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

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0
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1answer
153 views

Graphing a Riemann Sum

I'm supposed to use the Riemann sum to approximate the area under the graph of $f(x)= 2e^{-x}$ on the interval $1$ to $2$ using $n=5$ sub-intervals with the selected points as the right end points Is ...
30
votes
1answer
822 views

Prove $\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}\mathrm dx=\frac{\pi^2}8-\frac12$

How can I prove the following identity? $$\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}\mathrm dx=\frac{\pi^2}8-\frac12$$
0
votes
1answer
106 views

Which of these norms are equivalent to the canonical one

Regarding the space of continuously differentiable functions $C^1([0,1])$, I am wondering which of these norms are equivalent to the norm $||x||= ||x||_{\infty} + ||x'||_{\infty}$. The candidates are ...
1
vote
2answers
91 views

$\lim\limits_{n \to \infty } ({X_{n + 1}} - {X_n}) = c$ implies $\lim\limits_{n \to \infty } {{{X_n}} \over n} = c $

Let $\{X_n\}$ be a sequence such that: $$\lim\limits_{n \to \infty } ({X_{n + 1}} - {X_n}) = c$$ Prove that: $$\lim\limits_{n \to \infty } {{{X_n}} \over n} = c$$ I've tried many ...
0
votes
1answer
68 views

Mean Value Theorem with a constant [closed]

Consider a function $f$ and a point $a$. Suppose that there is a number $K$ such that $g(x) = f(a) + K(x − a)$ satisfies $$\lim_{x \to a} \frac{f(x) − g(x)}{x-a} = 0 $$ Prove that $f$ is ...
1
vote
0answers
78 views

Find $\int 1/(1+\sin(x) \cos(x))\,dx$

Integrate $$\int \frac{1}{1+\sin(x) \cos(x)}dx$$ Now, I have the correct solution in my book, but I can't see where I go wrong. Attempt:
0
votes
3answers
1k views

farmers pen part a part b

A) A rectangular pen is built with one side against a barn, 200 meters of fencing are used for the other three sides of the pen. What dimensions maximize the area of the pen? B) A rancher plans to ...
0
votes
3answers
47 views

Does a differentiable $f[g(x)]$ imply a differentiable $g(x)$ or the reverse?

Does a differentiable $f[g(x)]$ imply a differentiable $g(x)$? Does a differentiable $g(x)$ imply a differentiable $f[g(x)]$? Thanks in advance
6
votes
3answers
307 views

Compute $\int_0^\infty \frac{dx}{1+x^3}$

Problem Compute $$\displaystyle \int_0^\infty \frac{dx}{1+x^3}.$$ Solution I do partial fractions $$\frac{1}{x^3+1}= \frac{2-x}{3 \left( x^{2}-x+1 \right)}+\frac{1}{3 \left( x+1 \right)}.$$ But we ...
1
vote
3answers
72 views

Find any asymptotes for $f(x)=\frac{x}{\sqrt[3]{x^2-1}}$

Find any asymptotes for the function $f(x)$. $$f(x)=\frac{x}{\sqrt[3]{x^2-1}}$$ I don't even know how to start. Your help would be very appreciated.
33
votes
2answers
2k views

Integral $\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$$ It also can be represented as $$I=\int_0^{\pi/4}\frac{\phi^2}{\cos \phi\,\sqrt{\cos ...
0
votes
3answers
751 views

Rate of change of radius and volume

They put a gas bubble in someone's eye. The volume of a gas bubble changes from $0.4$ $cc$ to $1.6$ $cc$ in $74$ hours. Assuming that the rate of change of the radius is constant, find (a) The rate ...
0
votes
1answer
48 views

How prove this $12hkS(h,k)+12khS(n,k)=h^2+k^2-3hk+1$

let $h,k$ is positive interger number,and such $gcd(h,k)=1$,let $$S(h,k)=\sum_{r=1}^{k-1}\dfrac{r}{k}\left(\dfrac{hr}{k}-\left[\dfrac{hr}{k}\right]-\dfrac{1}{2}\right)$$ where $[x]$ is the largest ...
1
vote
1answer
323 views

