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.

  • Circles as Conic Sections

    • You already know the formula for a line: y=mx+b.
    • Diameter: the longest chord, a line segment whose endpoints lie on the circle and which passes through the center; or the length of such a segment, which is the largest distance between any two points on the circle.
    • Radius: a line segment joining the center of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.
    • Tangent: a straight line that touches the circle at a single point.
    • Secant: an extended chord, a straight line cutting the circle at two points.

  • 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.

  • Secant and the Trigonometric Cofunctions

    • The secant function is the reciprocal of the cosine function, and is abbreviated as $\sec$.
    • It is easy to calculate secant with values in the unit circle.
    • Therefore, the secant function for that angle is
    • As with secant, cosecant can be calculated with values in the unit circle.

  • Trigonometric Symmetry Identities

    • We will now consider each of the trigonometric functions and their cofunctions (secant, cosecant, and cotangent), and observe symmetry in their graphs.
    • The cosine and secant functions are symmetric about the y-axis.
    • Notice that only two of the trigonometric identities are even functions: cosine and secant.
    • Cosine and secant are even functions, with symmetry around the $y$-axis.

  • 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

  • Non-relativistic Shocks

    • so the Mach numbers on each side of the shock are given by the ratio of the slope of the secant to the slope of the tangent.
    • Because all of the adiabats are concave up in the $p-V-$plane, the slope of the secant must be larger than that of the tangent at $(p_1,V_1)$, so the flow enters the shock supersonically.
    • Conversely at $(p_2,V_2)$the slope of the secant must be small than that of the tangent, so the flow exits the shock subsonically.

  • Qualities of Line

    • Quality of line refers to the character that is embedded in the way a line presents itself.
    • Hard-edged, jagged lines present a staccato visual movement, while sinuous, flowing lines create a more comfortable feeling.
    • Horizontal, diagonal, and vertical lines describe a line's orientation.
    • Contour lines define the outer edges of an object.
    • Hatch lines are defined as parallel lines which are repeated short intervals generally in one direction.