. . Applications Example . A truck travels on a toll road with a speed limit of km/hr.
The truck completes a km journey in hours. At the end of the toll road the trucker is issued with a speed violation notice. Justify this using the Mean Value Theorem. Let f t ( ) be the distance travelled by the trucker in ' ' t hours.
This is a continuous function in [ , ] and differentiable in ( , ) . Now, f ( ) and f ( ) . By an application of the Mean Value Theorem, there exists a time c such that, ′ > c ( ) . Therefore at some point of time, during the travel in hours the trucker must have travelled with a speed more than km/hr which justifies the issuance of a speed violation notice.
Example . Suppose f x ( ) is a differentiable function for all x with ′ and f ( ) . What is the maximum value of f ( ) ? By the mean value theorem we have, there exists ' ' c ∈ such that, = ′ c ( ) .
Hence, f ( ) ≤× = Therefore, the maximum value of f ( ) is . Example . Prove that | sin | | |, α β α β α β ∈ using mean value theorem. Let f x which is a differentiable function in any open interval.
Consider an interval [ , ] α β . Applying the mean value theorem there exists c ∈ ( , α β such that, β α β α = ′ cos( ) Therefore, sin α β α β = cos( ) c £ Hence, | sin | α β £ | | α β Remark If we take β = in the above problem, we get | sin | | | α α Applications of Differential Calculus Example . A thermometer was taken from a freezer and placed in a boiling water. It took seconds for the thermometer to raise from − ° C to ° C .
Show that the rate of change of temperature at some time t is ° C per second. Let f t ( ) be the temperature at time t . By the mean value theorem, we have ¢ f c ( ) = f b f a = −− = = ° C per second. Hence the instantaneous rate of change of temperature at some time t is ° C per second.