18.02 - Practice Final A - Spring 2006
Problem 1. Let P = (0, 1, 0), Q = (2, 1, 3), R = (1, 1, 2). Compute P Q P R and nd
the equation of the plane through P , Q, and R, in the form ax + by + cz = d.
Problem 2. Find the p oint of intersection of the line
Least Squares Interpolation
1. The least-squares line.
Suppose you have a large number n of experimentally determined points, through which
you want to pass a curve. There is a formula (the Lagrange interpolation formula) producing
a polynomial curve of d
M.1
The exponential and logarithm functions
In this section, we study the exponential and logarithm functions and derive
their pr0perties.
We also dene ab for a > 0 and b arbitrary, and we verify the laws of expo-
nents.
As we did for the trig functions,
Critical Points
Critical points:
A standard question in calculus, with applications to many elds, is to nd the points where
a function reaches its relative maxima and minima.
Just as in single variable calculus we will look for maxima and minima (collecti
0.1
Chapter 7 Taylor's formula.
If f(x) has derivatives of orders l,.,n at the point
x = a, then the polynomial function
Tn(x) = a0 + al(x-a) + o-o + an(x-a)n,
where
(m)
_ f (a)
am f m!
th
for each m, is called the n order Tailor approximation to
f at a
13.92 Practice Fina-J 3 11:5.
Problem 1- Given the points F: [1.1. -1].L?t [1.9.11]. H t {HT-r. 2.231 and
at] Ft? it FR 1:] a plane a: -l- [:1- + c: = at through 33.113.
1 I] 1: I [I - I I
FrohlemlLtt= 2 oil , :e= y I t]: [I ., A: a - - _
1 1 2 1 El - K -
1. (20pts) For what values of A, ,LL and 1/ does the function f: R3 > IR,
f(33, y, Z) = A332 + #339 + y2 + V22,
have a nondegenerate local minimum at (0, O, 0)?
DE : (QXHVWQGWRVQ
) EH: LAW); X3: lv-(kL
mvmm A > O) A? o) A: 7 0
80 X >50 9 2. (20pts) Let f
Directional Derivatives
Directional derivative
Like all derivatives the directional derivative can be thought of as a ratio. Fix a unit vector
u and a point P0 in the plane. The directional derivative of w at P0 in the direction u
is dened as
dw
w
= lim
.
Lagrange Multipliers
We will give the argument for why Lagrange multipliers work later. Here, well look at
where and how to use them. Lagrange multipliers are used to solve constrained optimization
problems. That is, suppose you have a function, say f (x,
Non-independent Variables
1. Partial dierentiation with non-independent variables.
Up to now in calculating partial derivatives of functions like w = f (x, y) or w = f (x, y, z),
we have assumed the variables x, y (or x, y, z) were independent. However in
Proof of Lagrange Multipliers
Here we will give two arguments, one geometric and one analytic for why Lagrange multipliers work.
Critical points
For the function w = f (x, y, z) constrained by g(x, y, z) = c (c a constant) the critical points
are dened as
Chain rule
Now we will formulate the chain rule when there is more than one independent variable.
We suppose w is a function of x, y and that x, y are functions of u, v. That is,
w = f (x, y) and x = x(u, v), y = y(u, v).
The use of the term chain comes b
Gradient: denition and properties
Denition of the gradient
@w
@w
If w = f (x, y), then
and
are the rates of change of w in the i and j directions.
@x
@y
It will be quite useful to put these two derivatives together in a vector called the gradient
of w.
@w
Second Derivative Test
1. The Second Derivative Test
We begin by recalling the situation for twice dierentiable functions f (x) of one variable.
To nd their local (or relative) maxima and minima, we
1. nd the critical points, i.e., the solutions of f (x)
Least Squares Interpolation
1. Use the method of least squares to t a line to the four data points
(0, 0),
(1, 2),
(2, 1),
(3, 4).
Answer: We are looking for the line y = ax + b that best models the data. The deviation
of a data point (xi , yi ) from the
1. (20pts) Find the shortest distance between the line 1 given paramet
rieally by
(37,952) = _3+t12 _t)1
and the intersection of the two planes H1 and H2 given by the equations
$+y+z=3 and m2y+32=2.
0 Lune vol/um "W1 (A/J T; Wager is
(mlezb) 3 £11,114) +