Here y = x^{2}  2
\ y +
D y = (
x + D x )^{2}  2
= x^{2} + 2 x
D x + (D
x)^{2}  2
\ D
y = ( x^{2} + 2 x D
x + (D x)^{2}  2)  (x^{2}
 2)
= 2 x D
x + (D x)^{2}
and
= 2 x + (D x)
Interpretation In the figure above, PS is
parallel to x  axis and QS is parallel to y  axis. If the secant
PQ has inclination µ
with positive of x  axis, then tan µ
which is the slope of secant PQ.
Now,
Interpretation The limiting position of
the secant PQ at P is the tangent to the curve y = x^{2}
 2 at P. i.e.
Q ®
P then
which is the slope of tangent at P to y = x^{2} 
2
Example 9 If s = 3 t^{2} + 7 is
the distance moved by a body along a straight line from a fixed
point o in time ‘t’ (1) find Ds
in s when t varies from t = t_{0} to t = t_{0} +
D t
(2) find
and
and interpret the results.
(1) Here s = 3 t^{2} + 7,
then s + D s = 3(t
+ D
t)^{2} + 7
= 3 t^{2} + 6 t D
t + 3(D
t)^{2} + 7
and Ds
= 6 t D
t + 3 (D t)^{2 }
= 6 t + 3 D
t
Since Ds is the distance moved by a body in Dt , is the average rate of change of distance with respect to time. In other words it is the average velocity of the body in t to
