Mean Value Theorem - Formal Statement

Formal Statement

Topics in calculus
  • Fundamental theorem
  • Limits of functions
  • Continuity
  • Mean value theorem
  • Rolle's theorem
Differential calculus
  • Derivative
  • Second derivative
  • Third derivative
  • Change of variables
  • Implicit differentiation
  • Taylor's theorem
  • Related rates
  • Rules and identities
    Power rule
    Product rule
    Quotient rule
    Sum rule
    Chain rule
Integral calculus
  • Lists of integrals
  • Improper integral
  • Multiple integral
  • Integration by
    parts
    disks
    cylindrical shells
    substitution
    trigonometric substitution
    partial fractions
    changing order
Vector calculus
  • Gradient
  • Divergence
  • Curl
  • Laplacian
  • Gradient theorem
  • Green's theorem
  • Stokes' theorem
  • Divergence theorem
Multivariable calculus
  • Matrix calculus
  • Partial derivative
  • Multiple integral
  • Line integral
  • Surface integral
  • Volume integral
  • Jacobian

Let f : → R be a continuous function on the closed interval, and differentiable on the open interval (a, b), where a < b. Then there exists some c in (a, b) such that

The mean value theorem is a generalization of Rolle's theorem, which assumes f(a) = f(b), so that the right-hand side above is zero.

The mean value theorem is still valid in a slightly more general setting. One only needs to assume that f : → R is continuous on, and that for every x in (a, b) the limit

exists as a finite number or equals +∞ or −∞. If finite, that limit equals f′(x). An example where this version of the theorem applies is given by the real-valued cube root function mapping x to x1/3, whose derivative tends to infinity at the origin.

Note that the theorem, as stated, is false if a differentiable function is complex-valued instead of real-valued. For example, define f(x) = eix for all real x. Then

f(2π) − f(0) = 0 = 0(2π − 0)

while |f′(x)| = 1.

Read more about this topic:  Mean Value Theorem

Famous quotes containing the words formal and/or statement:

    “This is no argument against teaching manners to the young. On the contrary, it is a fine old tradition that ought to be resurrected from its current mothballs and put to work...In fact, children are much more comfortable when they know the guide rules for handling the social amenities. It’s no more fun for a child to be introduced to a strange adult and have no idea what to say or do than it is for a grownup to go to a formal dinner and have no idea what fork to use.”
    —Leontine Young (20th century)

    “Eloquence must be grounded on the plainest narrative. Afterwards, it may warm itself until it exhales symbols of every kind and color, speaks only through the most poetic forms; but first and last, it must still be at bottom a biblical statement of fact.”
    —Ralph Waldo Emerson (1803–1882)