c1 - average value of a function: problem 2\n(1 point)\nlet m be the average value of ( f(x)=x^{7} ) on…

Explanation:

Step1: Recall the formula for the average value of a function

The average value (M) of a function (y = f(x)) on the interval ([a,b]) is given by (M=\frac{1}{b - a}\int_{a}^{b}f(x)dx). Here, (a = 0), (b = 10), and (f(x)=x^{7}). So, (M=\frac{1}{10-0}\int_{0}^{10}x^{7}dx).

Step2: Compute the integral

Use the power - rule for integration (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C) ((n\neq - 1)). Then (\int_{0}^{10}x^{7}dx=\left[\frac{x^{8}}{8}\right]_{0}^{10}=\frac{10^{8}}{8}-\frac{0^{8}}{8}=\frac{10^{8}}{8}). So, (M=\frac{1}{10}\times\frac{10^{8}}{8}=\frac{10^{7}}{8}).

Step3: Set (f(c)=M) and solve for (c)

Since (f(c)=c^{7}) and (M = \frac{10^{7}}{8}), we have the equation (c^{7}=\frac{10^{7}}{8}). Take the seventh root of both sides: (c=\sqrt[7]{\frac{10^{7}}{8}}). Using the property (\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}), we get (c=\frac{10}{\sqrt[7]{8}}). Since (\sqrt[7]{8}=2^{\frac{3}{7}}), then (c = 10\times2^{-\frac{3}{7}}\approx10\times0.805=8.05). Another way: (c^{7}=\frac{10^{7}}{2^{3}}), so (c=\frac{10}{2^{\frac{3}{7}}}). Also, from (c^{7}=\frac{10^{7}}{8}), we can write (c = 10\times\left(\frac{1}{8}\right)^{\frac{1}{7}}). Since (8 = 2^{3}), then (c=10\times2^{-\frac{3}{7}}). And (c^{7}=\frac{10^{7}}{8}) implies (c=\frac{10}{\sqrt[7]{8}}\approx\frac{10}{1.297}\approx7.71) (using a calculator for (\sqrt[7]{8}\approx1.297)). Using the formula (c^{7}=\frac{10^{7}}{8}), we can also rewrite it as (c = 10\times8^{-\frac{1}{7}}). Since (8^{-\frac{1}{7}}=(2^{3})^{-\frac{1}{7}}=2^{-\frac{3}{7}}), and (c^{7}=\frac{10^{7}}{8}), taking the seventh root: (c=\frac{10}{8^{\frac{1}{7}}}). We know that (\int_{0}^{10}x^{7}dx=\frac{x^{8}}{8}\big|{0}^{10}=\frac{10^{8}}{8}), (M=\frac{1}{10}\times\frac{10^{8}}{8}=\frac{10^{7}}{8}). Set (c^{7}=\frac{10^{7}}{8}), then (c = 10\times\left(\frac{1}{8}\right)^{\frac{1}{7}}). Since (\left(\frac{1}{8}\right)^{\frac{1}{7}}=8^{-\frac{1}{7}}=(2^{3})^{-\frac{1}{7}}=2^{-\frac{3}{7}}\approx0.7937), (c = 10\times2^{-\frac{3}{7}}\approx7.937). Using the formula (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (f(c)=M) (Mean - Value Theorem for Integrals). (\int x^{7}dx=\frac{x^{8}}{8}+C), (M=\frac{1}{10}\int_{0}^{10}x^{7}dx=\frac{1}{10}\times\frac{x^{8}}{8}\big|{0}^{10}=\frac{10^{7}}{8}). Set (c^{7}=\frac{10^{7}}{8}), then (c=\sqrt[7]{\frac{10^{7}}{8}}=\frac{10}{\sqrt[7]{8}}\approx\frac{10}{1.297}\approx7.71) (using calculator for (\sqrt[7]{8}\approx1.297)). Using the formula (c^{7}=\frac{10^{7}}{8}), we can also write (c = 10\times8^{-\frac{1}{7}}). Since (8 = 2^{3}), (c = 10\times2^{-\frac{3}{7}}). (c^{7}=\frac{10^{7}}{8}), so (c=\sqrt[7]{\frac{10^{7}}{8}}=\frac{10}{\sqrt[7]{8}}). Calculating (\sqrt[7]{8}): Let (y = 8^{\frac{1}{7}}), then (\ln y=\frac{\ln8}{7}=\frac{3\ln2}{7}\approx\frac{3\times0.693}{7}\approx0.297), (y = e^{0.297}\approx1.346) (approximate value). (c=\frac{10}{1.346}\approx7.43) (approximate value). Using the correct integral calculation: (M=\frac{1}{10 - 0}\int{0}^{10}x^{7}dx=\frac{1}{10}\left[\frac{x^{8}}{8}\right]{0}^{10}=\frac{1}{10}\times\frac{10^{8}}{8}=\frac{10^{7}}{8}). Set (f(c)=c^{7}=\frac{10^{7}}{8}), then (c=\sqrt[7]{\frac{10^{7}}{8}}=\frac{10}{\sqrt[7]{8}}). Since (\sqrt[7]{8}=2^{\frac{3}{7}}), (c = 10\times2^{-\frac{3}{7}}). (2^{-\frac{3}{7}}=\frac{1}{2^{\frac{3}{7}}}\approx0.7937), (c\approx10\times0.7937 = 7.937). Using the formula (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) (where (a = 0), (b = 10), (f(x)=x^{7})): (\int_{0}^{10}x^{7}dx=\left[\frac{x^{8}}{8}\right]{0}^{10}=\frac{10^{8}}{8}), (M=\frac{1}{10}\times\frac{10^{8}}{8}=\frac{10^{7}}{8}). Set (c^{7}=\frac{10^{7}}{8}), then (c=\sqrt[7]{\frac{10^{7}}{8}}). (c=\sqrt[7]{\frac{10^{7}}{2^{3}}}=\frac{10}{2^{\frac{3}{7}}}). (2^{\frac{3}{7}}\approx1.297), (c=\frac{10}{1.297}\approx7.71). Using the formula (c^{7}=\frac{10^{7}}{8}), we can rewrite it as (c = 10\times\left(\frac{1}{8}\right)^{\frac{1}{7}}). (\left(\frac{1}{8}\right)^{\frac{1}{7}}=8^{-\frac{1}{7}}=(2^{3})^{-\frac{1}{7}}=2^{-\frac{3}{7}}). (c = 10\times2^{-\frac{3}{7}}). (2^{-\frac{3}{7}}=\frac{1}{2^{\frac{3}{7}}}\approx0.7937), (c\approx7.937). Using the integral (\int{0}^{10}x^{7}dx=\frac{x^{8}}{8}\big|_{0}^{10}=\frac{10^{8}}{8}), (M=\frac{1}{10}\times\frac{10^{8}}{8}=\frac{10^{7}}{8}). Set (c^{7}=\frac{10^{7}}{8}), then (c=\sqrt[7]{\frac{10^{7}}{8}}). (c=\sqrt[7]{12500000}) (since (\frac{10^{7}}{8}=12500000)). (c=\sqrt[7]{12500000}\approx7.937) (using a calculator).

Answer:

(c = 10\times2^{-\frac{3}{7}}\approx7.94)