if m ≤ f(x) ≤ m for a ≤ x ≤ b, where m is the absolute minimum and m is the absolute maximum of f on the…

if m ≤ f(x) ≤ m for a ≤ x ≤ b, where m is the absolute minimum and m is the absolute maximum of f on the interval a, b, then m(b - a) ≤ ∫a to b f(x) dx ≤ m(b - a). use this property to estimate the value of the integral. ∫π/12 to π/9 5 tan(3x) dx smaller value larger value

if m ≤ f(x) ≤ m for a ≤ x ≤ b, where m is the absolute minimum and m is the absolute maximum of f on the interval a, b, then m(b - a) ≤ ∫a to b f(x) dx ≤ m(b - a). use this property to estimate the value of the integral. ∫π/12 to π/9 5 tan(3x) dx smaller value larger value

Answer

Explanation:

Step1: Encontrar m y M

La función es $y = f(x)=5\tan(3x)$ en el intervalo $[\frac{\pi}{12},\frac{\pi}{9}]$. La derivada de $y = 5\tan(3x)$ es $y'=15\sec^{2}(3x)>0$ para todo $x$ en el dominio real donde está definida la función. Entonces, la función es creciente en el intervalo $[\frac{\pi}{12},\frac{\pi}{9}]$. Evaluamos la función en los extremos del intervalo: $f(\frac{\pi}{12}) = 5\tan(3\times\frac{\pi}{12})=5\tan(\frac{\pi}{4}) = 5$ $f(\frac{\pi}{9}) = 5\tan(3\times\frac{\pi}{9})=5\tan(\frac{\pi}{3}) = 5\sqrt{3}$ Así, $m = 5$ y $M = 5\sqrt{3}$.

Step2: Calcular $b - a$

Tenemos $a=\frac{\pi}{12}$ y $b=\frac{\pi}{9}$, entonces $b - a=\frac{\pi}{9}-\frac{\pi}{12}=\frac{4\pi - 3\pi}{36}=\frac{\pi}{36}$.

Step3: Aplicar la desigualdad

Usando la desigualdad $m(b - a)\leq\int_{a}^{b}f(x)dx\leq M(b - a)$: El valor más pequeño es $m(b - a)=5\times\frac{\pi}{36}=\frac{5\pi}{36}$ El valor más grande es $M(b - a)=5\sqrt{3}\times\frac{\pi}{36}=\frac{5\sqrt{3}\pi}{36}$

Answer:

smaller value: $\frac{5\pi}{36}$ larger value: $\frac{5\sqrt{3}\pi}{36}$