### CONTINUITY AND DIFFERENTIABILITY

1. A real valued function is continuous at a point in its domain if the limit of the function at that point equals the value of the function at that point. A function is continuous if it is continuous on the whole of its domain.

2. Sum, difference, product and quotient of continuous functions are continuous. i.e., if f and g are continuous functions, then

3. (f ± g) (x) = f (x) ± g(x) is continuous.

4. (f . g) (x) = f (x) . g (x) is continuous.

5. Every differentiable function is continuous, but the converse is not true.
6. Following are some of the standard derivatives (in appropriate domains):

7.Logarithmic differentiation is a powerful technique to differentiate functions of the form f (x) = [u (x)]v(x) . Here both f (x) and u(x) need to be positive for this technique to make sense. v (x)

8.Rolle’s Theorem: If f : [a, b] → R is continuous on [a, b] and differentiable on (a, b) such that f (a) = f (b), then there exists some c in (a, b) such that f ′(c) = 0.

### APPLICATION OF DERIVATIVES

1.If a quantity y varies with another quantity x, satisfying some rule y = f(x) the dy/dx (or f(x)) represents the rate of change of y with respect to x

2.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.

3.A function f is said to be
(a) increasing on an interval (a, b) if x1 < x2 in (a,b) ⇒ f(x1) ≤ f(x2) for all x1 , x2 ∈ (a,b)
Alternatively, if f ′(x) ≥ 0 for each x in (a, b)
(b) decreasing on (a,b) if x1 < x2 in (a, b) ⇒ f (x 1 ) ≥ f (x 2 ) for all x 1 , x 2 ∈ (a,b)

4.If dy/dx does not exist at the point (x0,y0) then the tangent at this point is parallel to the y-axis and its equation is x = xo.

1. If dy/dx at the point (x0,y0) is zero, then equation of the normal is x = xo.

2. If dy/dx at the point (x0,y0) does not exist , then then the normal is parallel to x-axis and its equation is y =yo

3. Let y = f (x), △ x be a small increment in x and △y be the increment in y corresponding to the increment in x, i.e., △y = f (x + △x) – f (x). Then dy given by below equaion and is a good approximation of △y when dx =△x is relatively small and we denote it by dy ≈ △y

1. A point c in the domain of a function f at which either f ′(c) = 0 or f is not differentiable is called a critical point of f.

2. First Derivative Test Let f be a function defined on an open interval I. Let f be continuous at a critical point c in I. Then

1. If f ′(x) changes sign from positive to negative as x increases through c, i.e., if f ′(x) > 0 at every point sufficiently close to and to the left of c, and f ′(x) < 0 at every point sufficiently close to and to the right of c, then c is a point of local maxima.

2. If f ′(x) changes sign from negative to positive as x increases through c, i.e., if f ′(x) < 0 at every point sufficiently close to and to the left of c, and f ′(x) > 0 at every point sufficiently close to and to the right of c, then c is a point of local minima.

3. If f ′(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima. Infact, such a point is called point of inflexion.

3. Second Derivative Test Let f be a function defined on an interval I and c ∈ I. Let f be twice differentiable at c. Then

1. x = c is a point of local maxima if f ′(c) = 0 and f ″(c) < 0 The values f (c) is local maximum value of f .

2. x = c is a point of local minima if f ′(c) = 0 and f ″(c) > 0 In this case, f (c) is local minimum value of f .

3. 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 maxima, minima or a point of inflexion.

4. Working rule for finding absolute maxima and/or absolute minima

1. Step 1: Find all critical points of f in the interval, i.e., find points x where either f ′(x) = 0 or f is not differentiable.

2. Step 2:Take the end points of the interval

3. Step 3: At all these points (listed in Step 1 and 2), calculate the values of f .

4. Step 4:Identify the maximum and minimum values of f out of the values calculated in Step 3. This maximum value will be the absolute maximum value of f and the minimum value will be the absolute minimum value of f .

1. Integration by partial fractions

1. Integration by substitution

1.  tan x dx = log |sec x| + C

2.  cot x dx = log |sin x| + C

3.  sec x dx = log |sec x + tan x| + C

4.  cosec x dx = log |cosec x – cot x| + C