Integration of $\int\frac{x^2-1}{\sqrt{x^4+1}} \, dx$

Integration of $\displaystyle \int\frac{x^2-1}{\sqrt{x^4+1}} \,dx$ $\bf{My\; Try}$:: Let $x^2=\tan \theta$ and $\displaystyle 2xdx = \sec^2 \theta \, d\theta\Rightarrow dx = \frac{\sec^2 ...
1
vote
0answers
41 views

integration until infinity

I need to evaluate this integral: $$\int_4^\infty \frac{0.697(x-4)}{x^{3.5}} \, dx$$ I first bring out the constants, so it becomes $\lim\limits_{t\rightarrow\infty}0.697\int\limits_4^t ...
3
votes
1answer
294 views

On finding polynomials that approximate a function and its derivative (extensions of Stone-Weierstrass?)

The Stone-Weierstrass Theorem tells us that we can approximate any continuous $f:\mathbb{R}^n\to\mathbb{R}$ arbitrary well on a compact subset of $\mathbb{R}^n$ by some polynomial. Suppose that $f$ is ...
8
votes
3answers
235 views

Proving $\int_a^b \frac {x dx}{\sqrt{(x^2-a^2)(b^2-x^2)}} = \frac {\pi}{2} $

Can anybody here help me to prove that $$ \int_a^b \frac {x\, \mathrm dx}{\sqrt{(x^2-a^2)(b^2-x^2)}} = \frac {\pi}{2} $$ Thanks for your help.
0
votes
2answers
17 views

Help with the chain rule $h(t)=f(t, X(t))$

Assume we have the function $h(t)=f(t, X(t)): \mathbb{R}\rightarrow \mathbb{R}$. How to I calculate $h'$? I thought of letting $g:t \rightarrow(t,X(t))$ and then $h' = g'(t)f'(g(t)) = (1, ...
0
votes
2answers
27 views

Calculating derivative

In order to solve a mathematical problem I have to calculate the following derivative: $\frac{\delta}{\delta k}\frac{11 + \sum_{i = 0}^{k-1}i}{k}$ Does anyone know this derivative?
1
vote
2answers
29 views

Function range problem

$(sin^2\theta +\sin\theta -1)/(sin^2\theta -\sin\theta+ 2)$ . It is asked to find the range of this function. Here i was assumed a variable $z=sin\theta$ so that i get $-1<=z<=1 $ and then ...
1
vote
1answer
146 views

$\int\dfrac{\cos^5x\sin^3x}{1+\cos2x}dx = \dfrac{\sin^4x}{8}-\dfrac{\sin^6x}{12} +C$ or $\dfrac{\cos^6x}{12}-\dfrac{\cos^4x}{8} +C$?

$\int\dfrac{\cos ^5x\sin ^3x}{1+\cos 2x}dx = \dfrac{\sin ^4x}{8}-\dfrac{\sin ^6x}{12} +C$ or $\dfrac{\cos ^6x}{12}-\dfrac{\sin ^4x}{8} +C$? $\int\dfrac{\cos ^5x\sin ^3x}{1+\cos 2x}dx$ = ...
1
vote
1answer
57 views

Show that $\{ 1, 1-x , 1-2x + {1 \over 2} x^2\}$ a orthogonal system

I have to show that the following set: $$A = \left\{ 1, 1-x , 1-2x + {1 \over 2} x^2\right\}$$ is orthogonal system in relative to the inner product $$\langle f, g\rangle = \int ^\infty_0 ...
1
vote
4answers
316 views

Show that $1<\sqrt{1+x^3}<1+x^3$ for $x>0$

The problem: Show that $1<\sqrt{1+x^3}<1+x^3$ for $x>0$ How do I solve this using the Fundamental Theorem of Calculus
1
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2answers
261 views

Proving a 2nd order Mean-Value theorem [closed]

