Answer:
the answer is C: Angle Bisector Theorem
Step-by-step explanation:
Answer: x = 1
Step-by-step explanation:
(any line with an undefined slope is a vertical line)
x = 1 has an undefined slope and passes through (1,8)
Answer:
Your answer would be number 4, "A and D"
Step-by-step explanation:
I already know that if you multiply a number by less than 1 it will give a product less than the original number so that makes B and C not the correct answer leaving a and d
I hope this helps :D
It is best to label everything. I color coded it for you to see better and understand.
Answer:
See Below.
Step-by-step explanation:
We want to estimate the definite integral:
![\displaystyle \int_1^47\sqrt{\ln(x)}\, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_1%5E47%5Csqrt%7B%5Cln%28x%29%7D%5C%2C%20dx)
Using the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule with six equal subdivisions.
1)
The trapezoidal rule is given by:
![\displaystyle \int_{a}^bf(x)\, dx\approx\frac{\Delta x}{2}\Big(f(x_0)+2f(x_1)+...+2f(x_{n-1})+f(x_n)\Big)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_%7Ba%7D%5Ebf%28x%29%5C%2C%20dx%5Capprox%5Cfrac%7B%5CDelta%20x%7D%7B2%7D%5CBig%28f%28x_0%29%2B2f%28x_1%29%2B...%2B2f%28x_%7Bn-1%7D%29%2Bf%28x_n%29%5CBig%29)
Our limits of integration are from x = 1 to x = 4. With six equal subdivisions, each subdivision will measure:
![\displaystyle \Delta x=\frac{4-1}{6}=\frac{1}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5CDelta%20x%3D%5Cfrac%7B4-1%7D%7B6%7D%3D%5Cfrac%7B1%7D%7B2%7D)
Therefore, the trapezoidal approximation is:
![\displaystyle =\frac{1/2}{2}\Big(f(1)+2f(1.5)+2f(2)+2f(2.5)+2f(3)+2f(3.5)+2f(4)\Big)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%5Cfrac%7B1%2F2%7D%7B2%7D%5CBig%28f%281%29%2B2f%281.5%29%2B2f%282%29%2B2f%282.5%29%2B2f%283%29%2B2f%283.5%29%2B2f%284%29%5CBig%29)
Evaluate:
![\displaystyle =\frac{1}{4}(7)(\sqrt{\ln(1)}+2\sqrt{\ln(1.5)}+...+2\sqrt{\ln(3.5)}+\sqrt{\ln(4)})\\\\\approx18.139337](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%5Cfrac%7B1%7D%7B4%7D%287%29%28%5Csqrt%7B%5Cln%281%29%7D%2B2%5Csqrt%7B%5Cln%281.5%29%7D%2B...%2B2%5Csqrt%7B%5Cln%283.5%29%7D%2B%5Csqrt%7B%5Cln%284%29%7D%29%5C%5C%5C%5C%5Capprox18.139337)
2)
The midpoint rule is given by:
![\displaystyle \int_a^bf(x)\, dx\approx\sum_{i=1}^nf\Big(\frac{x_{i-1}+x_i}{2}\Big)\Delta x](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_a%5Ebf%28x%29%5C%2C%20dx%5Capprox%5Csum_%7Bi%3D1%7D%5Enf%5CBig%28%5Cfrac%7Bx_%7Bi-1%7D%2Bx_i%7D%7B2%7D%5CBig%29%5CDelta%20x)
Thus:
![\displaystyle =\frac{1}{2}\Big(f\Big(\frac{1+1.5}{2}\Big)+f\Big(\frac{1.5+2}{2}\Big)+...+f\Big(\frac{3+3.5}{2}\Big)+f\Big(\frac{3.5+4}{2}\Big)\Big)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%5Cfrac%7B1%7D%7B2%7D%5CBig%28f%5CBig%28%5Cfrac%7B1%2B1.5%7D%7B2%7D%5CBig%29%2Bf%5CBig%28%5Cfrac%7B1.5%2B2%7D%7B2%7D%5CBig%29%2B...%2Bf%5CBig%28%5Cfrac%7B3%2B3.5%7D%7B2%7D%5CBig%29%2Bf%5CBig%28%5Cfrac%7B3.5%2B4%7D%7B2%7D%5CBig%29%5CBig%29)
Simplify:
![\displaystyle =\frac{1}{2}(7)\Big(f(1.25)+f(1.75)+...+f(3.25)+f(3.75)\Big)\\\\ =\frac{1}{2}(7) (\sqrt{\ln(1.25)}+\sqrt{\ln(1.75)}+...+\sqrt{\ln(3.25)}+\sqrt{\ln(3.75)})\\\\\approx 18.767319](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%5Cfrac%7B1%7D%7B2%7D%287%29%5CBig%28f%281.25%29%2Bf%281.75%29%2B...%2Bf%283.25%29%2Bf%283.75%29%5CBig%29%5C%5C%5C%5C%20%3D%5Cfrac%7B1%7D%7B2%7D%287%29%20%28%5Csqrt%7B%5Cln%281.25%29%7D%2B%5Csqrt%7B%5Cln%281.75%29%7D%2B...%2B%5Csqrt%7B%5Cln%283.25%29%7D%2B%5Csqrt%7B%5Cln%283.75%29%7D%29%5C%5C%5C%5C%5Capprox%2018.767319)
3)
Simpson's Rule is given by:
![\displaystyle \int_a^b f(x)\, dx\approx\frac{\Delta x}{3}\Big(f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+...+4f(x_{n-1})+f(x_n)\Big)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_a%5Eb%20f%28x%29%5C%2C%20dx%5Capprox%5Cfrac%7B%5CDelta%20x%7D%7B3%7D%5CBig%28f%28x_0%29%2B4f%28x_1%29%2B2f%28x_2%29%2B4f%28x_3%29%2B...%2B4f%28x_%7Bn-1%7D%29%2Bf%28x_n%29%5CBig%29)
So:
![\displaystyle =\frac{1/2}{3}\Big((f(1)+4f(1.5)+2f(2)+4f(2.5)+...+4f(3.5)+f(4)\Big)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%5Cfrac%7B1%2F2%7D%7B3%7D%5CBig%28%28f%281%29%2B4f%281.5%29%2B2f%282%29%2B4f%282.5%29%2B...%2B4f%283.5%29%2Bf%284%29%5CBig%29)
Simplify:
![\displaystyle =\frac{1}{6}(7)(\sqrt{\ln(1)}+4\sqrt{\ln(1.5)}+2\sqrt{\ln(2)}+4\sqrt{\ln(2.5)}+...+4\sqrt{\ln(3.5)}+\sqrt{\ln(4)})\\\\\approx 18.423834](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%5Cfrac%7B1%7D%7B6%7D%287%29%28%5Csqrt%7B%5Cln%281%29%7D%2B4%5Csqrt%7B%5Cln%281.5%29%7D%2B2%5Csqrt%7B%5Cln%282%29%7D%2B4%5Csqrt%7B%5Cln%282.5%29%7D%2B...%2B4%5Csqrt%7B%5Cln%283.5%29%7D%2B%5Csqrt%7B%5Cln%284%29%7D%29%5C%5C%5C%5C%5Capprox%2018.423834)