Calculus is a fundamental branch of mathematics that focuses on the study of change. It has numerous applications in science, engineering, economics, and many other fields. This tutorial will provide an introduction to the basics of calculus, including limits, derivatives, and integrals.
Basic Concepts
Limits
A limit is the value that a function approaches as the input approaches a certain value. In calculus, limits are used to define derivatives and integrals.
- Formal Definition: If ( f(x) ) is a function and ( c ) is a real number, then the limit of ( f(x) ) as ( x ) approaches ( c ) is ( L ) if for every positive number ( \epsilon ), there exists a positive number ( \delta ) such that if ( 0 < |x - c| < \delta ), then ( |f(x) - L| < \epsilon ).
Derivatives
A derivative is a measure of the rate at which one quantity changes with respect to another. In calculus, derivatives are used to determine the slope of a curve at a given point.
- Power Rule: The derivative of ( x^n ) with respect to ( x ) is ( nx^{n-1} ).
Integrals
An integral is a function that represents the area under the curve of another function. In calculus, integrals are used to find the area, volume, and other quantities.
- Fundamental Theorem of Calculus: If ( f(x) ) is continuous on the interval ( [a, b] ), then the definite integral of ( f(x) ) from ( a ) to ( b ) is given by ( F(b) - F(a) ), where ( F(x) ) is any antiderivative of ( f(x) ).
Practice Resources
For further reading and practice, check out our Calculus Practice Problems.
[center]
[center]