Calculus For Engineers (FIC 103) Assignment Sheet 2 PDF

Calculus For Engineers (FIC 103) : Assignment Sheet - 2 Topic- Maxima, Minima, Montonic Functions, Concavity, Curve Sketching,Optimization Problems & Area between curves SRM University AP, Amaravati, India 1. Find the critical points...

Calculus For Engineers (FIC 103) : Assignment Sheet - 2 Topic- Maxima, Minima, Montonic Functions, Concavity, Curve Sketching,Optimization Problems & Area between curves SRM University AP, Amaravati, India 1. Find the critical points of the following functions on its domain: 1 2 (i) f (x) = 2x3 − 3x2 − 36x (iii) f (x) = x 3 − x− 3. x−1 (ii) f (x) = 2 x +4 2. Find the absolute maximum and absolute minimum values of f on the given interval h πi (i) f (x) = 12 + 4x − x2 , [0, 5] (iii) f (x) = 2 cos x + sin 2x, 0, 2 2 p 2 (ii) f (x) = x 4 − x , [−1, 2] (iv) f (x) = 3x 3 , [−27, 8]. 3. If a and b are positive numbers, find the maximum value of f (x) = xa (1 − xb ), 0 ≤ x ≤ 1. 4. Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers that satisfy the conclusion of Rolle’s Theorem. √ (i) f (x) = 5 − 12x + 3x2 , [1, 3] (ii) f (x) = x − 31 x, [0, 9] 2 5. Let f (x) = 1 − x 3. Show that f (−1) = f (1) but there is no number c in (−1, 1) such that f ′ (c) = 0. Why does this not contradict Rolle’s Theorem? 6. Find the value or values of c that satisfy the equation f (b) − f (a) = f ′ (c) b−a in the conclusion of Mean Value theorem for the functions and intervals in the following: ( (i) f (x) = x3 − 3x + 2, [−2, 2] x3 −2 ≤ x ≤ 0   (iii) g(x) = 2 1 1 −x 0 < x ≤ 2 (ii) f (x) = x + , , 2 x 2 7. For what values of a, m and b does the function  3  x=0 f (x) = −x2 + 3x + a 0