Maximum and Minimum Values
Maximum and Minimum Values
As we saw for functions of a single variable, one of the main uses
of ordinary derivatives is in nding maximum and minimum values
(extreme values). In this lecture we see how to use partial
derivatives
6.2.3 The Fundamental Theorem of Calculus (Part II)
1. Quote.Each problem that I solved became a rule, which served
afterwards to solve other problems
(Rene Decartes, 1596-1650)
2. The Fundamental Theorem of Calculus, Part 2.
If f is continuous on [a, b],
6.1.3 Properties of the Riemann Integral
1. Quote. No human investigation can be called real science if it cannot
be demonstrated mathematically
(Leonardo da Vinci (1452-1519)
2. Two Special Properties of the Integral.
(a) If a > b then
Z
b
Z
f (x)dx =
a
7.1.1 The Substitution Rule for Indefinite Integrals
1. Quote. Persuasion is often more effectual than force.
(Aesop, Greek fabulist, 6th century BC)
2. Problem. Find
Z
2
2xex dx
3. Hint. What if we think of the dx above as a differential? If
2
u = ex , w
6.2.1 The Fundamental Theorem of Calculus (Part 1) and
6.2.2 Antiderivatives and Indefinite Integrals
1. Quote. If you cant run, then walk. And if you cant walk,
then crawl. Do what you have to do. Just keep moving forward
and never, ever give up
(Dean Ka
M ATH 155
C HAPTER 1
Question: Which of the following infinite series sums to
2
?
6
A.
1
1
+ 21 + 13 + 14 + 51 + . . .
(reciprocals of positive integers)
B.
1
2
+ 31 + 15 + 17 +
(reciprocals of primes)
C.
1
1
+ 41 + 19 +
1
16
+
1
25
+ .
(reciprocals of sq
M ATH 155
C HAPTER 2
Question. Which statement best describes the mathematical concept of area
for sets in the plane?
A. Every set in the plane has an area which is easy to calculate.
B. Every set in the plane has an area, but it might be difficult to com
M ATH 155
C HAPTER 1
Carl Friedrich Gau
Gauss is one of the all time great mathematicians. In fact, many historians
regard him as the greatest of all time. His impact on present day mathematics is extraordinary and would be hard to overstate.
Even when he
M ATH 155
C HAPTER 3
Review/Preview
1. We have defined the integral of a function in an attempt to formalize the
calculation of areas of regions in the planean applied problem.
2. Next we will develop some mathematical theory of the integral in an
attempt
M ATH 155
C HAPTER 3
Reminder. The definite integral was introduced to compute area. It is defined by the rule
b
Z
f (x)dx = lim
N
a
N
X
f (xk )(x).
k=1
P
The sum N
k=1 f (xk )(x) gives the area of an N -rectangle approximation to
the true area of the re
6.3.2 Cummulative Change and 6.3.3 Average Values
1. Quote. The real danger is not that computers will begin to
think like men, but that men will begin to think like computers
(Sydney J. Harris, American Journalist, 1917-1986)
2. The Cumulative or Net Cha
7.1.2 The Substitution Rule for Definite Integrals
1. Quote. In the depth of winter I finally learned that there was
in me an invincible summer.
(Albert Camus, 1913-1960, French philosopher)
2. Substitution Rule for Definite Integrals.
If g 0 is continuou
Midterm 2 Review
Integration by Parts
Understand how integration by parts can be used to evaluate
indenite and denite integrals
Integration by Parts
Example
1
Calculate
0
tan1 x dx.
Integration of Rational Functions by Partial Fractions
Understand how to
Functions of Several Variables
Functions of Several Variables
So far we have dealt with the calculus of functions of a single
variable. But, in the real world, physical quantities often depend
on two or more variables, so we now turn our attention to
func
6.1.2 Riemann Integrals
1. Quote. “Mathematics, rightly Viewed, possesses not only truth,
but supreme beauty”
(Betrand Russell, 1872—1970)
2. A more general formulation of area.
Ingredients: A function f that is continuous on a closed interval
[(1, bl 1.“
6.3.1 Areas
1. Quote. It is not worth an intelligent mans time to be in the majority.
By definition, there are already enough people to do that.
(G.H. Hardy)
2. Goal: To find the area between two curves.
3. Theorem: The area A of the region bounded by the
M ATH 155
C HAPTER 1
Although calculus is was developed in the 17th century, its roots go back
much further in time. One precursor of calculus is the special number
3.14159265358979323846264338327950288419716939937510 . . .
This number was recognized by
M ATH 155
C HAPTER 2
Question. What is the surface area of Greenland?
A. 500,000 km2
B. 1,000,000 km2
C. 2,000,000 km2
D. 4,000,000 km2
E. 8,000,000 km2
C.
2,000,000 km2
Note: This is a significant parameter for global warming concerns due to the
fact tha
K24 h. N
W
+w>[n)].z:
H
-1
3 0r- 3 L-
#13.3+g+22 :1: a 2 _
5 g- E 3 K23 K+2 i=5 6
h h 7 '- h _, n(n+|(2 H) J
#23. 2 (warm): 2 (K719) a 2K E I! ~ -i-~421
K31 KL, ICC-g K71
6 Mil
h(h+l)(2n+l) _ 1m. _._ h(2n+3n-23)
" 6 , 6
13 k 9 o
#30. it, =51) 1:01?sz #5
5%
ssignmem it? 5'
1!
9.
z 3/: dz:
M? W of 5
g, 4. _
an); 1,: km; on - JM/JroW #7943ng
em C
A q 0/9: :7 , 70 da:
1 . _. r
(334,943 (am 2" (0240/3 9-7:; (364993
'5): "J is 2107 connaaxo M m. 52 W-
I 2.1 mu .5
f / (Ho/5 15/): fmlum! 0/ mziymm, I [D
HSSIgnmenf #?
8.2
2 if 23(2o-cu),wo
2%
6) i9 _3Job" i b) a, c1g=6m(20.me )320
20-C cfw_7.990 cfw_mam
._ cfw_injgoc a 3i: +53 335%
-3+ 1 ck/gm . to Mouse
90" - 9.9 I g i eawam
3% . t
amaze cm I 3322/3
9:. so 14 i * 323 z gag/3
fhhs: 5*:20-cz)cza15 cfw_I :