Page 31 - Mathematics Class - XII

Unit III Calculus
Calculus

TOPIC - 3: Continuity and Differentiability

ACTIVITY 3.1

OBJECTIVE
To sketch the graphs of a and log x, a > 0, a ≠ 1 and to examine that they are mirror images of each other.
x
a
MATERIAL REQUIRED

Y Drawing board Y Sketch pens Y Eraser
Y Geometrical instruments Y Thick white paper Y A plane mirror
Y Drawing pins Y Adhesive Y Graph paper

Y Thin wires Y Pencil

PRE-REQUISITE KNOWLEDGE
1. Knowledge of exponential functions.

2. Knowledge of logarithm functions and their properties.

PROCEDURE
1. On the drawing board, fix a thick paper sheet of convenient size with
adhesive and fix a graph paper on it as shown in Fig. (a).
2. On the graph sheet, take two perpendicular lines XOXˊ and YOYˊ,
depicting coordinate axes.
3. Find some ordered pairs satisfying y = a and y = log x. Plot these points
x
a
corresponding to the ordered pairs and join them by free hand curves in
both the cases. Fix thin wires along these curves using drawing pins.

4. Draw the graph of y = x, and fix a wire along the graph, using drawing pins.

Fig. (a)
DEMONSTRATION
1. For a (let a = 2) and find the ordered pairs ( y = 2 ) for different value of x.
x
x
x 0 1 –1 2 –2 3 –3 4 –4
y = 2 x 1 2 0.5 4 0.25 8 0.125 16 0.0625

2. Plot these ordered pairs on the graph paper and fix a drawing pin at each point as shown in Fig. (b).
3. Join the bases of drawing pins with a thin wire. This will represent the graph of y = 2 as shown in Fig. (b).
x
4. Now, consider y = log x or x = 2 . Find some value of x and y, satisfying the function x = 2 .
y
y
2
x = 2 1 2 0.5 4 0.25 8 0.125 16 0.0625
y
y = log x 0 1 –1 2 –2 3 –3 4 –4
2
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