Mean Value Theorem: Applications - Applications of Differential Calculus

**Applications**

** **

A truck
travels on a toll road with a speed limit of 80 km/hr. The truck completes a
164 km journey in 2 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 [0, 2] and
differentiable in (0, 2) . Now, *f* (0)
=
0 and *f* (2) =
164 . By an application of the Mean Value Theorem, there exists a time *c* such that,

*f*â€²(c) = 164 âˆ’ 0 / 2-0 = 82 > 80 .

Therefore
at some point of time, during the travel in 2 hours the trucker must have
travelled with a speed more than 80 km/hr which justifies the issuance of a
speed violation notice.

** **

**Example 7.27**

Suppose *f (x)* is a differentiable function for
all* x *with *f (x)* â‰¤ 29 and* f *(2) = 17
. What is the maximum value of* f *(7)
?

**Solution**

By the
mean value theorem we have, there exists ' *c*
' âˆˆ (2, 7) such that,

* f*
(7) âˆ’* f *(2) / (7â€“2) =* f â€˜*(c) â‰¤ 29.

Hence,* f *(7) â‰¤ 5Ã— 29 +17 = 162

Therefore,
the maximum value of* f *(7) is 162 .

** **

Prove,
using mean value theorem, that

| sin *Î±* âˆ’ sin *Î²*
| â‰¤
| *Î±* âˆ’ *Î²*
|, *Î±* , *Î²*
âˆˆ .

**Solution**

Let *f* (*x*)
=
sin *x* which is a differentiable
function in any open interval. Consider an interval [*Î±* , *Î²*
] . Applying the mean value theorem there exists *c* âˆˆ(*Î±* , *Î²*
) such that,

Hence, |
sin *Î±* âˆ’ sin *Î²*
| â‰¤
| *Î±* âˆ’ *Î²*
| .

If we
take *Î²* = 0 in the above problem, we get |
sin *Î±* | â‰¤ | *Î±*
| .

** **

A
thermometer was taken from a freezer and placed in a boiling water. It took 22
seconds for the thermometer to raise from âˆ’ 10Â°C to 100Â°C
. Show that the rate of change of temperature at some time *t* is 5Â°C per second.

Let *f *(*t
*)* *be the temperature at time* t*. By the mean value theorem, we have

= 5Â°C per second.

Hence
the instantaneous rate of change of temperature at some time *t* is 5Â°C per second.

Tags : Applications of Differential Calculus | Mathematics , 12th Maths : UNIT 7 : Applications of Differential Calculus

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12th Maths : UNIT 7 : Applications of Differential Calculus : Applications - Mean Value Theorem | Applications of Differential Calculus | Mathematics

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