Unit 4 Homework Set: Problem 4 (3 points) Find the...

Unit 4 Homework Set: Problem 4 (3 points) Find the least-squares regression line $\hat{y} = b_0 + b_1x$ through the points $(-2, 1), (0, 9), (6, 15), (7, 20), (9, 25)$, and then use it to find point estimates $\hat{y}$ corresponding to $x = 4$ and $x = 6$. For this problem and to give you practice for the test, use the shortcut method to find $b_0$ given that $b_1 = 1.92222222222222$ For $x = 4$, $\hat{y} = $ For $x = 6$, $\hat{y} = $

Transcript

00:01 So, we have given here equation that is 4th derivative of y minus 8 y 3rd derivative of y and plus 16 2nd derivative of y that is equal to 0.

00:17 And we have given some condition here at y at 0 gives a value that is 10, y dash 0 gives a value 13, y double dash gives a value that is at 0 gives a value 16 and y triple dash gives a value at 0 that is equal to 0.

00:45 Now, we need to write the auxiliary equation.

00:49 So, it become m to the power 4 minus 8 m cube plus 16 m square that is equal to 0.

00:58 Taking m square outside and it become m square minus 8 m plus 16 that is equal to 0.

01:09 So, after factorization of this it become m square into m minus 4 whole square that is equal to 0.

01:18 So, m is equal to 0 comma 0 comma 4 comma 4.

01:25 So, our general solution y is equal to because this is homogeneous equation.

01:30 So, y is equal to c1 e to the power 0x plus c2 x e to the power 0x plus c3 e to the power 4x plus c4 x e to the power e to the power 4x.

01:52 Now, then our equation become y is equal to c1 plus c2 x plus c3 e to the power 4x plus c4 x e to the power x 4x.

02:12 Now, let's differentiate this function with respect to x.

02:16 So, it become c2 plus 4 c3 e to the power 4x plus c4 e to the power 4x and plus c4 4 c4 x e to the power 4x.

02:40 Now, we need to differentiate this again with respect to x.

02:45 So, it become 16 c3 e to the power 4x plus 4 c4 e to the power 4x and plus 4 c4 e to the power 4x and plus we need to write here that is 16 c4 x e to the power 4x.

03:22 So, again we need to differentiate this with respect to x.

03:27 So, it become from there 64 c3 e to the power 4x and after we need to add these two quantities.

03:38 So, it become 8 c4 e to the power x would become 32 c4 e to the power 4x and plus 16 c4 e to the power 4x and it is plus 64 c4 x e to the power 4x.

04:05 So, after simplify this we will get that is a 64 c3 e to the power 4x plus 48 c4 e to the power 4x and plus 64 c4 x e to the power 4x.

04:29 This is y triple dash.

04:33 Now, we need to put some here initial condition.

04:36 So, that is we have here at y is equal to 0 at x is equal to 0 y gives a value 10.

04:45 So, that is equal to c3 plus c1 is equal to 20...

Question

Please give Ace some feedback