Let $f\in C^1([a,b])$ have 2nd-order derivative in $(a,b)$. Prove that there exists $c\in (a,b)$ such that $$f(b)-2 f\left(\frac{a+b}{2}\right)+f(a)=\frac{1}{4} (b-a)^2 f''(c)$$
0
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1answer
340 views

Second Order Linear Differential Equations with Constant Coefficients Containing Trigonometric Functions

I'm having trouble applying the method of undetermined coefficients, as explained in Apostol's Calculus, to second order linear differential equations with constant coefficients containing ...
1
vote
3answers
121 views

Evaluation of $\int^1_0 \cos^2\frac{(m+n)\pi x}{2}\sin^2\frac{(n-m)\pi x}{2}dx$.

Just wondering whether the following integration is something special. To be more specific, is it equal to some constant real number, please? I found a integration table involving sine and cosine but ...
-2
votes
1answer
38 views

Sum of the Series , Calculus Homework

I put this into WolfRam and got e^(3/5), but I am trying to figure out how to arrive to that answer?
2
votes
0answers
44 views

Calculate the Maclaurin series, using binomial series

Calculate the Maclaurin series for the following function This is a note from my teacher through email "Question 5a (this question) is now a bonus question, since it requires binomial series that we ...
0
votes
2answers
33 views

how can i simplify this function to find the zeros?

$(r^2-1)x-r^2(r+1)x^2+2r^3x^3-r^3x^4=0$ All of the variables just have a one digit exponent.. So the biggest exponent is $X^4$
0
votes
1answer
114 views

Compute the 10th derivative

$f(x) = (\cos(5x^2) - 1 )/ x^2 $ at $x = 0$ We were given the hint to use the MacLaurin series for f(x). I get how to do it if it was just $\cos(5x^2)$ but what would I do with the other values in ...
1
vote
2answers
42 views

Integer Part of sequence convergence

I was trying to solve the following exercise. If $(a_n) \in \mathbb{R}$ and $(a_n)\rightarrow {1}/{2}$ show that $[a_n] \rightarrow 0$ , where $[~]$ the integer part. I was trying to solve it using ...
0
votes
1answer
16 views

Optimization and Modeling using Derivatives

I am having trouble getting started on a problem. I know the answer is m=50 n=50, but I am having troubles getting there. ...
0
votes
2answers
38 views

The integral $\int_0^1 \frac{z^j (1-z)^k}{j! k!} \, dz$

I am seeking help for a definite integral. $\int_0^1 \frac{z^j (1-z)^k}{j! k!} \, dz$
1
vote
1answer
937 views

Find the Maclaurin series of the function f(x) = (7 x^2) sin (2 x)

Find the Maclaurin series of the function $f(x) = (7 x^2) sin (2 x)$ $(f(x) = \sum_{n=0}^{\infty} c_n x^n) $ That is what is given on the question, we have to fill in 5 blanks $c_3$ to $c_7$ The ...
1
vote
0answers
46 views

differentiable sets of an arbitrary function

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an arbitrary function. I have proved that the set $E$ of points where $f$ is continuous is a Borel set. For $k\in \mathbb{N}$ and $p,q,r\in \mathbb{Q}$, ...
0
votes
2answers
247 views

Dirchlet Riemann Integrable in certain interval

Considering the Dirichlet function f : f (x) = { 1 if x is rational 0 if x is irrational } I want to know if this function can be Riemann Integrable in ...
6
votes
2answers
1k views

If $f(a) = g(a)$ and $f'(x) < g'(x)$ for all $x \in (a,b)$, then $f(b) < g(b)$

Assume that $f$ and $g$ are continuous on $[a, b]$ and differentiable on $(a, b)$. Prove that if $f(a) = g(a)$ and $f'(x) < g'(x)$ for all $x \in (a,b)$, then $f(b) < g(b)$. I ...
2
votes
4answers
981 views

