Mathematics Revision of Application Of Derivatives
Click for Only Video

Rate of Change of Quantities

\color{green} ✍️ Further, if two variables x and y are varying with respect to another variable t, i.e., if x = f (t) and y = g(t ),

Then by Chain Rule :

color{blue}{(dy) / (dx) =((dy)/(dt))/ ((dx)/(dt))  If (dx)/(dt) ne 0

Increasing and Decreasing Functions

=> Let I be an open interval contained in the domain of a real valued function f. Then f is said to be

1. Increasing on I if color{red}{x_1 < x_2} in I => color{blue}{f(x_1) <= f(x_2)} for all x_1, x_2 in I

2. Strictly increasing on I if color{red}{x_1 < x_2} in I=>color{blue}{f(x_1) < f(x_2)} for all x_1, x_2 in I

3. Decreasing on I if color{red}{x_1 < x_2} in I => color{blue}{f(x_1) >= f(x_2)} for all x_1, x_2 in I

4. Strictly increasing on I if color{red}{x_1 < x_2} in I => color{blue}{f(x_1) > f(x_2)} for all x_1, x_2 in I

First derivative test for increasing and decreasing functions

\color{green} ✍️ Let f be continuous on [a, b] and differentiable
on the open interval (a,b). Then

1. color{blue}{ f \ \ "is increasing in" [a,b] "if" \ \ f′(x) > 0 \ \ "for each"\ \ x ∈ (a, b)}
2. color{blue}{ f \ \ "is decreasing in" \ \ [a,b] \ \ "if" \ \ f ′(x) < 0 \ \ "for each" \ \ x ∈ (a, b)}
3. color{blue}{ f \ \ "is a constant function in" \ \[a,b] "if" \ \ f ′(x) = 0 \ \ "for each" \ \ x ∈ (a, b)}

Tangents and Normals

=> The equation of the tangent at (x_0, y_0) to the curve y = f (x) is given by

color{green}{y – y_0 = f ′(x_0)(x – x_0)}

=> The equation of Normal to the curve y = f (x) at (x_0, y_0)  is given by

color{green}{y - y_0 = (-1)/( f' (x_0 ) ) (x - x_0)}

Approximations

\color{green} ✍️ Let Δx denote a small increment in x. Recall that the increment in y corresponding to the increment in x, denoted by Δy, is given by color{blue}{Δy = f (x + Δx) – f (x).}

We define the following
(i) The differential of x, denoted by dx, is defined by dx = Δx.

(ii) The differential of y, denoted by dy, is defined by dy = f′(x) dx or

color{red}{dy = ( (dy)/(dx) ) Δx}

Second Derivative Test For Maxima/Minima

\color{green} ✍️ Let f be a function defined on an interval I and c ∈ I. Let f be twice differentiable at c. Then

(i)color{orange}{x = c\ \ "is a point of local maxima if" \ \ f ′(c) = 0}
color{orange}{"and" f ″(c) < 0}
=> The value f (c) is local maximum value of f .

(ii)color{orange}{ x = c \ \ "is a point of local minima if" \ \ f ′(c) = 0}
color{orange}{"and" f ″(c) > 0}
=> In this case, f (c) is local minimum value of f .

(iii) color{orange}{"The test fails if" \ \ f ′(c) = 0 "and" f ″(c) = 0.}

In this case, we go back to the first derivative test and find whether c is a point of local maxima, local minima or a point of inflexion.