Learning goals for section 12.6 (Cylinders and Quadrics):
At the end of this lecture you should be able to:
- identify a given surface as a plane, cylinder or quadric
- distinguish between a curve in R2 and a cylinder in R3 : 1, 2
- describe and sketch a
Quiz 14.4
(If you don know the answers to these questions, you should look them
t
up in your lecture notes, and textbook, and you should know them before
you start working problems for the exam).
1. If a surface is described as z = f (x; y), write the for
Worked Examples 14.4: Tangent Planes and Linear
Approximations
1. Find an equation of the tangent plane to the given surface at the specified point.
z = 6x272, (1,1.1).
2. Explain why the function is differentiable at the given point. Then find the linear
Learning goals for section 14.5: The Chain Rule
At the end of this section you should be able to:
- use the chain rule and tree diagrams to calculate and evaluate total and
partial derivatives of functions of several variables: 1-6, 7-12, 13-14, 17-20,
21
Quiz 14.5
(If you don know the answers to these questions, you should look them
t
up in your lecture notes, and textbook, and you should know them before
you start working problems for the exam).
1. State the Chain Rule (write formulas) for the case z = f
Worked Examples 14.5: The Chain Rule
1. Use the Chain Rule to find dw/dt:
w =xy +yzz; x = e,y = esint, z = ecost
2. Use the Chain Rule to find 62/6,; and east.
2 =x/y,x =Se,y = 1+Se".
3. Use the Chain Rule to find au/ax, aura», (Btu/5r whenx = Ly = 2, t =
Learning goals for section 14.6: Directional Derivative and
the Gradient Vector
At the end of this section you should be able to:
- calculate and evaluate gradients of functions of several variables
- nd the directional derivative of a given function at a
Worked Examples 14.6: Directional Derivatives and the Gradient
Vector
1. Consider the CobbDouglas production function Q : 4K34L14.
a) Suppose that the inputs K and L vary with time t and. the interest 1», via the
expressions
K(t,r) L 192 and Mar) = 612 t
Quiz 14.6
(If you don know the answers to these questions, you should look them
t
up in your lecture notes, and textbook, and you should know them before
you start working problems for the exam).
1. Write the formula for the directional derivative of a di
Learning goals for section 14.4: Tangent Planes and Linear
Approximations
At the end of this section you should be able to:
- nd the normal vector of the tangent plane and the equation of the
tangent plane to a surface given as an explicit function z = f
Worked Examples 14.3: Partial Derivatives
1. |ff(x,y) = 16 w 4x2 y2, findfx(1,2) andf,(1,2). illustrate with either hand-drawn
sketches or computer plots.
2. Find the first partial derivatives of the functions:
a) u = few"
b) f(x, y) = cos(t2)dt
3. Flnd,(
Learning goals for section 14.1 (Functions of Several Variables):
At the end of this lecture you should be able to:
- nd and sketch domains of functions of two and three variables: 8-12,
13-22
- nd the range of functions of two and three variables: 9, 10
Worked Examples 14.1: Functions of Several Variables
1. Find and sketch the domain of the function
f(x,y) = ,fo +y2 1 +1n(4x2 y2)
2. Sketch the graph of the function f(x, )2) 2 1 + 2x2 + 2y2.
3. Draw a contour map of the function x, y) = a showing several
Learning goals for section 14.2 (Limits and Continuity):
At the end of this lecture you should be able to:
- nd the limit of a function of two variables if it exists, by either plugging
in the coordinates of the point, by simplifying the function, or by u
Worked Examples 14.2: Limits and Continuity
1. Find the limit if it exists, or show that the limit doesn't exist:
4
lim 4y 4
(wt40,0) x + 3y
2. Find the limit if it exists, or show that the limit doesnt exist:
lim xy cosy
(gyms) 3:2 +322
3: Find the l
LIMITS AND CONTINUITY
Functions of one variable
0- Consider a function of one variabfex). LetD be the domain of the function,
and a a point such thatD inciudes points arbitrarily close to a (a needs not be in
the domain off). ' -
We say that:
lim x) = L,
Learning goals for section 14.3: Partial Derivatives
At the end of this section you should be able to:
- estimate signs and values of partial derivatives based on a table of values,
a graph, or a contour map: 3, 4, 5-8, 10, 73, 74
- nd rst partial derivat
11.3: PARTIAL DERIVATIVES
Functions of one variable
0 Consider a functionx).
f(x) miim mx + hi)? fm .
M
o Other notations: 7}:- or x).
o f (a) measures the instantaneous rate of change of x), the slope of the line
tangent to the graph of x) at x = a.
o Th
I1I.7: MAXIMUM AND MINIMUM VALUES
Constrained Optimization
FunCtionsiof One Variable
Consider a'function of one variable x). I
J ' /
; o f is defined on a closed interval [a,b]
0 f(x) is not defined ab: = a andx = b
o In this case, after finding all the
Learning goals for section 14.7: Maximum and Minimum Values in Local and Global Optimization
At the end of this section you should be able to:
- classify quadratic forms and their matrices as positive denite, negative
denite, indenite, positive semidenite
Deniteness of Quadratic Forms and Matrices
Positive Denite
A quadratic form Q (x1 ; :; xn ) is positive denite if Q (x1 ; :; xn ) > 0
for all (x1 ; :; xn ) 6= (0; 0; :; 0) :
example: Q (x; y) = x2 + y 2
Q (x) = xT Ax, with A a symmetric matrix. A is posi
Math 256 Lecture 2
Test 2 Solutions
25 March 2015
Name:
Andrew ID:
Please circle your section:
Section F
8:30am
Section G
3:30pm
Question Points Score
1
25
2
25
3
50
4
50
5
50
Total:
200
If answering a question requires calculations, then you must show yo
12.4: THE CROSS PRODUCT
Denition:
! ! !
i
j k
!
a
a1 a2 a3
b1 b2 b3
!
!
a3 b2 ) i + (a3 b1 a1 b3 ) j + (a1 b2
!
b =
!
a
!
b = (a2 b3
Magnitude: !
a
!
a2 b1 ) k :
!
!
b = j!j b sin
a
!
Direction: Perpendicular to both ! and b , orientation given by the
a
r
Span of a set of vectors
Consider n vectors in Rm : S = fv1 ; v2 ; :; vn g.
The span of the (set of) vectors v1 ; v2 ; :; vn is dened to be the set
of all possible linear combinations of the vectors v1 ; v2 ; :; vn :
W = span(S) = fc1 v1 + c2 v2 + : + cn
' Examples:
Matrices and matrix operations
A matrix is a rectangular array of elements arranged in horizontal rows
and vertical columns.
411 «5
QB g j] b)3 2 1 c) Jr
42
0 19.5
The size of the matrix is written as m x n, where m is the number of rows,
and
14.8: Lagrange Multipliers: The Kuhn-Tucker Theorem
Problem: Maximize f (x) subject to constraints:
g1 (x)
x1
b1 ; :; gk (x) bk ;
0; :; xn 0
where x = (x1 ; :; xn ):
Note: Here all the variables are required to be 0. Since this a maximization problem, ine
What are matrices used for?
- organize information:
Stere 1 of a three store chain has 3
refrigerators, 5 stoves, 3 washing machines,
and 4 dryers in stock.
Store 2: no refrigerators, stoves, 9 washing
machines and 5 dryers
4390.)
NNU}
Store 3: 4 refriger
Exercises 42, 47, 52, page 890.
42. A contour map of a function is shown. Use it to make a rough
sketch of the graph of f .
Exercises 42, 47, 52, page 890.
Solution: think of the previous picture as the view from above of
the graph. The higher the value o