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.

$x_V = -\dfrac{-4}{2 \cdot 1} = 2$. (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: $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, $B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ — check $AB \ne BA$.)

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

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

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 $f(x) = 2\sin(x + \pi/3)$. Max value: $2$.