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# Applications - Mean Value Theorem

Mean Value Theorem: Applications - Applications of Differential Calculus

Applications

### Example 7.26

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.

### Solution

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,

fŌĆ▓(c) = 164 ŌłÆ 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) Ōēż 29 and f (2) = 17 . What is the maximum value of f (7) ?

Solution

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

f (7) ŌłÆ f (2) / (7ŌĆō2) = f ŌĆś(c)  Ōēż 29.

Hence, f (7) Ōēż 5├Ś 29 +17 = 162

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

### Example 7.28

Prove, using mean value theorem, that

| sin ╬▒ ŌłÆ sin ╬▓ | Ōēż | ╬▒ ŌłÆ ╬▓ |, ╬▒ , ╬▓ Ōłł .

Solution

Let f (x) = sin x which is a differentiable function in any open interval. Consider an interval [╬▒ , ╬▓ ] . Applying the mean value theorem there exists c Ōłł(╬▒ , ╬▓ ) such that, Hence, | sin ╬▒ ŌłÆ sin ╬▓ | Ōēż | ╬▒ ŌłÆ ╬▓ | .

### Remark

If we take ╬▓ = 0 in the above problem, we get | sin ╬▒ | Ōēż | ╬▒ | .

### Example 7.29

A thermometer was taken from a freezer and placed in a boiling water. It took 22 seconds for the thermometer to raise from ŌłÆ 10┬░C to 100┬░C . Show that the rate of change of temperature at some time t is 5┬░C per second.

### Solution

Let f (t ) be the temperature at time t. By the mean value theorem, we have = 5┬░C per second.

Hence the instantaneous rate of change of temperature at some time t is 5┬░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