Please help me with this line problem?

Posted on 2015-10-29 06:55:00 in Math by FREDRIC

Fitting a Line to Data Using Least Squares (Linear Regression): Assume
you have n data points (x1, y1),(x2, y2), . . . ,(xn, yn). Let the equation of
the least squares be y = mx + b.
(Part a) Show that the square of the vertical distance between any of the
data points and the line is (yi ? (mxi + b))2
.
(Part b) Form the function f(b,m) which is the sum of the all of the n squared
distances, i.e.,
f(b, m) = ?n
i=1
(yi ? (mxi + b))2
.
(Part c) Find the partial derivatives ?f
?b ,
?f
?m .
(Part d) Find the critical points, hence the line equation.
2
Problem 3 (2 Points):
Prove using the epsilon-delta definition of limits that
lim
(x,y)?(1,1)
x
2 + xy + y = 3.
(Here is again that definition: Given ? > 0, find ? > 0 such that
0 < |
??u ?
??a | < ? implies |f(
??u ) ? 3| < ?;
where ??u = (x, y) and ??a = (1, 1).
Problem 4 (2 Points):
A mountain rising from a plateau (or flat ground) is modeled as
f(x, y) = {
3(1 ?
?
x
2 + y
2) : ?1 ? x, y ? 1
0 : otherwise
A road up the mountain is modeled as a spiral ?r(t) = at cos(b ln t)?i+atsin(b ln t)?j,
where 0 ? t ?
1
a
. (See Figure 3 on the next page).
(Part a) Find the directional derivative (commonly r

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