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Calculus
I. What is Calculus?
Calculus is the branch of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences.
a. Differential Calculus - concerned with the determination, properties, and application of derivatives and differentials.
b. Integral Calculus - concerned with the determination, properties, and application of integrals.
II. Brief History of Calculus
Calculus was created by Isaac Newton, a British scientist, as well as Gottfried Leibniz, a self-taught German mathematician, in the 17th century. It has been long disputed who should take credit for inventing calculus first, but both independently made discoveries that led to what we know now as calculus. Newton discovered the inverse relationship between the derivative (slope of a curve) and the integral (the area beneath it), which deemed him as the creator of calculus. Thereafter, calculus was actively used to solve the major scientific dilemmas of the time, such as:
a. calculating the slope of the tangent line to a curve at any point along its length
b. determining the velocity and acceleration of an object given a function describing its position, and designing such a position function given the object's velocity or acceleration
c. calculating arc lengths and the volume and surface area of solids
d. calculating the relative and absolute extrema of objects, especially projectiles
For Newton, the applications for calculus were geometrical and related to the physical world - such as describing the orbit of the planets around the sun. For Leibniz, calculus was more about analysis of change in graphs. Leibniz's work was just as important as Newton's, and many of his notations are used today, such as the notations for taking the derivative and the integral.
III. Applications of Calculus
With calculus, we have the ability to find the effects of changing conditions on a
I. What is Calculus?
Calculus is the branch of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences.
a. Differential Calculus - concerned with the determination, properties, and application of derivatives and differentials.
b. Integral Calculus - concerned with the determination, properties, and application of integrals.
II. Brief History of Calculus
Calculus was created by Isaac Newton, a British scientist, as well as Gottfried Leibniz, a self-taught German mathematician, in the 17th century. It has been long disputed who should take credit for inventing calculus first, but both independently made discoveries that led to what we know now as calculus. Newton discovered the inverse relationship between the derivative (slope of a curve) and the integral (the area beneath it), which deemed him as the creator of calculus. Thereafter, calculus was actively used to solve the major scientific dilemmas of the time, such as:
a. calculating the slope of the tangent line to a curve at any point along its length
b. determining the velocity and acceleration of an object given a function describing its position, and designing such a position function given the object's velocity or acceleration
c. calculating arc lengths and the volume and surface area of solids
d. calculating the relative and absolute extrema of objects, especially projectiles
For Newton, the applications for calculus were geometrical and related to the physical world - such as describing the orbit of the planets around the sun. For Leibniz, calculus was more about analysis of change in graphs. Leibniz's work was just as important as Newton's, and many of his notations are used today, such as the notations for taking the derivative and the integral.
III. Applications of Calculus
With calculus, we have the ability to find the effects of changing conditions on a