curve

Examples of curve in the following topics:

• Curve Sketching

  • Curve sketching is used to produce a rough idea of overall shape of a curve given its equation without computing a detailed plot.
  • Determine the symmetry of the curve.
  • If the exponent of $x$ is always even in the equation of the curve, then the $y$-axis is an axis of symmetry for the curve.
  • Determine the asymptotes of the curve.
  • Also determine from which side the curve approaches the asymptotes and where the asymptotes intersect the curve.

• Line Integrals

  • A line integral is an integral where the function to be integrated is evaluated along a curve.
  • A line integral (sometimes called a path integral, contour integral, or curve integral) is an integral where the function to be integrated is evaluated along a curve.
  • The value of the line integral is the sum of the values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve).
  • This can be visualized as the surface created by $z = f(x,y)$ and a curve $C$ in the $xy$-plane.
  • For some scalar field $f:U \subseteq R^n \to R$, the line integral along a piecewise smooth curve $C \subset U$ is defined as:

• The Derivative and Tangent Line Problem

  • The tangent line $t$ (or simply the tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.
  • Informally, it is a line through a pair of infinitely close points on the curve.
  • More precisely, a straight line is said to be a tangent of a curve $y = f(x)$ at a point $x = c$ on the curve if the line passes through the point $(c, f(c))$ on the curve and has slope $f'(c)$ where f' is the derivative of $f$.
  • Suppose that a curve is given as the graph of a function, $y = f(x)$.
  • Define a derivative as the slope of the tangent line to a point on a curve

• Direction Fields and Euler's Method

  • Here, a differential equation can be thought of as a formula by which the slope of the tangent line to the curve can be computed at any point on the curve, once the position of that point has been calculated.The idea is that while the curve is initially unknown, its starting point, which we denote by $A_0$, is known (see ).
  • Along this small step, the slope does not change too much $A_1$ will be close to the curve.
  • After several steps, a polygonal curve is computed.
  • In general, this curve does not diverge too far from the original unknown curve, and the error between the two curves can be made small if the step size is small enough and the interval of computation is finite.
  • The unknown curve is in blue and its polygonal approximation is in red.

• Area Between Curves

  • It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).
  • The area between a positive-valued curve and the horizontal axis, measured between two values $a$ and $b$ ($b$ is defined as the larger of the two values) on the horizontal axis, is given by the integral from $a$ to $b$ of the function that represents the curve.
  • Find the area between the two curves $f(x)=x$ and $f(x)= 0.5 \cdot x^2$ over the interval from $x=0$ to $x=2$.
  • Two curves, $y=x$ and $y = 0.5 \cdot x^2$, meet at the points $(x_0,y_0)=(0,0)$ and $(x_1,y_1)=(2,2)$.
  • Integration can be thought of as measuring the area under a curve, defined by $f(x)$, between two points (here, $a$ and $b$).

• Parametric Equations

  • In mathematics, a parametric equation of a curve is a representation of the curve through equations expressing the coordinates of the points of the curve as functions of a variable called a parameter.
  • The notion of parametric equation has been generalized to surfaces of higher dimension with a number of parameters equal to the dimension of the manifold (dimension one and one parameter for curves, dimension two and two parameters for surfaces, etc.)
  • This way of expressing curves is practical as well as efficient; for example, one can integrate and differentiate such curves term-wise.

• Area and Arc Length in Polar Coordinates

  • If you were to straighten a curved line out, the measured length would be the arc length.
  • The arc length of the curve defined by a polar function is found by the integration over the curve $r(\theta)$.
  • Let $L$ denote this length along the curve starting from points $A$ through to point $B$, where these points correspond to $\theta = a$ and $\theta = b$ such that $0 < b-a < 2 \pi$.
  • Let $R$ denote the region enclosed by a curve $r(\theta)$ and the rays $\theta = a$ and $\theta = b$, where $0 < b-a < 2 \pi$.
  • The curved lines bounding the region $R$ are arcs.

• Tangent and Velocity Problems

  • Iinstantaneous velocity can be obtained from a position-time curve of a moving object.
  • In this atom, we will learn that instantaneous velocity can be obtained from a position-time curve of a moving object by calculating derivatives of the curve.
  • The green line shows the tangential line of the position-time curve at a particular time.
  • Recognize that the slope of a tangent line to a curve gives the instantaneous velocity at that point in time

• Area of a Surface of Revolution

  • If the curve is described by the function $y = f(x) (a≤x≤b)$, the area $A_y$ is given by the integral $A_x = 2\pi\int_a^bf(x)\sqrt{1+\left(f'(x)\right)^2} \, dx$ for revolution around the $x$-axis.
  • A surface of revolution is a surface in Euclidean space created by rotating a curve around a straight line in its plane, known as the axis .
  • If the curve is described by the function $y = f(x)$, $a \leq x \leq b$, then the integral becomes:
  • The spherical surface with a radius $r$ is generated by the curve $x(t) =r \sin(t)$, $y(t) = r \cos(t)$, when $t$ ranges over $[0,\pi]$.
  • A portion of the curve $x=2+\cos z$ rotated around the $z$-axis (vertical in the figure).

• Arc Length and Speed

  • The length of a curve can be difficult to measure.
  • A curve may be thought of as an infinite number of infinitesimal straight line segments, each pointing in a slightly different direction to make up the curve.
  • The length of the curve is called the arc length.
  • The arc length is calculated by laying out an infinite number of infinitesimal right triangles along the curve.
  • The length of a curve can be approximated by using a series of right triangles with the hypotenuses lying along the curve.