Show $f$ is infinitely differentiable February 7

Assume that $E$ is a compact Lebesgue measurable subset of $\mathbb{R}$ and let, $$f(t) = \int_E \cos(tx) \, dx, \hspace{1mm} t \in \mathbb{R}$$Show that $f$ is infinitely differentiable.I have a solution using the mean value theorem, induction, and

Find the center of circle given two tangent lines (the lines are parallel) and a point. February 7    3

Find the center of circle given two tangent lines (the lines are parallel) and a point.
How to find the center of a circle if the circle is passing through $(-1,6)$ and tangent to the lines $x-2y+8=0$ and $2x+y+6=0$?Hint: The lines are perpendicular, and the center of the circl ...

finding the radius of the circle given a coordinate February 7    2

finding the radius of the circle given a coordinate
find the radius of the circle with center at (-1,2) if a chord of length 10 is bisected at (4,-3).(this is exactly what our professor given to us)im thinking of using the distance formula wh ...

Proving function is infinitely differentiable February 7

On some interval, there is a function $h(x)$ that is differentiable and continuous. Then, there is another function $c(x)$ that is smooth(infinitely differentiable). Given that $h'(x) = c(h(x))$, show that $h(x)$ is also smooth on the interval.

Externally tangent circle coordinates February 7

Externally tangent circle coordinates
I am trying to program following case as show in figure below. I have two circles at (x1,y1) and (x2,y2). Having coordinates, we can mention that the two circles are making an angle \theta_2 ...

Without computing, is the integral of (lower limit 0, upper limit 1) t(t-1)(t-2)dt positive or negative February 7    1

Without computing, is the integral of (lower limit 0, upper limit 1) t(t-1)(t-2)dt positive or negative
I have to graph the function, but I don't think I'm doing it right. A polynomial function is continuous and hence can only change signs at roots. To see whether the function is positive on $ ...

Zeros of this function February 7

Let $$f(z)=\gamma + z^{\beta_2-\beta_1}$$ where $\gamma\in \mathbb{R}$, $\beta_1\in \mathbb{Z}$, $\beta_2 \in \mathbb{Z}$ and $\beta_2 > \beta_1$. The variable $z$ takes complex values.Is there a way of expressing the zeros of $f(z)$ in terms of the

Finding an orthogonal vector February 7    3

How will I be able to do this question?Find a vector C $\in\mathbb{C}^3$ which is orthogonal to both A and B. Where $A = [2,0,i]^T$ and $B = [i,-1,2]^T$. The answer is $C =c[i/2, 5/2, 1]^T$ , but how did they get that? And does $\mathbb{C}^3$ mean a

find all vectors orthogonal to both: February 7    2

Find all vectors v = (x, y, z) orthogonal to both$u_1$ = (2, -1, 3)$u_2$ = (0, 0, 0) I'm not sure how to get to the answer of s(1, 2, 0) + t(0, 3, 1). I know how to find a vector orthogonal to just 1, getting confused with the both part.What vectors

Monomial Algebras problem : lcm February 7

I am trying to prove the following:Monomial Algebras (Second Edition) Rafael H. Villarreal - Exercise 6.1.22Let I and J be two ideals generated by finite sets of monomials F and G, respectively, prove that the intersection I ∩ J is generated by the se

How to solve $\left\frac{x+4}{ax+2}\right \frac1x$ February 7    2

How to solve: $$\left|\frac{x+4}{ax+2}\right| > \frac{1}{x}$$What I have done:I) $x < 0$:Obviously this part of the inequation is $x\in(-\infty, 0), x\neq \frac{-2}{a}$II) $x > 0$:$$\left|\frac{x+4}{ax+2}\right| > \frac1x$$ $$\frac{|x+4|}{|ax+

Having trouble proving transitivity February 7    1

We have a universal set of lowercase alphabet letters, $U = \{a, b, ... , z\}$ . For sets $A,B \subseteq U$ we can define a relation, $A \sim B$ as long as the number of elements that are in either $A$ or $B$, but not both, is even. Prove that $\sim$

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