PYQs / GATE ME / 2019 / Set 2 / Q38 GATE ME 2019 Set 2 — Question 38 Go beyond PYQs with Success Tracker AI-powered personalised practice and doubt support. Unlimited practice on eligible plans; AI usage limits apply. MCQ +2 / -0.67 Medium Error Analysis Numerical Methods Engineering Mathematics
Engineering Mathematics → Numerical Methods → Error Analysis
Last updated 5 September 2026
Question The derivative of
f ( x ) = cos ( x ) f(x) = \cos(x) f ( x ) = cos ( x ) can be estimated using the approximation
f ′ ( x ) = f ( x + h ) − f ( x − h ) 2 h f'(x) = \frac{f(x+h)-f(x-h)}{2h} f ′ ( x ) = 2 h f ( x + h ) − f ( x − h ) The percentage error is calculated as
( Exact value − Approximate value Exact value ) × 100 \left( \frac{\text{Exact value} - \text{Approximate value}}{\text{Exact value}} \right) \times 100 ( Exact value Exact value − Approximate value ) × 100 The percentage error in the derivative of
f ( x ) f(x) f ( x ) at
x = π / 6 x = \pi/6 x = π /6 radian, choosing
h = 0.1 h = 0.1 h = 0.1 radian, is
Correct answer (B) 0.1 \% and < 1 \%
Solution 1. Exact value : The derivative of f ( x ) = cos ( x ) f(x) = \cos(x) f ( x ) = cos ( x ) f ′ ( x ) = − sin ( x ) f'(x) = -\sin(x) f ′ ( x ) = − sin ( x ) x = π / 6 x = \pi/6 x = π /6 f ′ ( π / 6 ) = − sin ( π / 6 ) = − 0.5 f'(\pi/6) = -\sin(\pi/6) = -0.5 f ′ ( π /6 ) = − sin ( π /6 ) = − 0.5 2. Approximate value : Using the central difference formula with h = 0.1 h = 0.1 h = 0.1 f ′ ( π / 6 ) ≈ cos ( π / 6 + 0.1 ) − cos ( π / 6 − 0.1 ) 2 ( 0.1 ) f'(\pi/6) \approx \frac{\cos(\pi/6 + 0.1) - \cos(\pi/6 - 0.1)}{2(0.1)} f ′ ( π /6 ) ≈ 2 ( 0.1 ) cos ( π /6 + 0.1 ) − cos ( π /6 − 0.1 ) f ′ ( π / 6 ) ≈ cos ( 0.5236 + 0.1 ) − cos ( 0.5236 − 0.1 ) 0.2 = cos ( 0.6236 ) − cos ( 0.4236 ) 0.2 f'(\pi/6) \approx \frac{\cos(0.5236 + 0.1) - \cos(0.5236 - 0.1)}{0.2} = \frac{\cos(0.6236) - \cos(0.4236)}{0.2} f ′ ( π /6 ) ≈ 0.2 cos ( 0.5236 + 0.1 ) − cos ( 0.5236 − 0.1 ) = 0.2 cos ( 0.6236 ) − cos ( 0.4236 ) f ′ ( π / 6 ) ≈ 0.81178 − 0.91146 0.2 = − 0.09968 0.2 = − 0.4984 f'(\pi/6) \approx \frac{0.81178 - 0.91146}{0.2} = \frac{-0.09968}{0.2} = -0.4984 f ′ ( π /6 ) ≈ 0.2 0.81178 − 0.91146 = 0.2 − 0.09968 = − 0.4984 (More precisely, using
cos ( A ) − cos ( B ) = − 2 sin ( A + B 2 ) sin ( A − B 2 ) \cos(A) - \cos(B) = -2 \sin(\frac{A+B}{2}) \sin(\frac{A-B}{2}) cos ( A ) − cos ( B ) = − 2 sin ( 2 A + B ) sin ( 2 A − B ) , the approx value is
− sin ( x ) sin ( h ) h = − 0.5 × sin ( 0.1 ) 0.1 ≈ − 0.499167 -\sin(x) \frac{\sin(h)}{h} = -0.5 \times \frac{\sin(0.1)}{0.1} \approx -0.499167 − sin ( x ) h s i n ( h ) = − 0.5 × 0.1 s i n ( 0.1 ) ≈ − 0.499167 )
3. Percentage Error :
Error = ∣ − 0.5 − ( − 0.499167 ) − 0.5 ∣ × 100 = 0.000833 0.5 × 100 = 0.1666 % \text{Error} = \left| \frac{-0.5 - (-0.499167)}{-0.5} \right| \times 100 = \frac{0.000833}{0.5} \times 100 = 0.1666 \% Error = − 0.5 − 0.5 − ( − 0.499167 ) × 100 = 0.5 0.000833 × 100 = 0.1666% This value is between
0.1 % 0.1 \% 0.1% and
1 % 1 \% 1% .
