Rolle's theorem

Examples of Rolle's theorem in the following topics:

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

  • The theorem is used to prove global statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.
  • This theorem can be understood intuitively by applying it to motion: If a car travels one hundred miles in one hour, then its average speed during that time was 100 miles per hour.
  • 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).
  • Rolle's Theorem states that if a real-valued function $f$ is continuous on a closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and f(a) = f(b), then there exists a c in the open interval $(a, b)$ such that $f'(c)=0$.
  • Use the Mean Value Theorem and Rolle's Theorem to reach conclusions about points on continuous and differentiable functions

• Maximum and Minimum Values

  • 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).

• Trigonometry and Complex Numbers: De Moivre's Theorem

• Stokes' Theorem

  • Stokes' theorem relates the integral of the curl of a vector field over a surface to the line integral of the field around the boundary.
  • The generalized Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.
  • The Kelvin–Stokes theorem, also known as the curl theorem, is a theorem in vector calculus on $R^3$.
  • The Kelvin–Stokes theorem is a special case of the "generalized Stokes' theorem."
  • As we have seen in our previous atom on gradient theorem, this simply means:

• Student Learning Outcomes

• Green's Theorem

  • Green's theorem gives relationship between a line integral around closed curve $C$ and a double integral over plane region $D$ bounded by $C$.
  • Green's theorem is a special case of the Kelvin–Stokes theorem, when applied to a region in the $xy$-plane.
  • Considering only two-dimensional vector fields, Green's theorem is equivalent to the two-dimensional version of the divergence theorem.
  • Green's theorem can be used to compute area by line integral.
  • Explain the relationship between the Green's theorem, the Kelvin–Stokes theorem, and the divergence 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.
  • The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by differentiation.
  • We can see from this picture that the Fundamental Theorem of Calculus works.

• The Divergence Theorem

  • The divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.
  • In physics and engineering, the divergence theorem is usually applied in three dimensions.
  • In one dimension, it is equivalent to the fundamental theorem of calculus.
  • The theorem is a special case of the generalized Stokes' theorem.
  • Apply the divergence theorem to evaluate the outward flux of a vector field through a closed surface

• Right Triangles and the Pythagorean Theorem

  • The Pythagorean Theorem, ${\displaystyle a^{2}+b^{2}=c^{2},}$ can be used to find the length of any side of a right triangle.
  • 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

• The Coase Theorem

  • The Coase theorem states that private parties can find efficient solutions to externalities without government intervention.
  • The Coase Theorem, named after Nobel laureate Ronald Coase, states that in the presence of an externality, private parties will arrive at an efficient outcome without government intervention.
  • In practice, transaction costs are rarely low enough to allow for efficient bargaining and hence the theorem is almost always inapplicable to economic reality.
  • This graph exemplifies how Coase's Theorem functions in a practical manner, underlining the effects of an externality in an economic model.
  • According to the Coase theorem, two private parties will be able to bargain with each other and find an efficient solution to an externality problem.