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

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1answer
74 views

Find the limit $ \int_{0}^1 f(x^n) dx$ [closed]

Let $f\colon [0,1] \to [0,1]$ be a continuous functions. Find the value of $$ \lim_{n \to \infty} \int\limits_{0}^1 f(x^n) \ dx$$
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1answer
68 views

find $f(g(x))$ given $f(x)$ and $g(x)$

let $f:\mathbb R \to \mathbb R$ and $g:\mathbb R \to \mathbb R$ be defined $$f(x) = \begin{cases} -1 & x<\frac 12 \\ -\frac12, & \text{if} -\frac12\leq x<0 \\ 0 & \text{x=0}\\ 1 ...
1
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1answer
38 views

Computing directional derivatives of the max of three different functions

Consider $f_1,f_2,f_3: \mathbb{R} \to \mathbb{R}$ defined as $f_1(x):= (1/2)|x|, f_2(x):= (x-1)^2, f_3(x):=(1/2)[(1/2)x+1]$ and take $g(x):=\text{max}\{f_1(x),f_2(x),f_3(x)\}$ How do I compute the ...
0
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2answers
51 views

$f(1)=0, f'(x) > f(x) $ for all $x>1,$ then $f(x)$ is

if $f(1)=0, f'(x)> f(x)$ for all $x>1$ then $f(x)$ is positive valued for all $x>1$ negative valued for all $x>1$ positive valued on $(1,2)$ but negatiive valued on $[2,\infty]$ 4 none ...
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0answers
69 views

Integral change at jump discontinuity.

The above is the graph of $f(t)$. $q(x)$ is defined as: $$q(x) = \int_{0}^{2x - 1} f(t) dt$$ What is $q'(5/2)$ I would say it DOES NOT exist (discontinuity at $x = 4$) after Itake $q'(x)$
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2answers
148 views

f(x) takes only rational values and f(1)=1. then find f(2)

Let $f$ be real valued continuous function on $[0,3]$ suppose that $f(x)$ takes only rational values and $f(1)=1$. then find $f(2)$: $1)\, 2$ $2)\, 4$ $3)\, 8$ $4)\,$ none of the above.
2
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2answers
47 views

Differentiability of f(x)=x ($\sqrt{x}+\sqrt{x+9}$).

Let f(x)=x ($\sqrt{x}+\sqrt{x+9}$). Question is to check if it is differentiable or not at x=0. Edit : FOR ABOVE FUNCTION I HAVE TO CHOOSE OUT FOLLOWING 1. continously differentiable at x=0 ...
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0answers
37 views

Show that $\gamma$ is a generalized helix if and only if the binormal $b(s)$ has a constant angle $\theta$ with a fixed vector $a\neq 0$.

A regular curve $\gamma$ in $\mathbb R^3$ with curvature $\kappa(t) > 0$ is a generalized helix if the tangent vector $t$ makes a fixed angle $\theta$ with a fixed unit vector $a$. Now let ...
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0answers
12 views

Find a formula for $n_s'$ for a regular plane curve $\gamma$ of arbitrary speed.

I've already found that $n_s' = -\kappa_s t$, where $n_s$ is the signed normal, $\kappa_s$ is the curvature and $t$ is the tangent, when $\gamma$ is a unit speed curve. Here I used that $t' = ...
1
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1answer
30 views

$f_1=\frac{x}{x+1}$ define $ f_n (x)=f_1 (f_{n-1}(x))$

Let$ f_1=\frac{x}{x+1}$ define $ f_n (x)=f_1 (f_{n-1}(x)) $where n $\ge $2. Then $f_n(x)$is Decreasing in n 2 increasing in n 3 initially decreasing in n then increasing in n 4 initially ...
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1answer
27 views

Using Fundamental Theorem of Calculus during getting first order differential equation

I am having problem while solving a question below: I have looked into solution to understand the process of solving this problem, and encountered a part that I do not understand. The part I can't ...
6
votes
4answers
82 views

How to determine the curve?

