# Slope - Calculus

Calculus

The concept of a slope is central to differential calculus. For non-linear functions, the rate of change varies along the curve. The derivative of the function at a point is the slope of the line tangent to the curve at the point, and is thus equal to the rate of change of the function at that point.

If we let Δx and Δy be the distances (along the x and y axes, respectively) between two points on a curve, then the slope given by the above definition,

,

is the slope of a secant line to the curve. For a line, the secant between any two points is the line itself, but this is not the case for any other type of curve.

For example, the slope of the secant intersecting y = x2 at (0,0) and (3,9) is 3. (The slope of the tangent at x = 3⁄2 is also 3—a consequence of the mean value theorem.)

By moving the two points closer together so that Δy and Δx decrease, the secant line more closely approximates a tangent line to the curve, and as such the slope of the secant approaches that of the tangent. Using differential calculus, we can determine the limit, or the value that Δyx approaches as Δy and Δx get closer to zero; it follows that this limit is the exact slope of the tangent. If y is dependent on x, then it is sufficient to take the limit where only Δx approaches zero. Therefore, the slope of the tangent is the limit of Δyx as Δx approaches zero, or dy/dx. We call this limit the derivative.

### Other articles related to "calculus":

Lambda-mu Calculus
... logic and computer science, the lambda-mu calculus is an extension of the lambda calculus, and was introduced by M ... in computability theory and from the μ operator of modal μ-calculus) and the bracket operator ... One of the main goals of this extended calculus is to be able to describe expressions corresponding to theorems in classical logic ...
Calculus - Applications
... Calculus is used in every branch of the physical sciences, actuarial science, computer science, statistics, engineering, economics, business, medicine, demography ... Physics makes particular use of calculus all concepts in classical mechanics and electromagnetism are interrelated through calculus ... an object within a conservative field can be found by the use of calculus ...
Criticism Of Non-standard Analysis - Bishop's Criticism - Bishop's Review
... Bishop reviewed the book Elementary Calculus An Infinitesimal Approach by Keisler which presented elementary calculus using the methods of nonstandard analysis ... However, in applications of the calculus, it is helpful to imagine a line in physical space as a hyperreal line ... No invocation of Newton and Leibniz is going to justify developing calculus using axioms V* and VI*-on the grounds that the usual definition of a limit is too ...
List Of People Considered Father Or Mother Of A Scientific Field - Formal Sciences - Mathematics
... invention of the Cartesian Coordinate System Calculus Isaac Newton Gottfried Leibniz See Leibniz and Newton calculus controversy ... case when these are infinitely far away Tensor calculus Gregorio Ricci-Curbastro (founder) Book The Absolute Differential Calculus Trigonometry Aryabhata Hipparchus ... Vector algebra, vector calculus Willard Gibbs Oliver Heaviside (founders) For their development and use of vectors in algebra and calculus ...
Calculus The Musical
... "Calculus The Musical! is a comic “review” of the concepts and history of Calculus." The musical was originally created by Matheatre, composed of Marc Gutman and Sadie Bowman ... The musical used the story of the development of Calculus to tie together songs that Marc had created as mnemonics when he was a teacher ...

### Famous quotes containing the word calculus:

I try to make a rough music, a dance of the mind, a calculus of the emotions, a driving beat of praise out of the pain and mystery that surround me and become me. My poems are meant to make your mind get up and shout.
Judith Johnson Sherwin (b. 1936)