7 Questions
What is the name of the method developed by Newton for finding successively better approximations to the roots of a function?
Newton's Method
What is the notation used by Newton for the derivative of a function with respect to time?
ẋ
What is the fundamental theorem developed by Newton that relates the derivative of a function to the area under its curve?
Fundamental Theorem of Calculus
What is the name of the formulas developed by Newton for expressing the elementary symmetric functions of the roots of a polynomial equation in terms of the coefficients of the equation?
Newton's Identities
What is the degree of the equations that can be solved using Newton's Identities?
Equations of degree ≤ 4
What is the name of the method developed by Newton as a precursor to integration?
Method of indivisibles
What is the notation used by Newton for time derivatives?
ḟ
Study Notes
Newton's Contributions to Mathematics
Calculus
- Developed the method of "fluxions" (now known as derivatives)
- Introduced the concept of the "method of infinite series" for calculating π
- Developed the fundamental theorem of calculus, which relates the derivative of a function to the area under its curve
Newton's Notation
- Used dot notation for time derivatives (e.g. ḟ for the derivative of f)
- Introduced the notation of ẋ for the derivative of x with respect to time
Newton's Method
- A recursive method for finding successively better approximations to the roots of a function
- Based on the iterative formula: xₙ₊₁ = xₙ - f(xₙ) / f'(xₙ)
Newton's Identities
- A set of formulas for expressing the elementary symmetric functions of the roots of a polynomial equation in terms of the coefficients of the equation
- Can be used to solve equations of degree ≤ 4
Other Contributions
- Developed the "method of indivisibles" (a precursor to integration)
- Made significant contributions to the study of algebra, including the development of the "Newton-Girard" formulas for the symmetric functions of the roots of an equation.
Explore the significant contributions of Sir Isaac Newton to the field of mathematics, including the development of calculus, notation, and methods for solving equations. Learn about his work on infinite series, derivatives, and more.
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