$\sin 2\theta +\sin \theta =1$

I tried Wolframalpha to solve this equation. The solution is $\theta\approx 0.355$. Since once a wise guy at MSE told me not to trust machines, I would like to know what methods can be used to solve ...
1
vote
0answers
569 views

find critical point of a trig function on a closed interval

$F (x)=\sin (3x)$ on a closed interval $[-\pi/4, \pi/3]$ I am having trouble finding the critical point I find the first derivative set it to 0 and I conclude that the critical point would be ...
2
votes
2answers
268 views

Meeting point for 5 people with least distance travelled (interview question)

I had an interview today and I'm completely stumped on what they asked me. Essentially: if you are given 5 people on a 2D grid, and you need to meet at a point with the least amount of distance ...
2
votes
1answer
60 views

Examples of a problem solved by a well-chosen derivative equaling zero

What are examples of problems which are solved by taking a derivative of a well-chosen function, and finding that it is zero, therefore the function must be constant? I can think of a few: Show ...
0
votes
1answer
66 views

Show $f$ concave, $C^2$ implies $f''\leq 0$

Suppose I wanted to show that a concave function $f:(a,b) \to \mathbb{R}$ which is $C^2$ must have negative second derivative at each $x\in (a,b)$. I might try this by finite difference, noting that ...
2
votes
1answer
178 views

Weak implicit function theorem. Is my proof alright?

I want to prove that if we have $ f \in C(A \times U,Y)$, where $x_0 \in U$, $\lambda_0 \in A$ and A and U are open sets in Banach spaces and Y is a Banach space too and we have that: ...
1
vote
2answers
193 views

Evaluating $\lim\limits_{n \to \infty} \sum_{i=1}^{n}\sqrt{\frac{i}{n^{3}}}$ using Riemann Sums and FTC

Factoring out the $\frac {1}{n}$ out of the sigma, we get: $$\lim_{n \to \infty} \sum_{i=1}^{n}\sqrt{\frac{i}{n}}\cdot\frac{1}{n}$$ which looks awfully similar to $$\lim_{n \to \infty} ...
3
votes
0answers
43 views

Convergence of a integral

The question is: exists a natural number $n \geq 2$such that $$ \displaystyle\int_{0}^{+ \infty} \displaystyle\frac{\ln r}{(1 + r^2)^{n}} r dr< \infty ?$$ I am trying to do this : i know that ...
1
vote
1answer
67 views

Why is $\sum_{s=1}^{\infty }\left(-1\right)^{s+1} \left[\vphantom{\Large A}-2^{-s}\ \left(2^s-2\right)\ \zeta\left(s\right)\right]= 1-\log (2)$?

could you give and explanation why $$ \sum_{s=1}^{\infty }\left(-1\right)^{s+1} \left[\vphantom{\Large A}-2^{-s}\ \left(2^s-2\right)\ \zeta\left(s\right)\right] = 1-\log (2) $$ and $$\sum ...
1
vote
1answer
123 views

$\sum\frac{k^{k/2}}{k!}$ converge or diverge?

Does the following series converge or diverge? $\sum\frac{k^{k/2}}{k!}$ I did the ratio test $\frac{a_{n+1}}{a_n}$ I did $\frac{(k+1)^{(k+1)/2}}{(k+1)!}$ $\frac{k!}{k^{k/2}}$ Then I simplified ...
0
votes
1answer
78 views

Find permutation index of multiple lists where corresponding list indices match

I have several date time values: Mon 17h10 Tue 20h30 Wed 21h45 that maps to the following lists ...
6
votes
2answers
324 views

Find the limit: $\lim \limits_{x \to 1} \left( {\frac{x}{{x - 1}} - \frac{1}{{\ln x}}} \right)$

Find: $$\lim\limits_{x \to 1} \left( {\frac{x}{{x - 1}} - \frac{1}{{\ln x}}} \right) $$ Without using L'Hospital or Taylor approximations Thanks in advance
0
votes
1answer
111 views

The integral of a function that is 0 a.e is 0

I am working on this problem: Let $f = 0$ for all $x \in [a,b] $ except for $x$ in a set of Lebesgue measure zero. Then $\int_a^b f \,dx = 0$ if the integral exists. Here are my ideas: Split $f$ into ...