8/7/2019 EndsemCal Marking

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Marking scheme of End-semester Examination

Part II : Calculus

1. (a) First method: For showing 0 < xn < α for all n ≥ n0, where α < 1 : 2 marksApplication of sandwich theorem to get the answer : 1 markSecond method: For correct solution : 3 marks

(b) For correct solution (any method) : 2 marks(c) Using ratio test to show that the series converges absolutely for |x + 3| < 5 and diverges for|x + 3| > 5 : 1 markFor showing the divergence of the series at x = −8 : 1 markFor showing the conditional convergence at x = 2 : 1 mark

2. (a) For defining the function f  correctly to disprove the statement : 3 marks(Although the model solution for this question gives the details of proving the claim, that isnot required for getting 3 marks.)

(b) First method: For showing the existence of  x1, y1 satisfying |x1 − y1| = 1

2(b − a) and

f (x1) = f (y1) : 3 marksRemaining part to get the solution : 2 marksSecond method: For the argument that at least one of the points where f  attains maximum orminimum is in (a, b) (unless f  is constant) : 2 marksApplication of intermediate value theorem to get the answer : 3 marks

3. (a) For differentiability at x = 0 with the expression for f (x) for x = 0 : 1 markFor showing f (0) = 0 : 1 markFor showing continuity of  f  at 0 : 1 mark

(b) For defining g correctly : 1 mark

Applying Rolle’s theorem to get g(c) = 0 : 1 markGetting f (c) = f (c) : 1 mark

4. (a) For writing f (x) correctly in terms of  f (0), f (0), f (0) and f (ξ) (using Taylor’s theorem): 1 markFinding f (0), f (0), f (0) and f (ξ) : 1 markFor proving the inequality : 1 mark

(b) For the correct formula of  F  : 1 markFor showing that F  is not differentiable at 1 : 1 mark

5. (a) For defining F  correctly : 1 markFor showing that at most one b can exist (by showing that F (y) = 0 for all y ≥ 0) : 1 markFor showing the existence of  b (by using the intermediate value theorem) : 1 mark

(b) First method: For mentioning the inequality showing removal of cos3 x : 1 mark

For showing that∞ 

0

x√1+x5

dx is convergent : 1 mark

For the remaining part : 1 markSecond method: For showing the conditions on f  and g : 1 + 1 marksFor applying Dirichlet’s test to get the conclusion : 1 mark

6. (a) For correct integral expression for the required volume : 2 marks

For correct final answer : 1 mark

(b) For correct integral expression for the required area : 2 marksFor correct final answer : 1 mark