In calculus, a curve is a mathematical object similar to a line, but it doesn’t have to be straight. A curve describes the path of a continuously moving point. As a practical example, an ellipse is a curve describing the motion of planets around a star.
Calculus originates from the study of curves and their properties; without curves, there would be no calculus.
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Calculus Curves: Classes
Many curves studied in calculus belong to certain classes of curves, sharing certain features and properties. In a way, you can think of these curves as the “famous” math curves; they have been studied extensively in ancient or modern times.
Algebraic curves are any curves that can be defined by a polynomial equation [1]. Quartic curves are given by a fourth degree polynomial (e.g. f(x) = x4).
Plane curves lies in a two-dimensional plane. In other words, the points of the curve are all on the same plane; you could draw these curves on a sheet of paper, unlike a 3D shape like a cube (you could draw this on a sheet of paper, but to be accurate you would really need to create a 3D model). In comparison, a space curve’s points do not necessarily all lie on a single plane [2].
Special curves have well-known features. They include circles, ellipses, and hyperbolas. However, there are many more obscure curves that are special. These include the Peano Curve and Quadratix of Hippias.
The quadratix of Hippias is a special curve created by a uniform motion.
How To Articles
- Arc Length Formula
- Area Between Curve and Y-Axis
- Concave Up (Convex), Down
- Curve Sketching
- Find the Area Under the Curve (Excel, Definite Integrals
- Intersections
- Linearization and Linear Approximation
- Normal Line
- Parameterize a Curve
- Quadrifolium
- Riemann Sums
- Secant Line
- Slope of a Curve
- Tangent Lines
- Tangent Vector (Velocity Vector)
Calculus Curves: A to Z
- Algebraic Curves
- Alysoid
- Ampersand Curve
- Astroid Curve
- Bean Curve
- Gaussian / Bell Curve
- Bow Curve
- Bullet Nose Curve
- Butterfly curve
- Cassini Oval
- Cissoid of Diocles
- Conchoid of Nicomedes
- Cruciform Curve
- Damped Sine Wave
- Deltoid Curve
- Devil’s Curve
- Eight Curve (Leminiscate of Gerono)
- Equiangular Spiral, Spira Mirabilis (Logarithmic Spiral)
- Fish Curve
- Folium of Descartes (Noeud de Ruban)
- Hippopede
- Hypocycloid Curve
- Kampyle of Eudoxus
- Klein Quartic
- Lamé curve
- Level Curve
- Lissajous Curves
- Lituus Curve
- Natural Equation of a Curve
- Nodal Cubic
- Pedal Curve
- Piriform Curve
- Plateau Curve
- Pursuit Curve
- Rectifiable Curve
- Rhodonea Curve (Rose Curve)
- Sawtooth Function (Wave)
- Serpentine Curve
- Simple Closed Curve
- Smooth Curve
- Spiric Section
- Tagaki Curve
- Tractrix Curve
- Trident of Newton
- Trifolium Curve
- Witch of Agnesi
Calculus Curves: References
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[1] Degree of a Curve. Retrieved February 27, 2022 from: http://assets.press.princeton.edu/chapters/s9665.pdf
[2] Montoya, D. & Naves, D. On Plane and Space Curves. Retrieved January 15, 2022 from: https://web.ma.utexas.edu/users/drp/files/Spring2020Projects/DRP_Final_Project%20-%20Daniel%20Naves.pdf
Stephanie Glen. "Calculus Curves" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/calculus-curves/
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