Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Answer:
Average value: 2.30305455955
z ≈ 4.29945519514
Step-by-step explanation:
You want the value of z such that f(z) is the average value of f(x) = √(x+1) on the interval [1, 8].
Average value
The average value is the integral of the function, divided by the width of the interval.
[tex]\displaystyle\overline{y}=\dfrac{1}{x_2-x_1}\int_{x_1}^{x_2}{f(x)}\,dx=\dfrac{1}{7}\int_1^8{\sqrt{x+1}}\,dx\\\\\\\overline{y}=\dfrac{2}{21}\left((8+1)^\frac{3}{2}-(1+1)^\frac{3}{2}\right)=\dfrac{2(27-2\sqrt{2})}{21}\\\\\\\boxed{\overline{y}\approx 2.30305455955}[/tex]
The corresponding value of z so that f(z) = y is ...
[tex]z=\overline{y}^2-1\\\\\boxed{z \approx 4.29945519514}[/tex]
The average value of f(x)= √(x+1) on 1,8 is [tex]\frac{18}{7} - \frac{4\sqrt{2}}{21}[/tex] . For [tex]\[ z = \frac{2948 - 432\sqrt{2}}{441} - 1 \][/tex] on 1,8 does this average value equal f(z).
1. Integral calculation:
- [tex]\[ \int_1^8 \sqrt{x + 1} \, dx \][/tex]
- Using the substitution u = x + 1, du = dx:
[tex]\[ \int_2^9 \sqrt{u} \, du = \int_2^9 u^{1/2} \, du = \left. \frac{2}{3} u^{3/2} \right|_2^9 = \frac{2}{3} \left( 9^{3/2} - 2^{3/2} \right) \][/tex]
- Since [tex]\( 9^{3/2} = 27 \)[/tex] and [tex]\( 2^{3/2} = 2\sqrt{2} \)[/tex]:
[tex]\[ \frac{2}{3} \left( 27 - 2\sqrt{2} \right) = \frac{54 - 4\sqrt{2}}{3} = 18 - \frac{4\sqrt{2}}{3} \][/tex]
2. Average value calculation:
- Average value = [tex]\frac{1}{8 - 1} \int_1^8 \sqrt{x + 1} \, dx = \frac{1}{7} \left( 18 - \frac{4\sqrt{2}}{3} \right) = \frac{18}{7} - \frac{4\sqrt{2}}{21}[/tex].
3. Finding z such that f(z) equals the average value:
[tex]\[ \sqrt{z + 1} = \frac{18}{7} - \frac{4\sqrt{2}}{21} \][/tex]
- Squaring both sides:
[tex]\[ z + 1 = \left( \frac{18}{7} - \frac{4\sqrt{2}}{21} \right)^2 = \left( \frac{54 - 4\sqrt{2}}{21} \right)^2 = \frac{(54 - 4\sqrt{2})^2}{21^2} \][/tex]
- Calculate [tex]\( (54 - 4\sqrt{2})^2 \)[/tex]:
[tex]\[ (54 - 4\sqrt{2})^2 = 54^2 - 2 \cdot 54 \cdot 4\sqrt{2} + (4\sqrt{2})^2 = 2916 - 432\sqrt{2} + 32 = 2948 - 432\sqrt{2} \][/tex]
- So:
[tex]\[ z + 1 = \frac{2948 - 432\sqrt{2}}{441} \][/tex]
- Finally, solving for z:
[tex]\[ z = \frac{2948 - 432\sqrt{2}}{441} - 1 \][/tex].
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
