12th Maths - Geometrical interpretation of derivatives - JEE Main 2014 by ednexa.com

1. Geometrical Interpretation of Derivative
Let P be the point x = a on the curve y =f (x).
[This means that the x-coordinate of P is a].
∴ P is (a, f (a)).
Take a point Q (a + h, f (a + h)) on the curve. Then
slope of secant PQ

2. 
f a  h f a
f a h f a
f
h
a ha
Now as Q → P along the curve, the limiting position
of the secant PQ is called the tangent to the curve
at the point P, i.e. at the point x = a and the slope
of the tangent at P is called the gradient of the
curve at the point P.
Now as Q → P along the curve, h → 0
∴ Gradients of the curve at P = slope of the
tangent PT

lim
Q P
 lim 0
h
(Slope of secant PQ)
f a  h  f a
 f ' a.
h

3. Thus f’ (a) represents the slope of the tangent to
the curve
y = f(x) at the point (a, f (a)).
[i.e., the point x = a on the curve.]
Physical significance of a derivative:
Suppose a function given by y = f (x) is
differentiable at x = a. If x changes from a to a +
h, then f (x) changes from f (a) to f (a + h).
Thus, f (a + h) – f (a) is the change in f (x)
corresponding to the change h in x. The average
rate of change is f a h  f a in the interval [a, a + h],
h

4. if h is positive. If h is negative then it is the
average rate of change in the interval [a + h, a].
Taking the limit as h → 0, we have,
lim
h0
f a  h  f a
 f ' a .
h
This exists since f is differentiable at x = a.
It is called instantaneous rate of change of the
function f at x = a. Hence, we can say that the
derivative f’ (a) measures the rate of change of f at
x = a. Thus, the derivative of a function is a rate
measurer of the function.
Let a particle be moving along a straight line
such that its displacement s at time t is expressed
as a function of t, say s = f (t).

5. The rate of change of s w.r.t. t which is
ds
 f ' t
dt
called the velocity v of the particle at time t.
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