Limits: Definition, Properties, Evaluating, Examples

What is a Limit? Limit means value of a function that approaches another value. Limit is symbolized as It is read as “limit of function of x, as x approaches to a equal to L”. Look at the graph, Properties of Limits If there are two limits, Then Limits to Infinity limit to infinity is a


Calculus is the mathematical study of continuous change. The 2 branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite series & infinite sequences to a well-defined limit. Derivatives (Differential) Differentiable Common Derivatives Implicit Differentiation Homogeneous Differential Equations Integration (Integral) Integration


Definition of Differentiable Differentiable means:  the function has derivative the derivative f’(x) must exist for every value in the function’s domain Examples The use of differentiable function When a function is differentiable, we can use all the power of calculus when working with it. When a function is differentiable, it is continuous. Differentiable ⇒ Continuous

Integration by Substitution (Reverse Chain Rule)

Definition Integration by Substitution (also called “The Reverse Chain Rule” or “u-Substitution” ) is a method to find an integral, but only when it can be set up in a special way. It is an important method in mathematics. Integration by substitution is the counterpart to the chain rule for differentiation. When it is possible to perform


Definition Integration is used to find areas, volumes, central points and often is used to find the area underneath the graph of a function. Common Functions of Integration   Function Integral Constant ∫ a dx ax + c Variable ∫ x dx Square ∫ x2 dx   Reciprocal ∫ (1/x) dx ln|x| + c Exponential ∫

Common Derivatives

Derivative of a function of a real variable measures the sensitivity to change of the function value (output) with respect to a change in its argument (input). Derivatives are a fundamental tool of calculus. The derivative of a function of a single variable at a chosen input value (when it exists) is the slope of the tangent

L’hopital’s rule: Definition, Proof, Examples

What is l’hopital’s rule L’hospital rule is one of ways to make limit easier to solve. It was introduced by French mathematicians, Guillaume de l’Hôpital. It was 1969 in his book “Analyse des Infiniment Petits pour l’Intelligence des Lignes Courbes.” L’hospital rule use derivative concept. In calculus, L’hospital use derivative to determine limit value in

Homogeneous Differential Equations

A Differential Equation is an equation with a function and one or more of its derivatives: Example: an equation with the function y and its derivative Homogeneous Differential Equations A first order Differential Equation is Homogeneous when it can be in this form: We can solve it using Separation of Variables but first we create

Implicit Differentiation

Some equations in x and y in mathematics do not explicitly define y as a function x and cannot be easily manipulated to solve for y in terms of x, even though such a function may exist. If it happen, it is implied that there exists a function y = f( x) such that the given

Integration by Parts: Formula & Examples

Integration by Parts Formula Integration is one of calculus part. Integration is not only consisting of general formula, but also integration by substitution and integration by parts. Different problem needs different way to solve it. Integration by parts was discovered by mathematicians, Brook Taylor, in 1715. Integration by parts is used to solve integration problems