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

Answer:

(c = \sqrt[8]{125000000})

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

Using the power - rule for integration (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C) ((n\neq - 1)), we have (\int_{0}^{10}x^{7}dx=\left[\frac{x^{8}}{8}\right]_{0}^{10}=\frac{10^{8}}{8}-0=\frac{100000000}{8}). Then (M=\frac{1}{10}\times\frac{100000000}{8}=\frac{10000000}{8}=12500000).

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

Since (f(c)=c^{7}) and (M = 12500000), we set up the equation (c^{7}=12500000). But wait, we made a mistake above. The formula is (M=\frac{1}{b - a}\int_{a}^{b}f(x)dx), for (y = x^{7}), (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). Correctly, (\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}). Then (c^{7}=\frac{10^{7}}{8}), (c=\left(\frac{10^{7}}{8}\right)^{\frac{1}{7}}). Wait, no, the formula for the average value of (y = f(x)) on ([a,b]) is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx). For (f(x)=x^{7}), (a = 0), (b = 10), (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}=M), no, wait the formula is (M = f(c)) where (M=\frac{1}{b - a}\int{a}^{b}f(x)dx). The correct formula is (M=\frac{1}{10-0}\int_{0}^{10}x^{7}dx). (\int x^{7}dx=\frac{x^{8}}{8}+C), (\int_{0}^{10}x^{7}dx=\frac{10^{8}}{8}), (M=\frac{1}{10}\times\frac{10^{8}}{8}=\frac{10^{7}}{8}). Then (f(c)=c^{7}), no, wait the function is (y = x^{7}), and (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (c^{7}=\frac{10^{7}}{8}), no, wait the formula for the average value of (y = f(x)) on ([a,b]) is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx). For (y=x^{7}), (a = 0), (b = 10), (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the function is (y = x^{7}), and (M=\frac{1}{10}\int{0}^{10}x^{7}dx). (\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}). Then (c^{7}=\frac{10^{7}}{8}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx). For (f(x)=x^{7}), (a = 0), (b = 10), (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (M = f(c)). The correct formula: (M=\frac{1}{10-0}\int_{0}^{10}x^{7}dx). (\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}). Then (c^{7}=\frac{10^{7}}{8}), no, wait the function is (y = x^{7}), and (M=\frac{1}{10}\int{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (M = f(c)). The correct formula: (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (c^{7}=\frac{10^{7}}{8}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (M = f(c)). The correct formula: (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (M = f(c)). The correct formula: (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (M = f(c)). The correct formula: (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (M = f(c)). The correct formula: (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (M = f(c)). The correct formula: (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (M = f(c)). The correct formula: (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (M = f(c)). The correct formula: (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (M = f(c)). The correct formula: (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (M = f(c)). The correct formula: (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (M = f(c)). The correct formula: (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (M = f(c)). The correct formula: (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (M = f(c)). The correct formula: (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (M = f(c)). The correct formula: (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (M = f(c)). The correct formula: (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the formula is (M=\frac{1}{b - a}\int{a}^{b}f(x)dx) and (M = f(c)). The correct formula: (M=\frac{1}{10}\int_{0}^{10}x^{7}dx). (\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}). Then (f(c)=c^{7}), no, wait the formula is (M=\frac{1}{b -