intermediate value theorem

Examples of intermediate value theorem in the following topics:

• The Intermediate Value Theorem

  • For each value between the bounds of a continuous function, there is at least one point where the function maps to that value.
  • In plotting a continuous and smooth function between two points, all points on the function between the extremes are described and predicted by the Intermediate Value Theorem.
  • It meets the requirements of the Intermediate Value Theorem.
  • In what situation would it not meet the requirements for the theorem?
  • Explain what the intermediate value theorem means for the graphs of polynomials

• Intermediate Value Theorem

  • For a real-valued continuous function $f$ on the interval $[a,b]$ and a number $u$ between $f(a)$ and $f(b)$, there is a $c \in [a,b]$ such that $f(c)=u$.
  • The intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value.
  • The theorem depends on (and is actually equivalent to) the completeness of the real numbers.
  • The intermediate value theorem can be used to show that a polynomial has a solution.
  • Use the intermediate value theorem to determine whether a point exists on a continuous function

• Maximum and Minimum Values

  • The maximum and minimum of a function, known collectively as extrema, are the largest and smallest values that the function takes at a point either within a given neighborhood (local or relative extremum) or on the function domain in its entirety (global or absolute extremum).
  • For example, if a bounded differentiable function $f$ defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem).

• The Fundamental Theorem of Calculus

  • The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function to the concept of the integral.
  • The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function to the concept of the integral.
  • There are two parts to the theorem.
  • Let $f$ be a continuous real-valued function defined on a closed interval $[a,b]$.
  • Let $f$ and $F$ be real-valued functions defined on a closed interval $[a,b]$ such that the derivative of $F$ is $f$.

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

  • The MVT states that for a function continuous on an interval, the mean value of the function on the interval is a value of the function.
  • 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 .
  • Therefore, the Mean Value Theorem tells us that at some point during the journey, the car must have been traveling at exactly 100 miles per hour; that is, it was traveling at its average speed.
  • The mean value theorem follows from the more specific statement of Rolle's theorem, and can be used to prove the more general statement of Taylor's theorem (with Lagrange form of the remainder term).
  • Use the Mean Value Theorem and Rolle's Theorem to reach conclusions about points on continuous and differentiable functions

• Right Triangles and the Pythagorean Theorem

  • A right angle has a value of 90 degrees ($90^\circ$).
  • The Pythagorean Theorem, also known as Pythagoras' Theorem, is a fundamental relation in Euclidean geometry.
  • Although it is often said that the knowledge of the theorem predates him,[2] the theorem is named after the ancient Greek mathematician Pythagoras (c. 570 – c. 495 BC).
  • The Pythagorean Theorem can be used to find the value of a missing side length in a right triangle.
  • Use the Pythagorean Theorem to find the length of a side of a right triangle

• Sampling Distributions and the Central Limit Theorem

  • The central limit theorem has a number of variants.
  • $n$ is the number of values that are averaged together not the number of times the experiment is done.
  • Consider a sequence of independent and identically distributed random variables drawn from distributions of expected values given by $\mu$ and finite variances given by $\sigma^2$.
  • By the law of large numbers, the sample averages converge in probability and almost surely to the expected value $\mu$ as $n \rightarrow \infty$.
  • This figure demonstrates the central limit theorem.

• 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.
  • In physics this theorem is one of the ways of defining a "conservative force."
  • The gradient theorem also has an interesting converse: any conservative vector field can be expressed as the gradient of a scalar field.

• Parametric Surfaces and Surface Integrals

  • Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form.
  • Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values).
  • We will study surface integral of vector fields and related theorems in the following atoms.
  • An illustration of the Kelvin–Stokes theorem, with surface $\Sigma$, its boundary $\partial$, and the "normal" vector $\mathbf{n}$.

• Leverage Models

  • The basic theorem states that in the absence of taxes, bankruptcy costs, agency costs, and asymmetric information - and in an efficient market - the value of a firm is unaffected by how that firm is financed.
  • The Modigliani–Miller theorem is also often called the Capital Structure Irrelevance Principle.
  • In the above equation, VL is the value of a levered firm, VU is the value of an unlevered firm, TC is the corporate tax rate, and D is the value of the company's debt.
  • Further, value may be added by utilizing leverage.
  • The value of a levered firm equals the value of an unlevered firm plus the tax rate times the value of debt.