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MHTCET 15 May 2024 Shift 1 Paper

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Question : 144 of 150

Marks: +1, -0

The approximate value of

3^{2.001}

\log 3=1.0986

is

9.00898
9.0989
9.0898
9.00989

Solution:

Concept:

Use linear approximation: for a small change

h

,

f(a+h) \approx f(a) + h f'(a)

.

Explanation:

Let

f(x) = 3^x

.

Then

f'(x) = 3^x \log 3 = 3^x \ln 3

.

Take

a = 2

and

h = 0.001

, since

3^{2.001} = f(2+0.001)

.

Now,

f(2+0.001) \approx f(2) + 0.001 \cdot f'(2)

.

f(2) = 3^2 = 9

.

f'(2) = 3^2 \log 3 = 9 \times 1.0986 = 9.8874

.

So,

3^{2.001} \approx 9 + 0.001 \times 9.8874 = 9 + 0.0098874

.

Thus,

3^{2.001} \approx 9.0098874 \approx 9.00989

.

Answer:

The approximate value is

9.00989

, which matches option D.

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