Calculus is a branch of mathematics that deals with the study of rates of change and accumulation. It is divided into two main branches: differential calculus and integral calculus.
Differential calculus deals with the study of the rate of change of a function, also known as its derivative. The derivative of a function describes how the function is changing at a particular point. For example, the derivative of the position of a moving object with respect to time is its velocity.
Integral calculus deals with the study of the accumulation of a function, also known as its integral. The integral of a function describes the total change of the function over a certain interval. For example, the integral of the velocity of a moving object with respect to time is its displacement.
Calculus is used in many fields, including physics, engineering, economics, and medicine. It is also used in computer graphics and animation, as well as many other areas of mathematics.
Definitions: -Derivative: The rate of change of a function at a particular point. -Integral: The accumulation of a function over a certain interval. -Limit: A value that a function approaches as the input gets closer to a certain value. -Continuity: A function is continuous if its graph has no breaks or gaps. -Differentiation: The process of finding the derivative of a function. -Integration: The process of finding the integral of a function.
Examples:
- Find the derivative of the function y = x^2. Solution: The derivative of y = x^2 is y’ = 2x.
- Find the integral of the function y = x^3. Solution: The integral of y = x^3 is ? y dx = ? x^3 dx = (x^4)/4 + C, where C is the constant of integration.
- Find the limit of the function y = (x^2 – 4x + 4)/(x – 2) as x approaches 2. Solution: The limit of y as x approaches 2 is 0, because as x gets closer and closer to 2, the value of the function approaches 0.
- Determine if the function y = x^2 is continuous at x = 2. Solution: The function y = x^2 is continuous at x = 2, because there are no breaks or gaps in its graph at x = 2.
- Find the area between the function y = x^2 and the x-axis between x = 0 and x = 1. Solution: The area between the function y = x^2 and the x-axis between x = 0 and x = 1 is (1/3), because it is the area of the region between the graph of y = x^2 and the x-axis between x = 0 and x = 1.
Quiz:
- What is the derivative of the function y = x^2?
- What is the integral of the function y = x^3?
- What is the limit of the function y = (x^2 – 4x + 4)/(x – 2) as x approaches 2?
- Is the function y = x^2 continuous at x = 2?
- What is the area between the function y = x^2 and the x-axis between x = 0 and x = 1?
- What is the opposite of the derivative?
- What is the opposite of the integral?
- Explain what does it mean for a function to be continuous.
- What is the derivative of the function y = sin(x
