Here's a very quick outline or summary of the concepts introduced in this tutorial. Future tutorials, which appear further in this series of calculus tutorials, will focus on how these ideas are applied and we will solve interesting examples and problems where these concepts will be applied.
• Differentiability Of Functions & Basic Differentiation Formulas:
The condition for differentiability of functions will be introduced in the tutorial.
Addition, Subtraction, Linearity, Product, Quotient, Chain and Power Rules, Exponential and Logarithmic Rules - these will be introduced in the tutorial document.
Formulas for derivatives of sin x, cos x, tan x, sec x, cot x, cosec x and their inverse values: sin-1 x, cos-1 x, tan-1 x, sec-1 x, cot-1 x, cosec-1 x
• Successive Differentiation:
The derivative f' (x) of a derivable function f (x) is itself a function of x. We suppose that it also possesses a derivative, which is denoted by f'' (x) and called the second derivative of f (x).
The third derivative of f (x) which is the derivative of f'' (x) is denoted by f '''(x) and so on. Thus the successive derivatives of f (x) are represented by the symbols, f (x), f; (x), . . . , f n (x), . . .
where each term is the derivative of the previous one. Sometimes y1 , y2 , y3 , . . . , yn , . . . are used to denote the successive derivatives of y.
• Leibnitz’s Theorem
The nth derivative of the product of two functions: If u, v be the two functions possessing derivatives of the nth order, then (uv)n = un v +n C1 un−1 v1 +n C2 un−2 v2 + . . . +n Cr un−r vr + . . . + uvn .
• Rolle’s Theorem:
If a function f (x) is derivable in an interval [a, b], and also f (a) = f (b), then there exists atleast one value c of x lying within [a, b] such that f (c) = 0.
• Lagrange’s Mean Value Theorem:
If a function f (x) is derivable in an interval [a, b], then there exists atleast one value c of x lying within [a, b] such that
(f (b) − f (a)) / (b-a) = f(c)
Increasing and Decreasing Functions:
A function whose derivative is positive for every value of x in an interval is a monotonically increasing function of x in that interval, i.e,
If f (x) > 0 for every value of x in [a, b], then f (x) is an increasing function of x in that interval.
A function whose derivative is negative for every value of x in an interval is a monotonically decreasing function of x in that interval, i.e,
If f (x) < 0 for every value of x in [a, b], then f (x) is an decreasing function of x in that interval.
What Maxima and Minima mean :
Maximum Value of a function: f (c) is said to be a maximum value of f (x), if it is the greatest of all its values of x lying in some neighbourhood of c, i.e, f (c) is a maximum value of x if there exists a positive δ such that f (c) > f (c + h) or f (c) − f (c + h) > 0 for values of h lying between −δ and δ.
Minimum Value of a function: f (c) is said to be a minimum value of f (x), if it is the least of all its values of x lying in some neighbourhood of c, i.e, f (c) is a minimum value of x if there exists a positive δ such that f (c) < f (c + h) or f (c + h) − f (c) > 0 for values of h lying between −δ and δ.
Greatest and least values of a function in any interval
The greatest and least values of f (x) in any interval [a, b] are either f (a) and f (b), or are given by the values of x for which f (x) = 0.
Change of Sign
A function is said to change sign from positive to negative as x passes through a number c, if there exists some left-handed neighbourhood (c − h, c) of c for every point of which the function is positive, and also there exists some right-handed neighbourhood (c, c + h) of c for every point of which the function is negative.
Sufficient Criteria for extreme values
Prove that f (c) is an extreme value of f (x) if and only if f (x) changes sign as x passes through c, and to show that f (c) is a maximum value if the sign changes from positive to negative and a minimum value if the sign changes from negative to positive.
Minimum and Maximum Values :
Minimum Value: f (c) is a minimum value of f (x), if f '(c) = 0 and f'' (c) > 0.
Maximum Value: f (c) is a maximum value of f (x), if f '(c) = 0 and f ''(c) < 0.
Implicit Differentiation:
A relation F (x, y) = 0 is said to define the function y = f (x) implicitly if, for x in the domain of f , F (x, f (x)) = 0. Given a differentiable relation F (x, y) = 0 which defines the differential function y = f (x), it is usually possible to find the derivative f even in the case when you cannot symbolically find f . The method of finding the derivative is called implicit differentiation.
Criteria for concavity, convexity and inflexion:
Criteria to determine whether a curve y = f (x) is concave upwards, concave downwards, or has a point of inflexion at P [c, f (c)] are:
(i) the curve is concave upwards at P if f'' (c) > 0.
(ii) the curve is concave downwards at P if f'' (c) < 0.
(iii) the curve has inflexion at P if f'' (c) = 0 and f''' (c) = 0.
Complete Tutorial with Definitions and Formulas :
