Answer:
A
Step-by-step explanation:
The first thing you need to do is find the slope. Find two perfect points.
In this case I chose (0,0) and (4,1). Slope is basically rise over run. In other words, how much you go up, and how much you go right. We rise up one and to the right 4.
Answer: 38.5
Step-by-step explanation:
Answer:
(2,3)
Step-by-step explanation:
Hey there!
when it ask for the solution of a systems of equation it's essentially asking for where the two lines cross
the two lines cross at (2,3) therefore 2,3 is your answer
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)
The answer to your equation rounded is x = 3 14/41