Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 30
ADVERTISEMENT
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41Worksheet # 23: Approximating Area
1. Write each of following in summation notation:
(a) 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
(b) 2 + 4 + 6 + 8 + 10 + 12 + 14
(c) 2 + 4 + 8 + 16 + 32 + 64 + 128.
4
3
2. Compute
( + )
=1
=1
The following summation formulas will be useful below.
( + 1)
( + 1)(2 + 1)
2
=
=
2
6
=1
=1
3. Find the number
such that
= 78.
=1
4. Give the value of the following sums.
20
2
(a)
(2
+ 3)
=1
20
(b)
(3 + 2)
=11
5. The velocity of a train at several times is shown in the table below. Assume that the velocity changes
linearly between each time given.
t=time in minutes
0
3
6
9
v(t)=velocity in Km/h
20
80
100
140
(a) Plot the velocity of the train versus time.
(b) Compute the left and right-endpoint approximations to the area under the graph of .
(c) Explain why these approximate areas are also an approximation to the distance that the train
travels.
6. Let ( ) = 1
. Divide the interval [1 3] into five subintervals of equal length and compute
and
5
, the left and right endpoint approximations to the area under the graph of
in the interval [1 3].
5
Is
larger or smaller than the true area? Is
larger or smaller than the true area?
5
5
2
7. Let ( ) =
1
. Divide the interval [0 1] into four equal subintervals and compute
and
, the
4
4
left and right-endpoint approximations to the area under the graph of . Is
larger or smaller than
4
the true area? Is
larger or smaller than the true area? What can you conclude about the value ?
4
2
8. Let ( ) =
.
(a) If we divide the interval [0 2] into
equal intervals of equal length, how long is each interval?
(b) Write a sum which gives the right-endpoint approximation
to the the area under the graph of
on [0 2].
(c) Use one of the formulae for the sums of powers of
to find a closed form expression for
.
(d) Take the limit of
as
tends to infinity to find an exact value for the area.
ADVERTISEMENT
0 votes
Related Articles
Related forms
Related Categories
Parent category: Education