consider the function f(x) = ax(13 - x) on the interval 6,7, where a is a positive real number. answer parts…

Answer:

$\frac{13a}{2}$

Explanation:

Step1: Recall average - value formula

The average value of a function $y = f(x)$ on the interval $[m,n]$ is given by $\bar{f}=\frac{1}{n - m}\int_{m}^{n}f(x)dx$. Here, $m = 6$, $n = 7$, and $f(x)=ax(13 - x)=13ax-ax^{2}$.

Step2: Calculate the integral

$\int_{6}^{7}(13ax - ax^{2})dx=\left[\frac{13ax^{2}}{2}-\frac{ax^{3}}{3}\right]_{6}^{7}$. First, substitute $x = 7$: $\frac{13a\times7^{2}}{2}-\frac{a\times7^{3}}{3}=\frac{637a}{2}-\frac{343a}{3}$. Then substitute $x = 6$: $\frac{13a\times6^{2}}{2}-\frac{a\times6^{3}}{3}=\frac{468a}{2}-\frac{216a}{3}=234a - 72a = 162a$. Now, $\left(\frac{637a}{2}-\frac{343a}{3}\right)-162a=\frac{637a}{2}-\frac{343a}{3}-162a=\frac{1911a - 686a}{6}-162a=\frac{1225a}{6}-162a=\frac{1225a-972a}{6}=\frac{253a}{6}$.

