^{n}}$ is ...
2
votes
2answers
146 views

Infinity divided by infinity and dirac delta?

A dirac delta produces something that's infinitely long, and it could also be seen as infinitely thin. Why do we define the surface of a dirac delta to be 1. If length$\times$width $= \infty \cdot ...
2
votes
1answer
815 views

Definition of local maxima, local minima

Wikipedia says that: A real-valued function f defined on a real line is said to have a local (or relative) maximum point at the point x∗, if there exists some ε > 0 such that f(x∗) ≥ f(x) when |x − ...
0
votes
2answers
75 views

Create a Generating function

Let $P$ be the set of permutations all of whose cycles are of even length. Prove that the exponential generating function for $P$ is $\dfrac{1}{\sqrt{1-x^2}}$.
2
votes
3answers
1k views

$x^4 + 4x^3 - 2x^2 - 12x + k$ has 4 real roots. Find the condition on k.

The question is: $f(x) = x^4 + 4x^3 - 2x^2 - 12x + k$ has 4 real roots. What values can k take? Please drop a hint!
1
vote
1answer
116 views

Cross product with orthonormal basis

Let ${\{u_1,u_2,\ldots,u_n}\}$ be an ortohonormal basis of $\mathbb{R^n}$ and $f,g:\mathbb{R}^n\to\mathbb{R}$ differtinable functions at $p\in\mathbb{R^n}$. If $n=3$ does ...
4
votes
2answers
119 views

$x^4 + 4rx + 3s = 0$ has no real roots. Relate $r, s$.

It is given that $x^4 + 4rx + 3s = 0$ has no real roots. What can be said about r and s? a) $r^2 < s^3$ b) $r^2 > s^3$ c) $r^4 < s^3$ d) $r^4 > s^3$ How to even begin??
4
votes
4answers
158 views

How do I find this limit [duplicate]

$$ \lim_{x \to \infty} \sqrt{x^4-3x^2-1}-x^2 $$ The answer is $$ \frac{-3}{2} $$ according to Wolfram alpha.
0
votes
2answers
71 views

Determine if the integral converges: $\int_1^{\infty} \frac{\arctan (px)}{x^q}dx$

Determine if the integral converges: $$\int_1^{\infty} \frac{\arctan (px)}{x^q}dx$$ where $p,q\in\Bbb R$.
10
votes
2answers
837 views

Integrate $2\int x^2\, \sec^2x \,\tan x\, dx$

$$ 2\int x^2\, \sec^2x \,\tan x\, \mathrm{d}x $$ How to solve this using integration by parts? WolframAlpha can solve it, but is unable to give a step-by-step solution, and has a different answer to ...