Applications of Differentiation

Derivative as a rate of measure
1.$\frac{\mathrm{dy}}{\mathrm{dx}}$ is the rate of change of y w.r. to x

2. If s is the displacement of a particle then the velocity of the particle at time 't' is v= $\frac{\mathrm{ds}}{\mathrm{dt}}$,

Acceleration, a = $\frac{\mathrm{dv}}{\mathrm{dt}}$= $\frac{d^{2}s}{{\mathrm{dt}}^2}$

Rolle's Theorem
If f(x) is continuous in [a,b] derivable in (a, b) and f(a) = f(b) then there exist atleast one point c $\in$
(a,b) such that f' (c)= 0

Lagrange's Mean Valve Theorem
If f(x) is continuous in [a, b] and derivable in (a, b) then there exists at least one point c $\in$(a,b) such that f'(c) = $\frac{f(b)-f(a)}{b-a}$

Percentage Error

Let $\mathrm{\Delta }$x be an error in the variable, then $\frac{\mathrm{\Delta }x}{x}$ is called the relative error and $\left(\frac{\mathrm{\Delta }x}{x}\right)$
. 100 is called the percentage error in x.

Increasing and Decreasing Functions
1. A function f (x) is said to be a strictly increasing function on (a,b), if $x_1$$<$$x_2$

$\Rightarrow$ f($x_1)$$<$f($x_2)$ for all $x_1$, $x_2$$\in$ ( a,b)

2. A function f(x) is said to be a strictly decreasing function on (a,b) if $x_1$$<$$x_2$ $\Rightarrow$ f($x_1$) $>$ f($x_2$) for all $x_1$, $x_2$ $\in$ (a,b)

Monotonic Functions
A function f (x) is said to be monotonic on an interval (a,b) if it is either increasing or decreasing on (a,b)

Necessary and sufficient conditions for monotonicity of functions
(a) If f' (x) $\ge$ 0 for all x $\in$ D, then f(x) is monotonic increasing in D.

If f' (x) $\le$ 0, for all x $\in$ D, then f(x) is monotonic decreasing in D.
(b) If f' (x) $>$ 0 for all x $\in$ D, then f(x) is strictly monotonic increasing in D.

If f' (x) $<$ 0, for all x $\in$ D, then f(x) is strictly monotonic decreasing in D.


Maxima and Minima

1. Point of Maxima: Let f(x) be a function with domain D $\underset{\_}{\subset }$R. Then f(x) is said to be maximum at a point a $\in$D if f(x) $\le$ f (a) for all x $\in$D.
Here 'a' is called the point of Maxima and f(a) is called maximum valve or the greatest valve or the absolute maximum valve of f(x).

2. Point of Minima: Let f(x) be a function with domain $\underset{\_}{\subset }$R. Then f(x) is said to be minimum at a point 'a' $\in$D if f(x) $\ge$ f(a) for all x $\in$ D.
Here 'a' is called the point of Minima and f(a) is called Minimum valve or the least valve or the absolute minimum valve of f(x).

3. Local Maxima
A function f(x) is said to attain a local maximum at x= a if there exists a nbd. ( a- $\delta$, a+$\delta$) of 'a' such that f(x) $<$ f(a) for all x $\in$( a- $\delta$, a+$\delta$) [ x $\ne$ a] or f(x) - f(a) $<$ 0 for all x $\in$ ( a- $\delta$, a+$\delta$) , x $\ne$ a.
Here f(a) is called the local Maximum valve of f(x) at x = a.

4. Local Minima
A function f(x) is said to attain a local minimum at x= a if there exists a nbd. ( a- $\delta$, a+$\delta$) of 'a' such that f(x) $>$ f(a) for all x $\in$ ( a- $\delta$, a+$\delta$) (x $\ne$ a) or f(x) - f(a) $>$0 for all x $\in$ ( a- $\delta$, a+$\delta$), x $\ne$ a.
Here f(a) is called the local Minimum valve of f(x) at x= a.

5. Extreme Valves
The points at which a function attains either the local maximum valve or local minimum valve are called extreme valves of f(x).

6. Necessary condition for Extreme valves.
A necessary condition for f(a) to be an extreme valve of a function f(x) is that f' (a) = 0, if it exists.

7. Stationary values or Critical values of x.
The value of x for which f' (x) = 0 are called stationary values or critical values of x and the corresponding values of f(x) are called stationary or turning values of f(x). The points at which f' (x) does not exist are also known as critical points.

8. First Derivative Test for Local Maxima and Minima.
(a) Let f(x) be a function differentiable at x=a. Then x = a is a point of local maximum of f(x) if (i) f' (a) = 0 and (ii) f' (x) changes sign from +ve to -ve as x passes thro' a i.e f' (x) $>$0 at every point in the right nbd (a, a+$\delta$) of 'a'.
(b) x = a is a point of local minimum of f(x) if (i) f' (x) $<$0 at every point in the left nbd. (a-$\delta$, a) of 'a' and f' (x) $>$ 0 at every point in the right nbd (a, a+$\delta$) of a.
(c) If f' (a) = 0 but f' (x) does not change sign, then 'x = a' is neither a point of local maximum nor a point of local minimum.

9. Second order Derivative Test
Let f(x) be a function in x. Find $f^{\prime }$ (x) and equate to zero we get the critical points x= a, b, c....
Find $f^{″}$(x) and (1) if at x = a, $f^{″}$(x) is -ve, the function is maximum. (2) If at x = b, $f^{″}$(x) is =ve, the function is minimum. (3) If at x= c, $f^{″}$(x)= 0 the function is neither maximum nor minimum. In this case 'c' is known as point of inflexion.

10. Point of inflexion.
An arc of a curve y = f(x) is called concave upwards if at each of its points, the arc lies above the tangent at the point.

Geometrical Meaning of Derivative
$\left[\frac{\mathrm{dy}}{\mathrm{dx}}\right](x_1,y_1)$ is the slope of the tangent to the curve y= f(x) at the point $(x_1,y_1)$

1. The equation of the tangent at the point $(x_1,y_1)$ on the curve y= f(x) is y-$y_1$= ${\left[\frac{\mathrm{dy}}{\mathrm{dx}}\right]}_{(x_1,y_1)}$ (x- $x_1$)

2. The equation of the normal at the point $(x_1,y_1)$ on the curve y= f(x) is y - $y_1$= $\frac{-1}{{\left[\frac{\mathrm{dy}}{\mathrm{dx}}\right]}_{(x_1,y_1)}}$. (x- $x_1$)

Note:

a) If the tangent is parallel to X- axis, then $\frac{\mathrm{dy}}{\mathrm{dx}}$=0

b) If the tangent is parallel to Y- axis, then $\frac{\mathrm{dy}}{\mathrm{dx}}$=0

Angle between two curves
1. If $\theta$ is the angle between the two curves at a point of intersection $(x_1,y_1)$ then tan$\theta$= $\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$ where $m_1$= ${\left[\frac{\mathrm{dy}}{\mathrm{dx}}\right]}_{(x_1,y_1)}$ for y= f(x)

Length of the tangent = $\left|\frac{y\sqrt{1+y_1^2}}{y_1}\right|$

Length of the normal= $y\sqrt{1+y_1^2}$

Length of the subtangent= y/$y_1$
Length of the subnormal= ${\mathrm{yy}}_1$

Intercept of tangent on the axes of x= $|x-y\mathrm{.}\frac{\mathrm{dx}}{\mathrm{dy}}|$

Intercept of tangent on the axes of y= $|y-x\mathrm{.}\frac{\mathrm{dy}}{\mathrm{dx}}|$