MAT137Y1 Midterm: 2003 Test 2 solution

• School: UTSG
• Department: Mathematics
• Course: MAT137Y1
• Professor: N/ A

Document Summary

1. (6%) (i) let f (x) = 1 x2 . Find f 0(x) using the de nition of derivative. f 0(x) = lim h 0. 2xh h2 hx2(x + h)2 = lim h 0. 2x h x2(x + h)2 = 2x h x4 = 2 x3 . 1 (x+h)2 1 x2 x2 (x + h)2 hx2(x + h)2. = lim h 0 (6%) (ii) differentiate g(x) = 3sec x + x tan(p x) x5 + 1. Do not simplify your answer. x5 + 1[3sec x tan x + tan(p x) x sec2(p x)] 5x4 x5+1. [3sec x + x tan(p x)] g0(x) = when t = 1 if y = u 2 u + 1 dy dt. Differentiating we get (6%) (iii) find x5 + 1 u = (s3 + 1)2, s = t 1/2. dy du (u + 1) (u 2) (u + 1)2. = 2(s3 + 1) 3s2 = 6s2(s3 + 1), ds dt.