In the figure above, segment $PQ$ is determined by two points: $P: (t,0)$ and $Q: (1,t)$, where $t\in [0,1]$ continuously increases and decreases between $0$ and $1$. Then this gives a close region ...
0
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1answer
40 views

$\lim_{n\to \infty} \frac{1-x^{-2n}}{1+x^{-2n}} $

$ \lim_{n\to \infty} \frac{1-x^{-2n}}{1+x^{-2n}} $, x $\gt $0 1.1 2.-1 3.0 4 limit does not exist. My attempt: here if x is between o and 1 then limit is -1 and if x is more than 1 limit is 1. ...
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1answer
42 views

Properties of a continuous function $f:[0,1]\to [0,1]$

Consider any continuous function $f:[0,1]\to [0,1]$. Which one of following statement is incorrect? $f$ always has atleast one maximum in interval $[0,1]$ $f$ always has atleast one minimum in ...
0
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1answer
54 views

Variation of Delta function integrated

We all know: $$\int_0^\infty \delta(y) dy = 1.$$ How about $$\int_{-\infty}^\infty y\delta(y) dy .$$ The solution of this is $0$. I have no idea how to get this. thx,
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2answers
41 views

What is the point of the remainder term in the Taylor series

Take the first order taylor approximation: We say $f$ is differentiable at $x_0 \in S^o$ if $\exists \space \nabla f(x_0)$ and a function $R: \mathbb{R}^n \to \mathbb{R}$ such that $f(x) = f(x_0) + ...
0
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0answers
51 views

Maximum value, minimum value of $|x^2 +2x -3 | + 1.5 \ln x $

Find the maximum and minimum value of function: $$F(x)= |x^2 +2x -3 | + 1.5 \ln x $$ over the interval: $$[\frac{1}{2},4]$$ $(21 +3\ln 2, -1.5\ln 2)$ $(21 + \ln 1.5,0)$ $(21 + 3\ln 2,0)$ $(21 +\ln ...
1
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1answer
40 views

Fundamental Theorem of Calculus - Need Confirmation of my answer?

*Not homework, just extra practice. Find $g'(x)$ given that $$g(x) = \int_3^{x^4} t \sqrt{2+t} dt.$$ ... If I understand it correctly, would I simply replace the $t$'s with $x^4$, so $$g'(x) = ...
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2answers
40 views

Finding a local max with an integral?

This is not a homework question. This is a question from a past test, in an attempt to study, and I'm completely lost. Consider $$g(x) = \displaystyle \int_{0}^x (t^2 - 4)e^{t^2} dt$$ Find all values ...
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2answers
48 views

Use differentiation to find a power series representation for $f(x)=\frac{1}{(1+x)^2}$ [duplicate]

Need help on all steps. Use differentiation to find a power series representation for $$ f(x)=\frac{1}{(1+x)^2} $$
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0answers
19 views

definite integral of matrix expression

In animal population dynamics, we estimate the number of new individuals that enter a population between two surveys, some of which die between surveys as: $$Bnew = B*\left[\int_{0}^1 a^{\left( ...
1
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1answer
67 views

integration by parts transforming a vector integral to vector times divergence?

In Jackson's 'classical electrodynamics' he re-expresses a volume integral of a vector in terms of a moment like divergence: $$\int \mathbf{J} d^3 x = - \int \mathbf{x} ( \boldsymbol{\nabla} \cdot ...
0
votes
0answers
62 views

Orientation of a plane curve

Let $B(x_0)$ be an open unit ball in $\mathbb{R}^2$. Assume that $f:\overline{B(x_0)} \rightarrow \mathbb{R}^2$ is a diffeomorphism and $f(x_0)=y_0$. Then $f(\partial B(x_0))$ is a Jordan curve and ...
2
votes
1answer
49 views

Differentiation using the Chain Rule

$$y=\frac { \cos(1+x) }{ 1+\cos(x) } $$ Steps I took: $$y'=\frac { (-\sin(1+x))(1+\cos(x))-(\cos(1+x))(-\sin(x)) }{ (1+\cos(x))^2 } $$ $$y'=\frac { (\cos(1+x))(\sin(x)) }{ (1+\cos(x))^2 } -\frac { ...
0
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1answer
24 views

Approximating discontinous derivatives with continuous derivatives?

