Summary

SUMMARY

• If y = f (x) , then dy/ dx represents instantaneous rate of change of y with respect to x .

If y = f ( g (t )) , then dy/dt = f ′ ( g (t ))⋅ g′ (t ) which is called the chain rule.

• The equation of tangent at ( a , b) to the curve y = f (x) is given by y − b = (dy/dx)_(a,b)(x − a) or y-b = f ′ (a)(x − a) .

• Rolle’s Theorem

Let f ( x) be continuous in a closed interval [ a , b] and differentiable on the open interval ( a , b) . If f (a) = f (b) , then there is at least one point c ∈( a,b) where f ′ (c) = 0 .

• Lagrange’s Mean Value Theorem

Let f ( x) be continuous in a closed interval [ a , b] and differentiable on the open interval ( a , b) (where f (a) and f (b) are not necessarily equal). Then there is at least one point c ∈( a,b) such that f ′ (c) = f (b) − f (a) / b-a. 

• Taylor’s series

Let f ( x) be a function infinitely differentiable at x = a . Then f ( x) can be expanded as a series in an interval ( x − a , x + a) ,of the form

• Maclaurin’s series

In the Taylor’s series if a = 0 , then the expansion takes the form

• The l’Hôpital’s rule

Suppose f ( x) and g ( x) are differentiable functions and g ′ ( x) ≠ 0 with

• If the function f ( x) is differentiable in an open interval ( a , b) then we say, if d/dx ( f (x)) > 0 , ∀ x ∈( a,b) then f ( x) is strictly increasing in the interval (a,b).

• if d/dx ( f (x)) < 0 , ∀ x ∈( a,b) then f ( x) is strictly decreasing in the interval ( a , b)

• A procedure for finding the absolute extrema of a continous function f ( x) on a closed interval [a , b].

Step 1 : Find the critical numbers of f ( x) in ( a , b) .

Step 2 : Evaluate f ( x) at all critical numbers and at the endpoints a and b .

Step 3 : The largest and the smallest of the values in Step 2 is the absolute maximum and absolute minimum of f ( x) respectively on the closed interval [a , b] .

• First Derivative Test

Let ( c , f (c)) be a critical point of function f ( x) that is continuous on an open interval I containing c . If f ( x) is differentiable on the interval, except possibly at c , then f (c) can be classified as follows:(when moving across I from left to right)

(i) If f ′ ( x) changes from negative to positive at c , then f ( x) has a local minimum f(c).

(ii) If f ′ ( x) changes from positive to negative at c , then f ( x) has a local maximum f (c).

(iii) If f ′ ( x) is positive on both sides of c , or negative on both sides of c then f ( x) has neither a local minimum nor a local minimum.

• Second Derivative Test

• Suppose that c is a critical point at which f ′ (c) = 0 , that f ′′ ( x) exists in a neighbourhood of c , and that f ′ (c) exists. Then f has a relative maximum value at c if f ′′ (c) < 0 and a relative minimum value at c if f ′′ (c) > 0 . If f ′′ (c) = 0 , the test is not informative.