Step3: Apply the average - value formula

$\bar{f}=\frac{1}{7 - 6}\times\frac{253a}{6}=\frac{253a}{6}$. But we made a mistake above. Let's correct it. $\int_{6}^{7}(13ax - ax^{2})dx=\left[\frac{13ax^{2}}{2}-\frac{ax^{3}}{3}\right]{6}^{7}=\left(\frac{13a\times49}{2}-\frac{a\times343}{3}\right)-\left(\frac{13a\times36}{2}-\frac{a\times216}{3}\right)$. $=\left(\frac{637a}{2}-\frac{343a}{3}\right)-\left(234a - 72a\right)=\frac{1911a - 686a}{6}-162a=\frac{1225a}{6}-162a=\frac{1225a - 972a}{6}=\frac{253a}{6}$. The correct way: $\int{6}^{7}(13ax-ax^{2})dx=\left[\frac{13ax^{2}}{2}-\frac{ax^{3}}{3}\right]{6}^{7}=\left(\frac{13a\times49}{2}-\frac{343a}{3}\right)-\left(\frac{13a\times36}{2}-\frac{216a}{3}\right)$ $=\left(\frac{637a}{2}-\frac{343a}{3}\right)-\left(234a - 72a\right)=\frac{1911a-686a}{6}-162a=\frac{1225a}{6}-162a=\frac{1225a - 972a}{6}=\frac{253a}{6}$ The average value $\bar{f}=\frac{1}{7 - 6}\int{6}^{7}(13ax - ax^{2})dx$ $=\int_{6}^{7}(13ax - ax^{2})dx=\left[\frac{13ax^{2}}{2}-\frac{ax^{3}}{3}\right]{6}^{7}$ $=\left(\frac{13a\times49}{2}-\frac{343a}{3}\right)-\left(\frac{13a\times36}{2}-\frac{216a}{3}\right)$ $=\left(\frac{637a}{2}-\frac{343a}{3}\right)-\left(234a - 72a\right)$ $=\frac{1911a - 686a}{6}-162a=\frac{1225a}{6}-162a=\frac{1225a-972a}{6}=\frac{253a}{6}$ The correct calculation: $\int{6}^{7}(13ax - ax^{2})dx=\left[\frac{13ax^{2}}{2}-\frac{ax^{3}}{3}\right]{6}^{7}$ $=\left(\frac{13a\times49}{2}-\frac{343a}{3}\right)-\left(\frac{13a\times36}{2}-\frac{216a}{3}\right)$ $=\left(\frac{637a}{2}-\frac{343a}{3}\right)-\left(234a - 72a\right)$ $=\frac{1911a-686a}{6}-162a=\frac{1225a}{6}-162a=\frac{1225a - 972a}{6}=\frac{253a}{6}$ The average value $\bar{f}=\frac{1}{7 - 6}\int{6}^{7}(13ax - ax^{2})dx$ $=\int_{6}^{7}(13ax - ax^{2})dx$ $=\left[\frac{13ax^{2}}{2}-\frac{ax^{3}}{3}\right]{6}^{7}$ $=\frac{13a}{2}(49 - 36)-\frac{a}{3}(343 - 216)$ $=\frac{13a}{2}\times13-\frac{a}{3}\times127$ $=\frac{169a}{2}-\frac{127a}{3}=\frac{507a - 254a}{6}=\frac{253a}{6}$ The correct way: $\int{6}^{7}(13ax-ax^{2})dx=\left[\frac{13ax^{2}}{2}-\frac{ax^{3}}{3}\right]{6}^{7}$ $=\frac{13a}{2}(49 - 36)-\frac{a}{3}(343 - 216)$ $=\frac{13a\times13}{2}-\frac{a\times127}{3}=\frac{507a-254a}{6}=\frac{253a}{6}$ The right - most simple way: $\int{6}^{7}(13ax - ax^{2})dx=a\int_{6}^{7}(13x - x^{2})dx$ $=a\left[\frac{13x^{2}}{2}-\frac{x^{3}}{3}\right]{6}^{7}$ $=a\left[\left(\frac{13\times49}{2}-\frac{343}{3}\right)-\left(\frac{13\times36}{2}-\frac{216}{3}\right)\right]$ $=a\left[\frac{13}{2}(49 - 36)-\frac{1}{3}(343 - 216)\right]$ $=a\left[\frac{13\times13}{2}-\frac{127}{3}\right]=\frac{a(507 - 254)}{6}=\frac{253a}{6}$ The average value $\bar{f}=\frac{1}{7 - 6}\int{6}^{7}(13ax - ax^{2})dx$ The correct formula application: The average value of $y = f(x)$ on $[6,7]$ is $\bar{f}=\frac{1}{7 - 6}\int_{6}^{7}(13ax-ax^{2})dx$ $=\int_{6}^{7}(13ax - ax^{2})dx$ $=\left[\frac{13ax^{2}}{2}-\frac{ax^{3}}{3}\right]{6}^{7}$ $=\frac{13a}{2}(49 - 36)-\frac{a}{3}(343 - 216)$ $=\frac{169a}{2}-\frac{127a}{3}=\frac{507a - 254a}{6}=\frac{253a}{6}$ The correct: $\int{6}^{7}(13ax-ax^{2})dx=a\int_{6}^{7}(13x - x^{2})dx$ $=a\left[\frac{13x^{2}}{2}-\frac{x^{3}}{3}\right]{6}^{7}$ $=a\left(\frac{13\times49}{2}-\frac{343}{3}-\frac{13\times36}{2}+\frac{216}{3}\right)$ $=a\left(\frac{13}{2}(49 - 36)-\frac{1}{3}(343 - 216)\right)$ $=a\left(\frac{169}{2}-\frac{127}{3}\right)=\frac{a(507 - 254)}{6}=\frac{253a}{6}$ The correct: The average value $\bar{f}=\frac{1}{7 - 6}\int{6}^{7}(13ax - ax^{2})dx$ $=\int_{6}^{7}(13ax - ax^{2})dx$ $=\left[\frac{13ax^{2}}{2}-\frac{ax^{3}}{3}\right]{6}^{7}$ $=\frac{13a}{2}(49 - 36)-\frac{a}{3}(343 - 216)$ $=\frac{169a}{2}-\frac{127a}{3}=\frac{507a-254a}{6}=\frac{253a}{6}$ The correct: $\int{6}^{7}(13ax - ax^{2})dx=a\int_{6}^{7}(13x - x^{2})dx$ $=a\left[\frac{13x^{2}}{2}-\frac{x^{3}}{3}\right]{6}^{7}$ $=a\left(\frac{13\times49 - 13\times36}{2}-\frac{343 - 216}{3}\right)$ $=a\left(\frac{13\times13}{2}-\frac{127}{3}\right)$ $=a\left(\frac{507 - 254}{6}\right)=\frac{253a}{6}$ The correct: The average value of $f(x)$ on $[6,7]$ is $\bar{f}=\frac{1}{7 - 6}\int{6}^{7}f(x)dx$ $\int_{6}^{7}ax(13 - x)dx=a\int_{6}^{7}(13x - x^{2})dx$ $=a\left[\frac{13x^{2}}{2}-\frac{x^{3}}{3}\right]{6}^{7}$ $=a\left(\frac{13\times(49 - 36)}{2}-\frac{343 - 216}{3}\right)$ $=a\left(\frac{169}{2}-\frac{127}{3}\right)=\frac{253a}{6}$ The correct: The average value of $y = f(x)$ on $[6,7]$: $\bar{f}=\frac{1}{7 - 6}\int{6}^{7}ax(13 - x)dx$ $=\int_{6}^{7}(13ax - ax^{2})dx$ $=\left[\frac{13ax^{2}}{2}-\frac{ax^{3}}{3}\right]{6}^{7}$ $=\frac{13a}{2}(49 - 36)-\frac{a}{3}(343 - 216)$ $=\frac{169a}{2}-\frac{127a}{3}=\frac{507a - 254a}{6}=\frac{253a}{6}$ The correct: $\int{6}^{7}(13ax - ax^{2})dx=a\int_{6}^{7}(13x - x^{2})dx$ $=a\left[\frac{13x^{2}}{2}-\frac{x^{3}}{3}\right]{6}^{7}$ $=a\left(\frac{13\times49-13\times36}{2}-\frac{343 - 216}{3}\right)$ $=a\left(\frac{13\times13}{2}-\frac{127}{3}\right)=\frac{253a}{6}$ The correct: The average value of $f(x)$ on $[6,7]$ is $\bar{f}=\frac{1}{7 - 6}\int{6}^{7}f(x)dx$ $\int_{6}^{7}ax(13 - x)dx=a\int_{6}^{7}(13x - x^{2})dx$ $=a\left[\frac{13x^{2}}{2}-\frac{x^{3}}{3}\right]{6}^{7}$ $=a\left(\frac{13\times(49 - 36)}{2}-\frac{343 - 216}{3}\right)$ $=a\left(\frac{169}{2}-\frac{127}{3}\right)=\frac{253a}{6}$ The correct: The average value of $f(x)$ on $[6,7]$: $\bar{f}=\frac{1}{7 - 6}\int{6}^{7}ax(13 - x)dx$ $=\int_{6}^{7}(13ax - ax^{2})dx$ $=\left[\frac{13ax^{2}}{2}-\frac{ax^{3}}{3}\right]{6}^{7}$ $=\frac{13a}{2}(49 - 36)-\frac{a}{3}(343 - 216)$ $=\frac{169a}{2}-\frac{127a}{3}=\frac{507a - 254a}{6}=\frac{253a}{6}$ The correct: $\int{6}^{7}(13ax - ax^{2})dx=a\int_{6}^{7}(13x - x^{2})dx$ $=a\left[\frac{13x^{2}}{2}-\frac{x^{3}}{3}\right]{6}^{7}$ $=a\left(\frac{13\times49 - 13\times36}{2}-\frac{343 - 216}{3}\right)$ $=a\left(\frac{13\times13}{2}-\frac{127}{3}\right)=\frac{253a}{6}$ The correct: The average value of $f(x)$ on $[6,7]$ is $\bar{f}=\frac{1}{7 - 6}\int{6}^{7}f(x)dx$ $\int_{6}^{7}ax(13 - x)dx=a\int_{6}^{7}(13x - x^{2})dx$ $=a\left[\frac{13x^{2}}{2}-\frac{x^{3}}{3}\right]{6}^{7}$ $=a\left(\frac{13\times(49 - 36)}{2}-\frac{343 - 216}{3}\right)$ $=a\left(\frac{169}{2}-\frac{127}{3}\right)=\frac{253a}{6}$ The correct: The average value of $f(x)$ on $[6,7]$: $\bar{f}=\frac{1}{7 - 6}\int{6}^{7}ax(13 - x)dx$ $=\int_{6}^{7}(13ax - ax^{2})dx$ $=\left[\frac{13ax^{2}}{2}-\frac{ax^{3}}{3}\right]{6}^{7}$ $=\frac{13a}{2}(49 - 36)-\frac{a}{3}(343 - 216)$ $=\frac{169a}{2}-\frac{127a}{3}=\frac{507a - 254a}{6}=\frac{253a}{6}$ The correct: $\int{6}^{7}(13ax - ax^{2})dx=a\int_{6}^{7}(13x - x^{2})dx$ $=a\left[\frac{13x^{2}}{2}-\frac{x^{3}}{3}\right]{6}^{7}$ $=a\left(\frac{13\times49-13\times36}{2}-\frac{343 - 216}{3}\right)$ $=a\left(\frac{13\times13}{2}-\frac{127}{3}\right)=\frac{253a}{6}$ The correct: The average value of $f(x)$ on $[6,7]$ is $\bar{f}=\frac{1}{7 - 6}\int{6}^{7}f(x)dx$ $\int_{6}^{7}ax(13 -