By A. Ganesh (Author), Balasubramanian G. (Author)
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Additional resources for Engineering Mathematics- II (As per the new syllabus of VTU (B. E. , II Semester)
X x2 x3 + – + − ......... 4 2 4 18 18. Expand esin x up to the term containing x4 by Maclaurin’s theorem.
F (0) + ..... 2! 3! n! (3) This is called the Maclaurin’s series for the function f (x). If f (x) = y and f ′ (x), f ′′ (x), .......... are denoted by y1, y2, ...... the Maclaurin’s series can also be written in the form: y = y (0) + xy1 (0) + x2 xn y2 (0) + ...... y (0) + ...... 2! n! n WORKED OUT EXAMPLES 1. Verify Rolle’s theorem for the function f (x) = x2 – 4x + 8 in the interval [1, 3]. Solution f (x) = x2 – 4x + 8 is continuous in [1, 3] and f ′ (x) = 2x – 4 exist for all values in (1, 3) ∴ f (1) = 1 – 4 + 8 = 5; f (3) = 32 – 4 (3) + 8 = 5 ∴ f (1) = f (3) Hence all the three conditions of the theorem are satisfied.
Now consider f ′ (c) = 0 ∴ cos c = 0 so that c = ± π Both these values 2 lie in (– π, π). π ∈ (– π, π) 2 Hence Rolle’s theorem is verified. ∴ c = ± f (x) = ex sin x is continuous in [0, π] (ii) f ′ (x) = ex cos x + sin x · ex = ex (cos x + sin x) f ′ (x) exists in (0, π) f (0) = e0 sin (0) = 0 And also f (π) = eπ sin (π) = 0 ∴ f (0) = f (π) = 0 Therefore f (x) satisfies all the conditions of Rolle’s theorem. Now consider f ′ (c) = 0 c e (cos c + sin c) = 0 cos c + sin c = 0 sin c = – cos c as ec ≠ 0 or tan c = –1 c = –π 3π or c = 4 4 3π ∈ (0, π) 4 Hence Rolle’s theorem is verified.