The average value of a continuous function f(x) over an interval [a, b] is
![\displaystyle f_{\mathrm{ave}[a,b]} = \frac1{b-a}\int_a^b f(x)\,dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f_%7B%5Cmathrm%7Bave%7D%5Ba%2Cb%5D%7D%20%3D%20%5Cfrac1%7Bb-a%7D%5Cint_a%5Eb%20f%28x%29%5C%2Cdx)
We're given that
![\displaystyle f_{\rm ave[-1,2]} = \frac13 \int_{-1}^2 f(x) \, dx = -4](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f_%7B%5Crm%20ave%5B-1%2C2%5D%7D%20%3D%20%5Cfrac13%20%5Cint_%7B-1%7D%5E2%20f%28x%29%20%5C%2C%20dx%20%3D%20-4)
![\displaystyle f_{\rm ave[2,7]} = \frac15 \int_2^7 f(x) \, dx = 8](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f_%7B%5Crm%20ave%5B2%2C7%5D%7D%20%3D%20%5Cfrac15%20%5Cint_2%5E7%20f%28x%29%20%5C%2C%20dx%20%3D%208)
and we want to determine
![\displaystyle f_{\rm ave[-1,7]} = \frac18 \int_{-1}^7 f(x) \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f_%7B%5Crm%20ave%5B-1%2C7%5D%7D%20%3D%20%5Cfrac18%20%5Cint_%7B-1%7D%5E7%20f%28x%29%20%5C%2C%20dx)
By the additive property of definite integration, we have
so it follows that
![\displaystyle f_{\rm ave[-1,7]} = \frac18 \left(\int_{-1}^2 f(x)\,dx + \int_2^7 f(x)\,dx\right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f_%7B%5Crm%20ave%5B-1%2C7%5D%7D%20%3D%20%5Cfrac18%20%5Cleft%28%5Cint_%7B-1%7D%5E2%20f%28x%29%5C%2Cdx%20%2B%20%5Cint_2%5E7%20f%28x%29%5C%2Cdx%5Cright%29)
![\displaystyle f_{\rm ave[-1,7]} = \frac18 \left(3\times(-4) + 5\times8\right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f_%7B%5Crm%20ave%5B-1%2C7%5D%7D%20%3D%20%5Cfrac18%20%5Cleft%283%5Ctimes%28-4%29%20%2B%205%5Ctimes8%5Cright%29)
![\displaystyle f_{\rm ave[-1,7]} = \boxed{\frac72}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f_%7B%5Crm%20ave%5B-1%2C7%5D%7D%20%3D%20%5Cboxed%7B%5Cfrac72%7D)
Answer:
Brad drank 1/8 bottle more water.
Step-by-step explanation:
Answer:
The Highest common factor of 120,140,180 is ( HCF) = 20
Step-by-step explanation:
<u><em>Explanation:-</em></u>
Given numbers 120,140,180
120 = 60 × 2
= 30 × 2× 2
= 15 × 2×2×2
= 5×3× 2×2×2
140 = 70 × 2
= 10 × 7× 2
= 5× 2× 7× 2
180 = 90 × 2
= 30×3×2
= 5 × 6 × 3× 2
= 5 × 2× 3× 2 × 3
The Highest common factor of 120,140,180 is 5 ×2×2 = 20
