Daily math problems for October 22, 2026 — Descriptive Statistics & Sampling, Euclidean Geometry, Calculus & more

Sum $=2+4+4+6+9=25$.

Mean $={\displaystyle \frac{25}{5}}=5$.

A reflection keeps size and shape unchanged, so the image is congruent. Enlargements with scale factor $\mathrm{\ne }1$ change the size.

(a) A polynomial — continuous on all of $\mathbb{R}$ ✓.

(b) Discontinuous at $x=0$ (not defined).

(c) Discontinuous at $x=\frac{\pi }{2}+k\pi$.

(d) Defined only for $x\ge 0$, so not continuous on negative reals.

$3x-7=2x+5$

$3x-2x=5+7$

$x=12$.

General solution of $y^{\mathrm{\prime }}=3y$: $y(x)=C{e}^{3x}$.

Apply $y(0)=5$: $C{e}^0=C=5$.

So $y(x)=5e^{3x}$.

$2{\cos }^{2}x+\cos x-1=0\Rightarrow (2\cos x-1)(\cos x+1)=0$. Roots: $\cos x=\frac{1}{2}$ ($x=\frac{\pi }{3},\frac{5\pi }{3}$) and $\cos x=-1$ ($x=\pi$). Three solutions.

Apply $\phi$ to $a\cdot a^{-1}=e$: $\phi (a)\phi (a^{-1})=e_H$, so $\phi (a^{-1})=\phi (a{)}^{-1}$.

${\displaystyle \frac{1+i}{1-i}}={\displaystyle \frac{(1+i{)}^2}{(1-i)(1+i)}}={\displaystyle \frac{2i}{2}}=i$, whose argument is ${\displaystyle \frac{\pi }{2}}$.

Outer: ${\sec }^2(x^2)$. Inner derivative: $2x$. So $f^{\mathrm{\prime }}(x)=2x{\sec }^2(x^2)$.

Every element is its own inverse: $A\mathrm{△}A=\mathrm{\varnothing }$.