animation derivative integral - ibiblio · the following animation shows two variables graphed over...
TRANSCRIPT
Questions
Question 1
Animation: differentiation and integration (calculus)
This question consists of a series of images (one per page) that form an animation. Flip the pages withyour fingers to view this animation (or click on the “next” button on your viewer) frame-by-frame.
The following animation shows two variables graphed over time, one being the derivative of the other
and the other being the integral of the first.
1
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
2
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
3
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
4
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
5
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
6
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
7
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
8
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
9
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
10
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
11
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
12
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
13
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
14
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
15
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
16
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
17
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
18
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
19
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
20
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
21
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
22
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
23
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
24
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
25
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
26
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
27
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
28
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
29
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
30
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
31
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
32
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
33
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
34
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
35
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
36
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
37
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
38
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
39
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
40
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
41
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
42
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
43
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
44
QV
Flow meter
Volume meter
Liquid storage tank
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
45
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
46
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
47
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
48
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
49
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
50
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
51
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
52
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
53
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
54
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
55
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
56
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
57
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
58
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
59
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
60
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
61
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
62
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
63
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
64
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
65
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
66
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
67
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
68
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
69
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
70
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
71
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
72
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
73
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
74
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
75
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
76
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
77
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
78
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
79
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
80
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
81
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
82
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
83
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
84
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
85
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
86
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
87
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
88
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
89
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
90
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height of the flowgraph directly relates to the slopeof the volume graph . . .
91
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
92
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
93
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
94
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
95
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
96
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
97
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
98
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
99
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
100
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
101
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
102
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
103
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
104
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
105
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
106
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
107
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
108
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
109
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
110
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
111
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
112
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
113
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
114
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
115
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
116
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
117
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
118
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
119
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
120
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
121
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
122
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
123
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
124
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
125
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
126
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
127
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
128
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
129
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
130
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
131
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
132
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
133
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
134
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
135
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
136
Q
V
Flow rate (Q) is thederivative of volumewith respect to time
Q = dVdt
Volume (V) is theintegral of flow ratewith respect to time
V = ∫ Q dt
Differen
tiation In
teg
rati
on
0
Max.
0
+Max.
-Max.
Note how the height increase ofthe volume graph directly relatesto the area accumulated by theflow graph . . .
file 04090
137
Answers
Answer 1
Nothing to note here.
138
Notes
Notes 1
The purpose of this animation is to let students study the generation of characteristic curves and reach
their own conclusions. Similar to experimentation in the lab, except that here all the data collection is done
visually rather than through the use of test equipment, and the students are able to “see” things that are
invisible in real life.
139