Calculus is the mathematical study of continuous change. Originally called infinitesimal calculus, it has two major branches, differential calculus and integral calculus.
Differential Calculus
The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent to a curve. The derivative of a function f(x) with respect to x is
The above expression is also called as fundamental theorem of differentiation.
A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant. The del symbol is used to denote the partial derivative. The chain rule is used to find the derivative of the multivariable function.
Differential Calculus
The above expression is also called as fundamental theorem of differentiation.
A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant. The del symbol is used to denote the partial derivative. The chain rule is used to find the derivative of the multivariable function.
Integral Calculus
The concept of the integral was first started to calculate the area under a curve. The area under a curve can be thought of as a sum of rectangles of infinitesimal width. The integral of a function f(x) with respect to a variable x on an interval [a, b] is written as
Differentiating on both sides
Integral from x to x+Δx is just the area under f in that interval. For a very small value of Δx, we can assume f(x) remains constant. Thus this area is equal to f(x)Δx.
Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, calculus has widespread roots in every branch of physics and engineering.
If limits are specified, the integral is called a definite integral. When the limits are omitted, the integral is called an indefinite integral. There is no fundamental theorem of integration unlike differentiation.
Now let's look at the fundamental theorem of calculus which relates these two branches of calculus i.e. differential calculus and integral calculus. It states that the integral of f over an interval is equal to the antiderivative of f (a function whose derivative would be f). This result was kind of surprising and nonintuitive that somehow the problem of finding areas under a curve could be solved with differentials.
Now let's look at the fundamental theorem of calculus which relates these two branches of calculus i.e. differential calculus and integral calculus. It states that the integral of f over an interval is equal to the antiderivative of f (a function whose derivative would be f). This result was kind of surprising and nonintuitive that somehow the problem of finding areas under a curve could be solved with differentials.
Proof : For a given function f, define the function F(x) as
Differentiating on both sides
Integral from x to x+Δx is just the area under f in that interval. For a very small value of Δx, we can assume f(x) remains constant. Thus this area is equal to f(x)Δx.
Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, calculus has widespread roots in every branch of physics and engineering.