1. State the problem: Order the following mathematical expressions from least to greatest: $$\infty, \sqrt{21}, 4!, \frac{9}{11}, \frac{4\pi}{2}, \log_2(16), \int_2^7 x \, dx, e^3, \sum_{i=4}^5 i$$ 2. Evaluate each expression: - $$\infty$$ is infinite. - $$\sqrt{21}$$ is approximately $$\sqrt{21} \approx 4.5826$$. - $$4! = 4 \times 3 \times 2 \times 1 = 24$$. - $$\frac{9}{11} \approx 0.8182$$. - $$\frac{4\pi}{2} = 2\pi \approx 6.2832$$. - $$\log_2(16)$$ asks "to what power must 2 be raised to get 16?" Since $$2^4 = 16$$, $$\log_2(16) = 4$$. - The definite integral $$\int_2^7 x \, dx$$ can be calculated: $$\int x \, dx = \frac{x^2}{2} + C$$ So, $$\int_2^7 x \, dx = \left[ \frac{x^2}{2} \right]_2^7 = \frac{7^2}{2} - \frac{2^2}{2} = \frac{49}{2} - \frac{4}{2} = \frac{45}{2} = 22.5$$. - $$e^3$$ where $$e \approx 2.71828$$, so $$e^3 = 2.71828^3 \approx 20.0855$$. - The sum $$\sum_{i=4}^5 i = 4 + 5 = 9$$. 3. Now, list all values numerically: - $$\frac{9}{11} \approx 0.8182$$ - $$\log_2(16) = 4$$ - $$\sqrt{21} \approx 4.5826$$ - $$\sum_{i=4}^5 i = 9$$ - $$e^3 \approx 20.0855$$ - $$\int_2^7 x \, dx = 22.5$$ - $$4! = 24$$ - $$\frac{4\pi}{2} = 6.2832$$ (Note: previously misordered, correct this as 6.2832 is less than 9, so placing it correctly between 4.5826 and 9) Revised correct order: - $$\frac{9}{11} \approx 0.8182$$ - $$\log_2(16) = 4$$ - $$\sqrt{21} \approx 4.5826$$ - $$\frac{4\pi}{2} = 6.2832$$ - $$\sum_{i=4}^5 i = 9$$ - $$e^3 \approx 20.0855$$ - $$\int_2^7 x \, dx = 22.5$$ - $$4! = 24$$ - $$\infty$$ (infinite, greater than all finite numbers) 4. Final answer, ordered from least to greatest: $$\frac{9}{11} < \log_2(16) < \sqrt{21} < \frac{4\pi}{2} < \sum_{i=4}^5 i < e^3 < \int_2^7 x \, dx < 4! < \infty$$