3.
ba /20
i
, xi a ix
n
4
8
8
4
3
a) A R4 f ( xi )x cos cos cos
8 8
4
8
i 1
x
4
b) A L4 f ( xi 1 )x
i 1
cos 0.79 (underestim ate )
2
3
cos0 cos 8 cos 4 cos 8 cos 2 1.18 (overestimate )
8
4.
x
ba 40
1, xi a ix i
n
4
4
a) A R4 f (
15. Let u = sin, and the integral becomes:
(1 u 2 ) 2
u
du u 1 / 2 (1 2u 2 u 4 )du (u 1 / 2 2u 3 / 2 u 7 / 2 )du
2sin( )
1/ 2
37.
4
sin( )5 / 2 2 sin( )9 / 2 C
5
9
43. We use the suitable trig identity:
1
1
1
1
sin 8x cos 5x dx 2 [sin(8x 5x) sin(8x 5x)]
15. We let x = asint to get:
/2
2
2
4
a sin t a cos t a cos t dt a
0
/2
0
1 cos 2t 1 cos 2t
dt
2
2
4 /2
a
4
a4
4
1 cos
0
/2
0
2
(2t ) dt
a4
4
/2
sin
2
(2t ) dt
0
/2
1 cos 4t
a4 1
dt
t 4 sin(4t )
2
8
0
a 4
16
25. We first complete the square a
7.
x2
1
12
x 1 dx x 1 x 1 dx 2 x x ln x 1 C
15. Here the long division is easy.
x3 2x 2 4
4
4
C
A B
1 3
1 2
1 2
3
2
2
x 2
x 2x
x 2x
x ( x 2)
x x
We quickly get: B = -2, C = 1, and then we let x = 1 to get: -4 = A + B C, so A = -1.
1 2
x3 2x 2 4
1
2
3. Let u = sin x, then we get:
cos x
1 sin
2
x
dx
5.
7.
13.
Another solution:
1
du tan 1 u C tan 1 (sin x) C
2
1 u
15.
55. We let: u
dx
xx x
x , x u 2 , dx 2udu
2udu
1
1
1
2
du 2
du 2 ln x ln( x 1) C
2
3
u (u 1)
u u
u u 1
61.
67. We use the trig sub
The Relativity of Wrong
Isaac Asimov
e Skeptical Inquirer, 1989
I received a letter from a reader the other day. It was handwritten in crabbed penmanship so that it was very
dicult to read. Nevertheless, I tried to make it out just in case it might prove
What Is Science?
http:/www.fotuva.org/feynman/what_is_science.html
What is Science?
Presented at the fifteenth annual meeting of the National
Science Teachers Association, 1966 in New York City, and
reprinted from The Physics Teacher Vol. 7, issue 6, 1968
Science as Falsification
The following excerpt was originally published in Conjectures and Refutations (1963).
by Karl R. Popper
When I received the list of participants in this course and realized that I had been asked to speak to
philosophical colleague
From:
http:/www.geo.sunysb.edu/esp/files/scientif ic- met hod.ht ml
Last accessed Feb. 5, 2007
Scientific Thinking and the Scientific Method
by
Steven D. Schaf ers man
Depart ment of Geology
Miami Univers ity
January, 1997
The Definition of Science
Scienc
15. Special cast of Problem 94 (see below).
19.
27.
39. Solution 1: Write the expression as:
sin
cos 2
x
x x lim cos cos(0)
lim lim
x 1
x
x
1
x
2
x
x
Solution 2: Use u = 1/x:
49.
sin( u )
lim x sin lim
lim cos( u ) cos(0)
x
u 0
u
x u
(This is a slightly different version from the one in the text, just adjust the numbers:)
10. a) Use y(t) = y(0)ekt,with y(1) = 0.945y(0) to get k. For the half life T, use y(T) = 0.5y(0) and get T.
b) Find t for which y(t) = 0.2y(0).
(This is a slightly
1
1
x 1 , then du (1) x 2 dx 2 dx
x
x
1
sec 2
1
x
2
x 2 dx sec u (du) tan u C tan x C
6. Let u
13.
18.
22.
26. Let u cos t , then du sin t dt
sin t sec cos t dt sec u (du) tan u C tancos t C
2
2
27. Let u sin x , then du cos xdx , so
Also:
cos x
1
1
19. You can solve the equation h(x) = 6 for x, or you can just guess that h(4) = 6, so h-1(6) = 4.
40. Note that using Theorem 7, you need to find f-1(2) = x, i.e. f(x) = 2. We can just guess x = 1. Thus, the
answer is 1/f(1), which can be easily calculat
23. Since 1.001 > 1, the limit is infinity.
25. Divide up and down by e3x. Since the limit of e-6x is zero, the limit of the fraction evaluates to 1.
28. As x2 from the negative side (think x = 1.999), the power goes to infinity and the whole function
goe