Let $f(x)=|x|$, so that $f'(x)= \frac x{|x|}$. Let $g$ be a function with $$g'(x)=\frac x{\sqrt {x^2+\frac1c}}.$$ Then $\lim_{c\to\infty}g'(x)=f'(x)$. Both functions have the same domain, but $g$ is a ...
0
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1answer
74 views

Resources for improving algebra skills?

I took my first calculus test today and my experience during that test confirmed my fears leading up to the test: I'm terrible at Algebra. One of my weakest areas is factoring. Can you give me some ...
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5answers
65 views

How to find the limit of $\frac{\tan(2x)}{\sin(3x)}$ as $x$ approaches $0$ analytically?

I'm taking a calculus 1 course, and I'm running into issues when trig gets added into the mix. I keep trying to simplify $\frac{\tan(2x)}{\sin(3x)}$ so that I can either plug in $0$ or find one of the ...
1
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4answers
53 views

Investigate the convergence of $\sum_{n=1}^\infty$ $n+1\over (n^3+2)^{1\over 2}$.

Investigate the convergence of $\sum_{n=1}^\infty$ $n+1\over (n^3+2)^{1\over 2}$. Since $n+1<n^2$ and $(n^3+2)^{1\over 2}>n^{3\over 2}$ for all $n\ge2$, then we can compare the above seres to ...
2
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1answer
62 views

Loss functions for regression

[From PRML Bishop, p:46] The average or expected loss function is given by $$E[L] = \int\int (y(x)-t)^2 p(x,t)\ \ dx\ \ dt$$, where, the loss function $L = (y(x)-t)^2$, given x and the ...
1
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4answers
103 views

Limit and infinite sums. Finding $\lim_{x\rightarrow\infty}\sum^{\infty}_{k=1}\frac{1}{k^3 x-k^2}$

Could anyone help me with this problem. Compute $$\lim_{x\rightarrow\infty}\sum^{\infty}_{k=1}\dfrac{1}{k^3 x-k^2}$$ I don't know how to change a limit and a sum. Could you help me with this problem ...
0
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2answers
235 views

Prove that $\ln x$ is not uniformly continuous on $(0,1]$

Prove that $\ln x$ is not uniformly continuous on $(0,1]$. Got all confused on how to prove this kind of things... Any hints would be great!
3
votes
2answers
131 views

If $\sum_{n_0}^{\infty} a_n$ diverges prove that $\sum_{n_0}^{\infty} \frac{a_n}{a_1+a_2+…+a_n} = +\infty $

I've tried to use the limit test: $\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = C$ in the series of $a_n$ and $\frac{a_n}{a_1+a_2+...+a_n}$ but it was inconclusive. The question does not especify ...
2
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3answers
30 views

Finding all values of $p$ for which $\operatorname{div}\vec{F} = 0$

Here is the full question: Let $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$ and let $r = \|\vec{r}\|$. Let $\vec{F} = r^p\vec{r}$. Find all values of $p$ for which div $\vec{F} = 0$ I'm a bit confused on ...
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1answer
35 views

How to solve this differential equation or find a better expression for its solution.

Let $y:[0,1] \rightarrow \mathbb R$ be a differentiable function. Let $\alpha, \beta,\gamma \in \mathbb{R}$ be some fixed parameters. The equation is the following: $$y'(x) = y(x)\left( \alpha - ...
0
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1answer
37 views

Explanation for derivative of $x*e^x$

Hello could someone explain me why exactly is the derivative of $f(x)$ is the following: $ f(x) = x * e^x \rightarrow f'(x) = e^x + x e^x = e^x ( 1 + x)$ Any help is appreciated, ...
2
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4answers
32 views

Sum of entries made one

Assume that I have performed some loop and summed values of $x_i$ and the sum is as follows $$S= \sum_{i=1}^n (x_i *0.0001) = 0.9999$$ However I would like to have the sum to be equal to 1 i.e, ...
2
votes
2answers
86 views

Does $\int_{\pi}^{\infty}\frac{\sin(x)}{\log(x)}dx$ diverge? [closed]

