GATE CE 2026 Set 2 — Question 12
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Engineering Mathematics → Numerical Integration & ODEs → Numerical Differentiation
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Question
A fifth-degree polynomial in is defined for . All coefficients of the polynomial are positive. The first derivative of the polynomial is obtained numerically at a point by using the first-order forward as well as the first-order backward difference methods. Identical step lengths are used for both the methods.
Following statements are made.(I) Forward difference method underestimates the true derivative.
(II) Backward difference method overestimates the true derivative.Which one of the following options is CORRECT?
Following statements are made.(I) Forward difference method underestimates the true derivative.
(II) Backward difference method overestimates the true derivative.Which one of the following options is CORRECT?
Correct answer
(A) Both statements (I) and (II) are FALSE.
Solution
Let be the fifth-degree polynomial where . For , the second derivative is always positive. This means the function is concave up.
1.Forward difference method: . Using Taylor series expansion: for some . Thus, . Since , the forward difference overestimates the true derivative. Statement (I) is FALSE.
2.Backward difference method: . Using Taylor series expansion: for some . Thus, . Since , the backward difference underestimates the true derivative. Statement (II) is FALSE.
Therefore, both statements are FALSE.Continue learning with Success Tracker
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