Math 220 February 19 I. Find the derivative: 1. f (x) = π 2. f (x) = x + π 2 3. f (x) = x5 + e5 4. f (x) = ex 5. f (x) = 1 x 6. f (x) = 1 x5 7. f (x) = sin(x) 8. f (x) = cos(x) 9. f (x) = √ 1 1 x+ √ +√ 4 3 x x7 10. f (x) = ex+2 + sin(x + 3) 11. f (u) = 2 sin(u)u2 12. f (x) = x2 ex − 2xex x2 13. f (x) = x e +1 14. w(x) = 2 sin(2x) √ u2 + u3 + eu √ 15. s(u) = u 16. y = tan(x) 1 17. h(u) = csc(u) + sec(u) + cot(u) 18. f (x) = sin(x) + cos(x) x2 19. f (x) = xex + cos(x) x sin(x) 20. f (x) = cot(x)(π sin(x) + 52 x2 ) √ x3/2 (ex sin(x) + 2x) 21. f (x) = x+2 22. f (x) = (x + 1)(x + 2)(x + 3) 23. f (x) = (2x + 1) sin(x)ex 24. f (x) = e3x + sin(3x) 1 − cos2 (x) sin(x) √ x sin(x) − x2 26. f (x) = πex − tan(x) 25. f (x) = 27. f (x) = (x sin(x))2 II. Find an equation to the tangent line to the curve at the given point. 1. y = xex , (0, 0) 2. y = x2 + sin(x), (π, π 2 ) 3. y = sin(3x), (π/6, 1) 4. y = Cex , (0, C) 5. y = 3+x , (2, 1) 1 + x2 2 1 Solutions I. Find the derivative: 1. f (x) = π Answer: f 0 (x) = 0 2. f (x) = x + π 2 Answer: f 0 (x) = 1 3. f (x) = x5 + e5 Answer: f 0 (x) = 5x4 4. f (x) = ex Answer: f 0 (x) = ex 1 5. f (x) = x Answer: f 0 (x) = −1 x2 f 0 (x) = −5 x6 1 6. f (x) = 5 x Answer: 3 7. f (x) = sin(x) Answer: f 0 (x) = cos(x) 8. f (x) = cos(x) Answer: f 0 (x) = − sin(x) √ 1 1 +√ x+ √ 4 3 x x7 Answer: 9. f (x) = f (x) = x1/2 + x−1/3 + x−7/4 1 7 1 f 0 (x) = x−1/2 − x−4/3 − x−11/4 2 3 4 10. f (x) = ex+2 + sin(x + 3) Answer: f 0 (x) = ex e2 + sin(x) cos(3) + cos(x) sin(3) f 0 (x) = ex e2 + cos(x) cos(3) − sin(x) sin(3) = ex+2 + cos(x + 3) 11. f (u) = 2 sin(u)u2 Answer: 4 f (u) = 2u2 sin(u) f 0 (u) = 2(u2 )0 sin(u) + 2u2 (sin(u))0 = 2(2u) sin(u) + 2u2 cos(u) = 4u sin(u) + 2u2 cos(u) = 2u(2 + u cos(u)) 12. f (x) = x2 ex − 2xex Answer: f 0 (x) = (x2 )0 ex + x2 (ex )0 − [(2x)0 ex + 2x(ex )0 ] = 2xex + x2 ex − 2ex − 2xex x x = 2xe + x2 ex − 2ex − 2xe = x2 ex − 2ex x2 ex + 1 Answer: 13. f (x) = (x2 )0 (ex + 1) − x2 (ex + 1)0 f (x) = (ex + 1)2 2x(ex + 1) − x2 ex = (ex + 1)2 0 14. w(x) = 2 sin(2x) Answer: 5 w(x) = 2(2 sin(x) cos(x)) = 4 sin(x) cos(x) w0 (x) = 4(sin(x))0 cos(x) + 4 sin(x)(cos(x))0 = 4 cos(x) cos(x) + 4 sin(x)(− sin(x)) = 4 cos2 (x) − 4 sin2 (x) = 4(cos2 (x) − sin2 (x)) = 4 cos(2x) 15. s(u) = u2 + √ u3 + eu √ u Answer: u3/2 eu u2 + + u1/2 