endpoint

Examples of endpoint in the following topics:

• Interval Notation

  • Interval notation uses parentheses and brackets to describe sets of real numbers and their endpoints.
  • The two numbers are called the endpoints of the interval.
  • An open interval does not include its endpoints and is indicated with parentheses.
  • An interval is said to be bounded if both of its endpoints are real numbers.
  • Conversely, if neither endpoint is a real number, the interval is said to be unbounded.

• Improper Integrals

  • An Improper integral is the limit of a definite integral as an endpoint of the integral interval approaches either a real number or $\infty$ or $-\infty$.
  • An improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or $\infty$ or $-\infty$ or, in some cases, as both endpoints approach limits.
  • in which one takes a limit at one endpoint or the other (or sometimes both).
  • It is often necessary to use improper integrals in order to compute a value for integrals which may not exist in the conventional sense (as a Riemann integral, for instance) because of a singularity in the function, or an infinite endpoint of the domain of integration.

• Redox Titrations

  • It is therefore possible to see when the titration has reached its endpoint, because the solution will remain slightly purple from the unreacted KMnO4.
  • A student conducts the redox titration and reaches the endpoint after adding 25 mL of the titrant.
  • In this case, starch is used as an indicator; a blue starch-iodine complex is formed in the presence of excess iodine, signaling the endpoint.
  • Note how the endpoint is reached when the solution remains just slightly purple.

• 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.
  • Now suppose the domain $U$ of $\varphi$ contains the differentiable curve $\gamma$ with endpoints $p$ and $q$ (oriented in the direction from $p$ to $q$).

• 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 $c$.

• Weak Acid-Strong Base Titrations

  • The endpoint and the equivalence point are not exactly the same: the equivalence point is determined by the stoichiometry of the reaction, while the endpoint is just the color change from the indicator.

• Radiation Reaction

  • We can drop the term from the endpoints if for example the acceleration vanishes at $t=t_1$ and $t=t_2$ or if the acceleration and velocity of the particle are the same at $t=t_1$ and $t=t_2$.We can identify,

• The Distance Formula and Midpoints of Segments

  • In geometry, the midpoint is the middle point of a line segment, or the middle point of two points on a line, and thus is equidistant from both endpoints.
  • The equation for a midpoint of a line segment with endpoints $(x_{1},y_{1})$and $(x_{2},y_{2})$

• Line Integrals

  • where $r: [a, b] \to C$ is an arbitrary bijective parametrization of the curve $C$ such that $r(a)$ and $r(b)$ give the endpoints of $C$ and $a$.
  • where $\cdot$ is the dot product and $r: [a, b] \to C$ is a bijective parametrization of the curve $C$ such that $r(a)$ and $r(b)$ give the endpoints of $C$.

• Estimating a Population Variance