4.5: OPTIMIZATION
1. A rectangular box is to have a square base and a volume of 20 cubic ft. If the material for the base costs 30
cents/sq. ft, the material for the sides costs 10 cents/sq. ft., and the material for the top costs 20 cents/sq. ft.
Determi
Solutions to Odd-Numbered Exercises 407
2“", k=—1.—2,.
(c) In 2 < '2' < 3, Z akz", ah = 4 1
-00 3k+1_53+1! k=0,1,2,.
In 5 < |z| < oo, 25, a, = 2n-1—4(3)"—1+ 5H, n = 1,2, 3,
l
N
25. 00 is removable for f iff 0 is removable for g(z) = f(1/z) iff g is b
Solutions to Odd-Numbered Exercises 401
A = {t e [0, 1], u(z(t) = c}.
Since u is equal to c on some disc centered at 20, the interval [0, 6) lies in A for some 6 > 0;
6 is the radius of the disc. Since u is continuous at the point 2(5), we also have 6 e A
Appendix 2 A Table of Conformal Mappings 391
Mapping w-plane
1+z
w:i(1-z)
i —— w ,
z = _ IS the
1 + w
inverse mapping.
_22+2iz+1
w=—1 .
22—212+1 392 Appendix 2 A Table of Conformal Mappings
Mapping w_plane
inverse mapping.
0 and a ar
APPENDIX 2
A Table of Conformal Mappings
z = x + iy, w = u + iv. Lowercase letters are mapped to uppercase letters, cross-hatched regions onto
shaded regions.
z-plane Mapping w-plane
\
:\
389 390 Appendix 2 A Table of Conformal Mappings
Mappi
Appendix 1 Locating the Zeros of a Polynomial 387
To ﬁt the theorem to our context, we take p1 = p and p2 = p’, the derivative
of p. Simple calculus shows that condition (3) holds; that is, that the sign of pp’
changes from minus to plus as x passes throu
APPENDIX 3
A Table of Laplace Transforms
u(x); u(x) = 0 ifx < 0
. sin bx
4.
cos bx b2 + $2
. b
5. s1nh bx sz b2
5
6. cosh bx s2 b2
b
7. " si b
e n x (s — a)2 + b2
ax s - a
8. e cos bx (s _ “)2 + b2
2sb
9. ' b
x
2.6: THE DERIVATIVE
Activity: Finding slope of a secant line (a line passing through a curve at two points
a) Suppose P = (1, 3) and Q = (5, 7). Find the slope of the secant line
PQ
b) Suppose P = (1, 3) and Q is some point (x, f(x). Find the slope of the
2.4 & 2.5: LIMITS AND CONTINUITY
I. Limit of a function one-sided limit, limit at a number
II. Properties of limits
III. Computing limits algebraically
IV. Limits at infinity and infinite limits
V. Limits of piecewise functions
VI. Continuity
-I. WHAT IS
3.5: HIGER ORDER DERIVATIVE
Notations: For function, f(x)
1st derivative the rate of change of f(x), denoted by
dy
f '(x) y'
dx
2nd derivative ( derivative of the 1st derivative) - the rate of change of the first derivate (or the rate of
change of the ra
3.4: MARGINAL FUNCTIONS IN ECONOMICS
Suppose that the total weekly cost (in dollars) of manufacturing x iPads is given by the function
C(x) 8000 200x 0.2x2
(0 x 400)
a) Find the actual cost incurred for manufacturing the 89th iPad.
b) Find
C'(88)
and comp
3.1: BASIC RULES OF DIFFERENTIATION
Derivative of a Constant:
For any constant c,
d
(c) _
dx
Example: If f(x) = 5, then
f '(x)
_
The Power Rule:
Use the definition of derivative to find
Recall from section 2.6, we found
In general,
d n
(x )
dx
d
(x)
dx