Does $\displaystyle \int_{\pi}^{\infty}\frac{\sin(x)}{\log(x)}dx$ diverge? If so, how can we show it?
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1answer
15 views

Requirement for continuity of $f~'$ in the expansion of Taylors formula to measure error

Assume $f$ has a continuous second derivative $f~''$ in some neighborhood of $a$.Then, for every $x$ in this neighborhood, we have $f(x) = f(a) + f~'(a)(x-a) + E_1(x)$ , where $$E_1(x) = \int_a^x ...
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4answers
58 views

Prove this alternative formula for derivative $f'(x)$

Show that: $$f'(x) = \frac{f(x + h) - f(x - h)}{2h} \tag 1$$ Proof: If $(1)$ is true then $f'(x) = \displaystyle \frac{f(x + h) - f(x) + f(x) - f(x - h)}{2h} = \frac{f(x + h) - f(x)}{2h} - ...
0
votes
1answer
31 views

Transform a boundary condition into polar coordinates

Suppose I have the boundary condition $u_y(x,0) = 0$. How can I transform it into polar coordinates? $$ x= r\cos(\theta), y = r\sin(\theta), $$ so $$ \frac{\partial u}{\partial y} = ? $$ I'm a ...
2
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1answer
59 views

solve $y'=\frac{2x+y-1}{x+y+2}$

I need some help solving this differential equation, I'm doing everything as I was taught but I just need help at the last step, I can't seem to get this right. the ode we want to solve is ...
1
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4answers
216 views

Find all continuous functions satisfying $\int_0^xf=(f(x))^2+C$ for some constant $C \neq 0$.

Find all continuous functions $f$ satisfying $$\int_0^xf=(f(x))^2+C$$ for some constant $C \neq 0$, assuming that $f$ has at most one $0$. I have a question about the solution to this problem. It ...
0
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0answers
16 views

Is there some information missing in this rates of^2 change question am getting 213mph

A plane is flying at an altitude of 8 miles and passes over a radar station. When the plane is 12miles from the base of the radar, the radar detects that it horizontal distance is changing at 320 mph. ...
1
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2answers
79 views

How to integrate $36/(4x^2-12x+9)$?

I've just learned integration today and my teacher wasn't too helpful in explaining this. If anyone could help me here I would be most appreciative! Thank you! $$\int\frac{36}{4x^2-12x+9}dx$$
1
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1answer
51 views

Find all solutions of the 2'nd order ODE: $x'' + x =0$.

Find all solutions of the 2'nd order ODE: $x'' + x=0$. I've found by inspection of $x (t)=e^{at} $ that $\cos t $, $\sin t $ are solutions. How do I see, that all linear combinations of these are all ...
1
vote
3answers
58 views

Differentiability of $f (x)=\frac{x}{1+|x|}$

Discuss continuity and differentiability of following function $$f (x)=\frac{x}{1+|x|}$$ Where x is any real number My attempt: Being a rational funtion it is continous everywhere. Also its ...
0
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1answer
35 views

Solid of revolution about y axis

I am worried that my bounds are incorrect, so I am just making sure that my solution is valid Question Let R be the region in the first quadrant bounded by the curve $y = (x − 2)^2$. Compute the ...
1
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2answers
42 views

Fundamental Theorem of Calculus Integration

I've been trying to work on this problem for an hour and for some reason I'm apparently not inputting the correct answer. The problem was to find the derivative of $\int_{x^5}^{x^7} (2t-1)^3 dt$ and I ...
1
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2answers
142 views

Intuitive explanation why integral sin(x) is -cos(x)

I realize there's a bunch of similar questions, but for derivative. However, this is a little bit different. I understand pretty well why derivative of sin(x) is ...
0
votes
3answers
37 views

Find another linearly independent solution of $2$-order ODE: $x^{''} + 2tx^{'}+(1-\frac 3 {4t^2}) x = 0$ defined on $(0,\infty)$.

Find another linearly independent solution of $2$-order ODE: $x^{''} + 2tx^{'}+(1-\frac 3 {4t^2}) x = 0$ defined on $(0,\infty)$. I've the following 2'nd order ODE: $$x^{''} + 2tx^{'}+(1-\frac 3 ...