1 2014-2015:Semester-II Department of Mathematics MAL 100: Calculus Tutorial Sheet 2: Limits, Continuity and Differentiability 1. Discuss the following limits. If the limit exists and finite, find the limit. If the limit goes to infinity, Justify? x2/3 + x−1 sin x sin x (b) lim 2/3 (c) lim (d) lim x sin x (a) lim 2 x→∞ x x→∞ x x→∞ x→∞ [x] + cos x 2. Determine the points and nature of discontinuity of the following functions: √ 1 x ∈ Q 1 cos x x4 + 1 x tan x (a) − 3x (b) (c) (e) f (x) = (d) 2 0 x 6∈ Q x−2 x − π/2 x +1 1 + sin2 x 3. Find the asymptotes of the graph of (a) x2 − 3 2x − 4 (b) x2 − 4 x−1 (c) x3 + 1 x 4. Find the following limits and determine if the functions are continuous at the point. π (b) lim sec(y sec2 y − tan2 y − 1) (c) lim tan( cos(sin x1/3 )) y→1 x→0 4 (a) lim sin(x − sin x) x→π 5. Determine if the following equations admits solutions in the interval mentioned. π 2x + 1 (a) x5 − 3x2 = −1, [0, 1] (b) sin2 x − 2 cos x = −1, [0, ] (c) sin x − , [0, π] 2 x−2 6. Bisection Method for finding roots: Let f (x) be a continuous function on [a, b] such that f (a)f (b) < 0. Then show that f has a zero in [a, b]. Begin with a1 = a, b1 = b 1 and x1 = a1 +b . If f (x1 ) = 0, we are done. Otherwise, apply the method on (x1 , b1 ) or 2 2 (a1 , x1 ). Define x2 = a2 +b ....and so on..Show that {xn } converges. 2 7. Determine which of the following functions are uniformly continuous in the interval mentioned. 2 (a) ex sin(x2 ), (0, 1) (b) | sin x|, [0, ∞) (c) √ x sin x 8. Determine if the following functionsare differentiable at 0. Find f 0 (0) if exists e− x12 x 6= 0 x cos 1 x 6= 0 x (a) (b) e−|x| (c) 0 0 x=0 x=0 9. Determine if f 0 is continuousat 0 for the following functions: 1 1 x3 sin 1 x = 2 6 0 x cos x x 6= 0 x2 ln |x| x 6= 0 x (a) (b) (c) 0 0 0 x=0 x=0 x=0 2 http://web.iitd.ac.in/∼sreenadh 10. When a circular plate of metal is heated in an oven, its radius increases at the rate of 0.01cm/min. At what rate the plate’s area increasing when the radius is 50 cm? 11. Show the following inequalities: (a) for 0 < p < 1 and a, b ≥ 0, (a + b)p ≤ ap + bp . (b) for p > 1, and a, b ≥ 0, (a + b)p ≤ 2p−1 (ap + bp ). 12. Let f be differentiable on IR and sup |f 0 (x)| < 1. Select s0 ∈ IR and define sn = f (sn−1 ). IR Prove that {sn } is a convergent sequence. 0 13. Let f be a thrice differentiable function on (a, b) such that f (p) = 0, f (p) 6= 0 for some f (x)f 00 (x) < 1. The following sequences come from the recursion p ∈ (a, b) and sup 0 (x)2 f (a,b) formula for Newton’s method: xn+1 = xn − f (xn ) f 0 (xn ) Show that the sequence {xn } converges. In each case, begin by identifying the function f that generates the sequence and determine if the sequence converges: x0 = 1, (a) xn+1 = xn − xn 1 + 2 xn (b) xn+1 = xn − 1 (c) xn+1 = xn − tan xn − 1 sec2 xn 14. Let f be differentiable on IR and |f (x)−f (y)| ≤ (x−y)2 . Then show that f is constant. 15. Prove that if f, g are differentiable on IR, f 0 (x) ≤ g 0 (x) on IR and f (0) = g(0), then f (x) ≤ g(x) for x ≥ 0. 16. Find the first 3 terms of the Taylor’s expansion about the point x0 given 2 (a) sin x, x0 = π/2 (b) e−1/x , x0 = 0 (c) tan−1 x, x0 = 0 17. Show that Rn (x) → 0 as n → ∞ for the following functions (i) log(1 + x) (ii) sinh x (iii) cosh x (iv) ax (v) (1 + x)m , 0 < m ∈ Q, x ∈ (0, 1) 18. Find the domain of validity when sin x is approximated by x − (x3 /6) with an error of magnitude no greater than 5 × 10−4 19. Estimate the error in the approximation of sin hx = x + (x3 /3!) when |x| < 0.5. √ 20. Find the error while approximating 1 + x with 1 + (x/2) in |x| < 0.01.
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