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## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 7 Applications of Differential Calculus Ex 7.4

Question 1.

Write the Maclaurin series expansion of the following functions:

(i) e^{x}

(ii) sin x

(iii) cos x

(iv) log (1 – x); -1 ≤ x < 1

(v) tan^{-1} (x) ; -1 ≤ x ≤ 1

(vi) cos^{2} x

Solution:

(vi) f(x) = cos^{2} x

f(0) = 1

f'(x) = 2 cos x (- sin x) = – sin 2x

f'(0) = 0

f”(x) = (-cos 2x)(2)

f”(0) = -2

f”'(x) = -2[- sin 2x](2) = 4 sin 2x

f”'(0) = 0

f^{4} (x) = 4(cos 2x)(2) = 8 cos 2x

f^{4} (0) = 8

Question 2.

Write down the Taylor series expansion, of the function log x about x = 1 upto three non-zero terms for x > 0.

Solution:

Question 3.

Expand sin x in ascending powers x – \(\frac{\pi}{4}\) upto three non-zero terms.

Solution:

f (x) = sin x

Question 4.

Expand the polynomial f(x) = x^{2} – 3x + 2 in powers of x – 1

Solution:

f(x) = x^{2} – 3x + 2 = (x – 1) (x – 2)

f(1) = 0

f'(x) = 2x – 3 ; f'(1) = -1

f”(x) = 2 ; f”(1) = 2

### Samacheer Kalvi 12th Maths Solutions Chapter 7 Applications of Differential Calculus Ex 7.4 Additional Problems

Question 1.

The Taylor’s series expansion of f(x) = sin x about x = \(\frac{\pi}{2}\) is obtained by the following way.

Solution:

Question 2.

Obtain the Maclaurin’s series expansion for the following functions.

(i) e^{x}

(ii) sin^{2 }x

(iii) \(\frac{1}{1+x}\)

Solution:

(i)

(ii)

(iii)