The slope can be used as a measure to determine the increasing or decreasing nature of a line.

**Derivative as
slope**

**Slope or Gradient of a line: **Let** ***l*** **be any given non vertical line as in the Fig. 7.1. Taking a finite** **horizontal line segment of any length
with the starting point in the given line *l*
and the vertical line segment starting from the end of the horizontal line to
touch the given line. It can be observed that the ratio of the vertical length
to the horizontal length is always a constant. This ratio is called the slope
of the line *l* and it is denoted as *m* .

The
slope can be used as a measure to determine the increasing or decreasing nature
of a line. The line is said to be increasing or decreasing according as *m* > 0 or *m* < 0 respectively. When *m* = 0 , the value of *y* does not change. Recall that *y *=* mx
*+* c *represents a straight line in the *XY *plane where* m *denotes the slope of the line.

**Slope or Gradient of a curve: **Let** ***y*** **=** ***f*** **(** ***x*)**
**be a given curve. The slope of the line joining the two distinct points ( *x* , *f*
( *x*)) and the point ( *x* + *h*,
*f* ( *x* + *h*)) is

Taking
the limit as *h* â†’
0 , we get

which is
the slope of the curve at the point ( *x*
, *y*) or ( *x* , *f* ( *x*)) .

**Remark**

If *Î¸* is the angle made by the tangent to
the curve *y* =
*f* ( *x*) at the Point *( x , y)* , then
the slope of the curve at ( *x , y*) is
*fâ€™*(*x*) = tan Î¸ , where *Î¸* is measured in the anti-clockwise direction from the X -axis. Note that, *fâ€™*(*x*)
is also denoted by *dy/dx* and also
called *instantaneous
rate of change*. The average rate of change in an interval is calculated
using Newton* *quotient.

** **

**Example 7.1**

For the
function *f* (*x*) = *x*^{2}
, *x* âˆˆ[0, 2] compute the average rate of
changes in the subintervals [0, 0.5],[0.5,1],[1,1.5],[1.5, 2] and the
instantaneous rate of changes at the points *x*
=
0.5,1,1.5, 2 .

**Solution**

The
average rate of change in an interval [*a*,
*b*] is [ *f* (*b*) âˆ’
*f* (*a*) ] / [*b *â€“* a*] whereas, the instantaneous rate of change at a point *x* is *f*
â€²(
*x*) for the given function. They are
respectively, *b* +
*a* and 2*x* .

**Rate of changes**

Tags : Meaning of Derivatives | Mathematics , 12th Maths : UNIT 7 : Applications of Differential Calculus

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12th Maths : UNIT 7 : Applications of Differential Calculus : Derivative as slope | Meaning of Derivatives | Mathematics

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