intermediate value theorem

Examples of intermediate value theorem in the following topics:

• 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

• 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 $\mathbf{n}$.

• Indefinite Integrals and the Net Change Theorem

  • Therefore, all the antiderivatives of $x^2$ can be obtained by changing the value of $C$ in $F(x) = \left ( \frac{x^3}{3} \right ) + C$, where $C$ is an arbitrary constant known as the constant of integration.
  • Essentially, the graphs of antiderivatives of a given function are vertical translations of each other, with each graph's location depending upon the value of $C$.
  • Such a problem can be solved using the net change theorem, which states that the integral of a rate of change is the net change (displacement for position functions):
  • Basically, the theorem states that the integral of or $F'$ from $a$ to $b$ is the area between $a$ and $b$, or the difference in area from the position of $f(a)$ to the position of $f(b)$.
  • This can be applied to find values such as volume, concentration, density, population, cost, and velocity.

• Average Value of a Function

  • An average is a measure of the "middle" or "typical" value of a data set.
  • If $n$ numbers are given, each number denoted by $a_i$, where $i = 1, \cdots , n$, the arithmetic mean is the sum of all $a_i$ values divided by $n$:
  • The first mean value theorem for integration states that if $G : [a, b] \to R$ is a continuous function and $\varphi$ is an integrable function that does not change sign on the interval $(a, b)$, then there exists a number $x$ in $(a, b)$ such that:
  • The value $G(x)$ is the mean value of $G(t)$ on $[a, b] $ as we saw previously.
  • Evaluate the average value of a function over a closed interval using integration

• Linear Approximation

  • Linear approximation is achieved by using Taylor's theorem to approximate the value of a function at a point.
  • Given a twice continuously differentiable function $f$ of one real variable, Taylor's theorem states that:
  • where $R_2$ is the remainder term (the difference between the actual value of $f(x)$ and the approximation found by the addition of the first two terms).
  • For example, given a differentiable function with real values, one can approximate for close to by the following formula:

• Area and Distances

  • Integration is connected with differentiation through the fundamental theorem of calculus: if $f$ is a continuous real-valued function defined on a closed interval $[a,b]$, then, once an antiderivative F of f is known, the definite integral of $f$ over that interval is given by$\int_{a}^{b}f(x)dx = F(b) - F(a)$.
  • As it is, the true value of the integral must be somewhat less.
  • According to Pythagoras's theorem, $ds^2=dx^2+dy^2$ , from which we can determine:
  • the arc length integral for values of $t$ from $-1$ to $1$ is:
  • For a small piece of curve, $\Delta s$ can be approximated with the Pythagorean theorem.