secant line

(noun)

a line that (locally) intersects two points on the curve

Related Terms

Examples of secant line in the following topics:

  • 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.
    • The slope of the secant line passing through $p$ and $q$ is equal to the difference quotient
    • The line shows the tangent to the curve at the point represented by the dot.
    • Define a derivative as the slope of the tangent line to a point on a curve

  • Limit of a Function

    • It also appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.

  • Trigonometric Functions

    • Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle.
    • The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side: so called because it can be represented as a line segment tangent to the circle, that is the line that touches the circle, from Latin linea tangens or touching line (cf. tangere, to touch).
    • The sine, tangent, and secant functions of an angle constructed geometrically in terms of a unit circle.

  • The Mean Value Theorem, Rolle's Theorem, and Monotonicity

    • In calculus, the mean value theorem states, roughly: given a planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints .
    • For any function that is continuous on $[a, b]$ and differentiable on $(a, b)$ there exists some $c$ in the interval $(a, b)$ such that the secant joining the endpoints of the interval $[a, b]$ is parallel to the tangent at $c$.

  • Hyperbolic Functions

  • Trigonometric Integrals

  • Equations of Lines and Planes

    • A line is a vector which connects two points on a plane and the direction and magnitude of a line determine the plane on which it lies.
    • A line is described by a point on the line and its angle of inclination, or slope.
    • Every line lies in a plane which is determined by both the direction and slope of the line.
    • The components of equations of lines and planes are as follows:
    • Now, we can use all this information to form the equation of a line on plane $M$.

  • Line Integrals

    • A line integral is an integral where the function to be integrated is evaluated along a curve.
    • This weighting distinguishes the line integral from simpler integrals defined on intervals.
    • Many simple formulae in physics (for example, $W=F·s$) have natural continuous analogs in terms of line integrals ($W= \int_C F\cdot ds$).
    • The line integral finds the work done on an object moving through an electric or gravitational field, for example.
    • 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:

  • Linear and Quadratic Functions

    • Linear and quadratic functions make lines and a parabola, respectively, when graphed and are some of the simplest functional forms.
    • Linear and quadratic functions make lines and parabola, respectively, when graphed.
    • Although affine functions make lines when graphed, they do not satisfy the properties of linearity.
    • An affine transformation (from the Latin, affinis, "connected with") is a transformation which preserves straight lines (i.e., all points lying on a line initially still lie on a line after transformation) and ratios of distances between points lying on a straight line (e.g., the midpoint of a line segment remains the midpoint after transformation).
    • It does not necessarily preserve angles or lengths, but does have the property that sets of parallel lines will remain parallel to each other after an affine transformation.

  • Fundamental Theorem for Line Integrals

    • Gradient theorem says that a line integral through a gradient field can be evaluated from the field values at the endpoints of the curve.
    • The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
    • The gradient theorem implies that line integrals through irrotational vector fields are path-independent.
    • where the definition of the line integral is used in the first equality and the fundamental theorem of calculus is used in the third equality.
    • Electric field lines emanating from a point where positive electric charge is suspended over a negatively charged infinite sheet.