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Correct answer (B) 0.1 \% and < 1 \%
Solution 1. Exact value : The derivative of f ( x ) = cos ( x ) f(x) = \cos(x) f ( x ) = cos ( x ) f ′ ( x ) = − sin ( x ) f'(x) = -\sin(x) f ′ ( x ) = − sin ( x ) x = π / 6 x = \pi/6 x = π /6 f ′ ( π / 6 ) = − sin ( π / 6 ) = − 0.5 f'(\pi/6) = -\sin(\pi/6) = -0.5 f ′ ( π /6 ) = − sin ( π /6 ) = − 0.5 2. Approximate value : Using the central difference formula with h = 0.1 h = 0.1 h = 0.1 f ′ ( π / 6 ) ≈ cos ( π / 6 + 0.1 ) − cos ( π / 6 − 0.1 ) 2 ( 0.1 ) f'(\pi/6) \approx \frac{\cos(\pi/6 + 0.1) - \cos(\pi/6 - 0.1)}{2(0.1)} f ′ ( π /6 ) ≈ 2 ( 0.1 ) cos ( π /6 + 0.1 ) − cos ( π /6 − 0.1 ) f ′ ( π / 6 ) ≈ cos ( 0.5236 + 0.1 ) − cos ( 0.5236 − 0.1 ) 0.2 = cos ( 0.6236 ) − cos ( 0.4236 ) 0.2 f'(\pi/6) \approx \frac{\cos(0.5236 + 0.1) - \cos(0.5236 - 0.1)}{0.2} = \frac{\cos(0.6236) - \cos(0.4236)}{0.2} f ′ ( π /6 ) ≈ 0.2 cos ( 0.5236 + 0.1 ) − cos ( 0.5236 − 0.1 ) = 0.2 cos ( 0.6236 ) − cos ( 0.4236 ) f ′ ( π / 6 ) ≈ 0.81178 − 0.91146 0.2 = − 0.09968 0.2 = − 0.4984 f'(\pi/6) \approx \frac{0.81178 - 0.91146}{0.2} = \frac{-0.09968}{0.2} = -0.4984 f ′ ( π /6 ) ≈ 0.2 0.81178 − 0.91146 = 0.2 − 0.09968 = − 0.4984 (More precisely, using
cos ( A ) − cos ( B ) = − 2 sin ( A + B 2 ) sin ( A − B 2 ) \cos(A) - \cos(B) = -2 \sin(\frac{A+B}{2}) \sin(\frac{A-B}{2}) cos ( A ) − cos ( B ) = − 2 sin ( 2 A + B ) sin ( 2 A − B ) , the approx value is
− sin ( x ) sin ( h ) h = − 0.5 × sin ( 0.1 ) 0.1 ≈ − 0.499167 -\sin(x) \frac{\sin(h)}{h} = -0.5 \times \frac{\sin(0.1)}{0.1} \approx -0.499167 − sin ( x ) h s i n ( h ) = − 0.5 × 0.1 s i n ( 0.1 ) ≈ − 0.499167 )
3. Percentage Error :
Error = ∣ − 0.5 − ( − 0.499167 ) − 0.5 ∣ × 100 = 0.000833 0.5 × 100 = 0.1666 % \text{Error} = \left| \frac{-0.5 - (-0.499167)}{-0.5} \right| \times 100 = \frac{0.000833}{0.5} \times 100 = 0.1666 \% Error = − 0.5 − 0.5 − ( − 0.499167 ) × 100 = 0.5 0.000833 × 100 = 0.1666% This value is between
0.1 % 0.1 \% 0.1% and
1 % 1 \% 1% .
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