Daily math problems for August 16, 2026 — Calculus, Logic, Quadratic Function & more

The exponential is its own antiderivative: $\int e^{x}dx=e^x+C$.

The contrapositive of "$n^2$ even $\Rightarrow$ $n$ even" is "$n$ odd $\Rightarrow$ $n^2$ odd". A statement and its contrapositive always have the same truth value.

$x_V=-{\displaystyle \frac{-4}{2\cdot 1}}=2$. (And $f(2)=4-8+3=-1$ is the minimum.)

$P(-1)=0$, so $(X+1)$ is a factor. Polynomial division: $P(X)=(X+1)(X^2-4)=(X+1)(X-2)(X+2)$.

Matrix multiplication is not commutative in general. (Concrete example: $A=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$, $B=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)$ — check $AB\mathrm{\ne }BA$.)

$\tan x={\displaystyle \frac{\sin x}{\cos x}}$ has vertical asymptotes where $\cos x=0$, i.e. $x={\displaystyle \frac{\pi }{2}}+n\pi$.

$d=\sqrt{(-3{)}^2+4^2}=\sqrt{25}=5$.

With $Y=X^2$: $Y^2-5Y+4=0\Rightarrow Y\in \{1,4\}$. Then $X^2=1\Rightarrow X=\pm 1$ and $X^2=4\Rightarrow X=\pm 2$. Four real solutions.

The integral of a non-negative continuous function over $[a,b]$ is non-negative (representing the area under the curve).

$R=\sqrt{1+3}=2$, so $f(x)=2\sin (x+\pi \mathrm{/}3)$. Max value: $2$.