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November 15, 2005 Lecturer: Dr Martin Kurth Michaelmas Term 2005 Course 1E1 2005-2006 (JF Engineers & JF MSISS & JF MEMS) Problem Sheet 6 Due: in the Tutorials 25 November / 28 November Igitur sensus lemmatis est, ut, si quantitatum quarumcunque perpetuo motu crescentium vel decrescentium A, B, C, &c. momenta, vel his proportionales mutationum velocitates dicantur a, b, c, &c. momentum vel mutatio geniti rectanguli AB fuerit aB + bA, . . . (Wherefore the sense of the Lemma is, that if the moments of any quantities A, B, C, &c. increasing or decreasing by a perpetual flux, or the velocities of the mutations which are proportional to them, be called a, b, c, &c. the moment or mutation of the generated rectangle AB will be aB + bA, . . . ) from: Isaac Newton, Philosophiae Naturalis Principia Mathematica 1. Consider the following functions f1 (x) = x2 − 1, 1 f2 (x) = 2 (x − 4) f3 (x) = x + 3, f4 (x) = x. (|x| = 6 2), Calculate (c) d dx (f1 (x)f2 (x)) d dx (f3 (x)f4 (x)), d f3 (x) dx f2 (x) , (d) d f4 (x) dx f1 (x) (e) d dx (f1 (x)f2 (x)f3 (x)f4 (x)) (a) (b) (|x| = 6 2), (|x| = 6 1), (|x| = 6 2). (6 points) 2. Calculate (a) (b) d dx d dx tan x, cot x. (4 points) 3. Calculate d752 d999 sin x + 752 cos x. 999 dx dx (*) Questions 1 and 2 should be answered by all students, you will get points for them. Question 3 is more challenging and meant as an exercise for the more mathematically interested students.