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.

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

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

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

$(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$ (wherever g(x) ≠ 0) is continuous

Every differentiable function is continuous, but the converse is not true.

Chain rule is rule to differentiate composites of functions. If f = v . u, t = u (x) and if both $\frac{dt}{dx}$ and $\frac{dv}{dt}$ exist then $\frac{df}{dx} = \frac{dv}{dt} . \frac{dt}{dx}$

Following are some of the standard derivatives (in appropriate domains):

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)

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.

Mean Value Theorem: If f : [a, b] → R is continuous on [a, b] and differentiable on (a, b). Then there exists some c in (a, b) such that $f'(c) = \frac{f(b) - f(a)}{b - a}$

If a quantity y varies with another quantity x, satisfying some rule y = f(x) then $\frac{dy}{dx}$ (or f(x)) represents the rate of change of y with respect to x and $\left [ \frac{dy}{dx} \right ]_{x = x_0}$ (or f '( $x_0$ ) ) represents the rate of change of y with respect to x at x = $x_0$

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

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)

The equation of the tangent at (x₀, y₀) to the curve y = f (x) is given by

If $\frac{dy}{dx}$ 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 = $x_0$

If tangent to a curve y = f (x) at x = $x_0$ is parallel to x-axis, then $\left [ \frac{dy}{dx} \right ]_{x = x_0}$ = 0

Equation of the normal to the curve y = f (x) at a point (x₀,y₀) is given by

If $\frac{dy}{dx}$ at the point (x₀,y₀) is zero, then equation of the normal is x = $x_0$

If $\frac{dy}{dx}$ at the point (x₀,y₀) does not exist , then then the normal is parallel to x-axis and its equation is y = $y_0$

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.

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.

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

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

Working rule for finding absolute maxima and/or absolute minima

Integration is the inverse process of differentiation. In the differential calculus, we are given a function and we have to find the derivative or differential of this function, but in the integral calculus, we are to find a function whose differential is given. Thus, integration is a process which is the inverse of differentiation

Let $\frac{d}{dx} F(x) = f(x)$ . Then we write $\int$ f(x) dx = F(x) + C. These integrals are called indefinite integrals or general integrals, C is called constant of integration. All these integrals differ by a constant.

From the geometric point of view, an indefinite integral is collection of family of curves, each of which is obtained by translating one of the curves parallel to itself upwards or downwards along the y-axis.

Some properties of indefinite integrals are as follows:

Some standard integrals

Integration by partial fractions

Integration by substitution

Integrals of some special functions

Integration by parts For given functions f ₁ and f ₂ , we have integral of the product of two functions = first function × integral of the second function – integral of {differential coefficient of the first function × integral of the second function}. Care must be taken in choosing the first function and the second function. Obviously, we must take that function as the second function whose integral is well known to us.

Some special types of integrals

Integrals of the types $\int \frac{px + q dx }{ax^2 + bx + c}$ or $\int \frac{px + q dx }{\sqrt{ax^2 + bx + c}}$ can be transformed into standard form by expressing $px + q = A\frac{d}{dx} (ax^2+bx+c)$ + B = A(2ax + b) + B , where A and B are determined by comparing coefficients on both sides.

We have defined $\int_{a}^{b} f(x) dx$ as the area of the region bounded by the curve y = f (x), a ≤ x ≤ b, the x-axis and the ordinates x = a and x = b. Let x be a given point in [a, b]. Then $\int_{x}^{a} f(x) dx$ represents the Area function A(x). This concept of area function leads to the Fundamental Theorems of Integral Calculus.

First fundamental theorem of integral calculus

Second fundamental theorem of integral calculus