Page 36 - Mathematics Class - XII

x 1 2 3 4 5 6 7 8 9 10

log 1 log 2 log 3 log 4 log 5 log 6 log 7 log 8 log 9 log 10
e
e
e
e
e
e
e
e
e
e
y′ = log x
e
0 0.6931 1.0986 1.3863 1.6094 1.7918 1.9459 2.0794 2.1972 2.3026
3. Plot the points (1, 0), (2, 0.30), (3, 0.47), (4, 0.60), (5, 0.69), (6, 0.77), (7, 0.84), (8, 0.90), (9, 0.95) and
(10, 1) on graph paper and join these points to get the graph of y = log x.
10
4. Again, plot the points (1, 0), (2, 0.69), (3, 1.09), Y Y
(4, 1.38), (5, 1.60), (6, 1.79), (7, 1.94), (8, 2.07), 2.4
2.4
(9, 2.19) and (10, 2.30) on graph paper and join 2.2 y´ = log e x´ = log e x
2.2
y
these points to get the graph of y′ = log x. 2.0
2.0
e
1.8
y 1.8
5. Find the ratio .
y′ 1.6
1.6
6. Repeat the above steps for some more points 1.4
1.4
on the x-axis (with different values) and find 1.2
1.2
the corresponding ratios of the ordinates as in 1.0
1.0
Step 5. 0.8 y = log 10 x= log 10 x
y
0.8
7. Each of these ratios will nearly be the same 0.6
0.6
and equal to 0.4, which is approximately equal 0.4
0.4
1
to . 0.2
0.2
log 10
e X´ X X
X´
O O 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10
10 1111
Y´
Y´
Fig. (a)
OBSERVATION
y
S. No. Points on the x-axis y = log x y´ = log x Ratio (approximate)
10 e y´
1 x = 1 y = 0 y ′ = 0 Undefined
1
1
1
2 x = 2 y = 0.3010 y ′ = 0.6931 0.4343
2
2
2
3 x = 3 y = 0.4771 y ′ = 1.0986 0.4343
3
3
3

4 x = 4 y = 0.6021 y ′ = 1.3863 0.4343
4
4
4

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