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�MEAN VALUE THEOREM� AND � ROLLE’S THEOREM

GAURAV SINGH

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INSTITUTE OF MATHEMATICS AND APPLICATIONS

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Mean Value Theorem

x

y

y = f(x)

a

b

Example 2

x

y

y = f(x)

c

a

b

Example 1

There exists at least one point on the graph�at which tangent line is parallel to the secant line

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Mean Value Theorem

  • Slope of secant line is the slope of line through the points (a,f(a)) and (b,f(b)), so it is

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  • Slope of tangent line is f′(c)

x

y

y = f(x)

c

a

b

f(b) – f(a)� (b-a)

f′(c) =

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MVT: exact statement

  • Suppose f is continuous on [a,b] and differentiable on (a,b)
  • Then there exists at least one point c in (a,b) such that

x

y

y = f(x)

c

a

b

f(b) – f(a)� (b-a)

f′(c) =

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MVT: alternative formulations

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Interpretation of the MVT�using rate of change

  • Average rate of change��is equal to the instantaneous rate of change�f′(c) at some moment c
  • Example: suppose the cities A and B are connected by a straight road and the distance between them is 360 km. You departed from A at 1pm and arrived to B at 5:30pm. Then MVT implies that at some moment your velocity v(t) = s′(t) was:��(s(5.5) – s(1)) / (5.5 – 1) = 360 / 4.5 = 80 km / h

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Application of MVT

  • Estimation of functions
  • Connection between the sign of derivative and behavior of the function:

if f ′ > 0 function is increasing

if f ′ < 0 function is decreasing

  • Error bounds for Taylor polynomials

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Example

  • Suppose that f is differentiable for all x
  • If f (5) = 10 and f ′ (x) ≤ 3 for all x, how small can f(-1) be?

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Solution

  • MVT: f(b) – f(a) = f ′(c) (b – a) for some c in (a,b)
  • Applying MVT to the interval [ –1, 5], we get:
  • f(5) – f(–1) = f ′(c) (5 – (– 1)) = 6 f ′(c) ≤ 6∙3 = 18
  • Thus f(5) – f(-1) ≤ 18
  • Therefore f(-1) ≥ f(5) – 18 = 10 – 18 = – 8

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Rolle’s Theorem

x

y

y = f(x)

c

a

b

x

y

y = f(x)

c1

c2

a

b

f(a) = f(b)

c3

f ′ (c) = 0

Example 1

Example 2

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Rolles Theorem

  • Suppose f is a function such that
    • f is continuous on [a,b]
    • differentiable at least in (a,b)
  • If f(a) = f(b) then there exists at least one c in (a,b) such that f′(c) = 0

x

y

y = f(x)

c

a

b

f ′ (c) = 0

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Applications of Rolle’s Theorem

  • It is used in the prove of the�Mean Value Theorem
  • Together with the Intermediate Value Theorem, it helps to determine exact number of roots of an equation

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THANK YOU