15 hours
of
studying
could
earn you
15% or
more on
your next
test, so
GET OUT
2008 AP
EXAM
D
2. B
3. C
4. B
5. E
6. A
7. E
8. E
9. D
10. D
1.
11.A
12.E
13.B
14.C
15.D
16.E
17.D
18.B
19.C
20.A
21.B
22.C
23.A
24.D
25.D
26.A
27.A
28.E
76.A
77.C
78.B
79.A
80.B
81.D
82.E
83.B
84.A
85.C
86.C
87.D
88.C
89.B
90.D
91.E
92.D
Let’s Talk about Score
Composite
AP Grade
75-108
5
58-74
4
40-57
3
25-39
2
0-24
1
Multiple choice
Sub-score
Estimated
AP Grade (ISH)
32-45
5
24-31
4
16-23
3
10-15
2
0-9
1
Let’s Talk about Score
Composite
AP Grade
75-108
5
58-74
4
40-57
3
Multiple choice
Sub-score
Estimated
AP Grade
25-39
2
32-45
5
0-24
1
24-31
4
16-23
3
Our break
down
10-15
2
11 and up
5
0-9
1
8-10
4
6-8
3
3-5
2
0-2
1
Estimated
AP Grade
SWBAT: apply an integral
5.1 Types of Approximations p.22
Riemann sums (3 types)
 Trapezoidal

Riemann Sums
Rectangular approximations
LRAM
“Left side”
MRAM
“Midpoint”
RRAM
“Right side”
Put in 5.1 (but from 5.9)
Trapezoidal Approximation (TrAM*)
1
𝐴 = 𝑏2 + 𝑏1 ℎ
2
5.2 Definite Integrals
𝒃
𝒏
lim
𝒏→∞
p.23
𝒇 𝒙𝒊 ∆𝒙 =
𝒊=𝟏
𝒃−𝒂
∆𝒙 =
𝒏
𝒇 𝒙 𝒅𝒙
𝒂
𝒙𝒊 = 𝒂 + ∆𝒙𝒊
Integrals

Definite integrals

 6x
3

Indefinite integrals
9  x dx
2
4
dx
0
“definitely” know its
interval.
 yields area under a
curve


No interval
yields a function.
 Has a +c

5.3 Power rule for integrals p.24

Power Rule for integrals( xn for n ≠ –1)
n1
x
x
dx


c

n1
n
Basic Integral rules
Trig. Integrals
5.3 FTOC 2
p. 25
”Fundamental Theorem of Calculus Part 2”
b
 f ( x )dx  F (b)  F (a)
a

Add F(a) to both sides.
b
F (a )   f ( x )dx  F (b )
a

(sometimes this seems tricky!)
b
 Rate  A(b)  A(a)
a
OR, in terms of motion:
b
Velocity

Position
(
b
)

Position
(
a
)

a
Think integral is: “net change”
Mapping Motion
Copy this
Diagram:
 Displacement: “how much the position changed”
𝑏
𝑣(𝑡) 𝑑𝑡
𝑎
 Distance
traveled:
𝒃
𝒗(𝒕) 𝒅𝒕
𝒂
This requires a few more steps.
p. 26
5.4 FTOC1

General form:
g  x    f  t  dt
x
a
(this is viewed as an accumulator.)
 Note that g depends only on x (not t!)
“Area So Far”
FTOC 1, with derivative
d u
f
(
t
)
dt

f
(
u
)

u
'

f
(
v
)

v
'

dx v
Note: If v is a constant, then v’ is 0 )
How FTOC 1 fits in
𝑔 𝑥 =
𝑥
𝑓(𝑡)
𝑑𝑡
𝑎
𝑔′ 𝑥 = 𝑓 𝑥
𝑔′′ 𝑥 = 𝑓′ 𝑥
5.5 U-Substitution
When integrating, ask yourself 3 ?’s:
1. Does it fit a formula?
2. Can it be simplified?
3. Can you use U-Sub?
p. 27
5.5 U-Substitution

Let u  1  x
du
 2x
dx
or , du  2 xdx
2 x 1  x dx
2
Substituing in :

1
2
udu   u du
3
2
3
2 2
2
2
 u  C  (1  x )  C
3
3
2
U-Sub. With Definite Integrals

4
0
2 x  1dx :
Solution
ASGN

Work on ch. 5 at a glance
ASGN

2002 1,2,5,
 2002
3,4,6
Ch. 4 At a glance answers
y'
x 2  4x
x2
 x  2
x2
2
none
 ,0   4, 
0,2   2,4
0, 2
 4,6
 x  2  2x  4    x 2  4 x  2  x  2 
y '' 
4
 x  2
2
 2,
 ,2