u1/2 u1/2 eu = u3/2 + u + 1/2 u s(u) = 3 1/2 (eu )0 u1/2 − eu (u1/2 )0 ds 0 = s (u) = u + 1 + du 2 (u1/2 )2 eu u1/2 − eu ( 12 u−1/2 ) 3 = u1/2 + 1 + 2 u 3 1/2 eu eu = u + 1 + 1/2 − 3/2 2 u 2u 16. y = tan(x) Answer: 6 y= sin(x) cos(x) dy (sin(x))0 cos(x) − sin(x)(cos(x))0 = y0 = dx cos2 (x) cos(x) cos(x) − sin(x)(− sin(x)) = cos2 (x) cos2 (x) + sin2 (x) = cos2 (x) 1 = cos2 (x) = sec2 (x) 17. h(u) = csc(u) + sec(u) + cot(u) Answer: dh = h0 (u) = − csc(u) cot(u) + sec(u) tan(u) − csc2 (u) du sin(x) + cos(x) 18. f (x) = x2 Answer: (sin(x) + cos(x))0 x2 − (sin(x) + cos(x))(x2 )0 (x2 )2 (cos(x) − sin(x))x2 − (sin(x) + cos(x))2x = (x4 ) (x2 − 2x) cos(x) − (x2 + 2x) sin(x) = (x4 ) f 0 (x) = 7 xex + cos(x) x sin(x) Answer: 19. f (x) = f 0 (x) = = = = = = = = = = [xex + cos(x)]0 x sin(x) − [xex + cos(x)](x sin(x))0 (x sin(x))2 [(xex )0 − sin(x)]x sin(x) − [xex + cos(x)](sin(x) + x cos(x)) x2 sin2 (x) [(ex + xex ) − sin(x)]x sin(x) − [xex + cos(x)](sin(x) + x cos(x)) x2 sin2 (x) [x sin(x)ex + x2 sin(x)ex − x sin2 (x)] − [x sin(x)ex + x2 cos(x)ex + cos(x) sin(x) + x co x2 sin2 (x) 2 x 2 x x 2 x [ x sin(x)e x sin(x)e + x sin(x)e − x sin (x)] − [ + x cos(x)e + cos(x) sin(x) + x co x2 sin2 (x) [x2 sin(x)ex − x sin2 (x)] − [x2 cos(x)ex + cos(x) sin(x) + x cos2 (x)] x2 sin2 (x) x2 sin(x)ex − x sin2 (x) − x2 cos(x)ex − cos(x) sin(x) − x cos2 (x) x2 sin2 (x) x2 sin(x)ex − x2 cos(x)ex − x sin2 (x) − x cos2 (x) − cos(x) sin(x) x2 sin2 (x) x2 ex (sin(x) − cos(x)) − x(sin2 (x) + cos2 (x)) − cos(x) sin(x) x2 sin2 (x) x2 ex (sin(x) − cos(x)) − x − cos(x) sin(x) x2 sin2 (x) 20. f (x) = cot(x)(π sin(x) + 52 x2 ) Answer: f 0 (x) = (cot(x))0 (π sin(x) + 52 x2 ) + cot(x)(π sin(x) + 52 x2 )0 = − csc2 (x)(π sin(x) + 52 x2 ) + cot(x)(π cos(x) + 52 2x) = − csc2 (x)(π sin(x) + 25x2 ) + cot(x)(π cos(x) + 50x) 8 x3/2 (ex sin(x) + 21. f (x) = x+2 Answer: √ 2x) √ 0 3/2 x 2x)] (x + 2) − [x (e sin(x) + 2x)](x + 2)0 f 0 (x) = (x + 2)2 √ √ √ [(x3/2 )0 (ex sin(x) + 2x) + x3/2 (ex sin(x) + 2 x)0 ](x + 2) = (x + 2)2 √ [x3/2 (ex sin(x) + 2x)]1 − (x + 2)2 √ √ [ 32 x1/2 (ex sin(x) + 2x) + x3/2 (ex sin(x) + ex cos(x) + 2√2x )](x + 2) = (x + 2)2 √ x3/2 (ex sin(x) + 2x) − (x + 2)2 √ √ √ [ 23 x(ex sin(x) + 2x) + x3 (ex sin(x) + ex cos(x) + √12x )](x + 2) = (x + 2)2 √ √ x3 (ex sin(x) + 2x) − (x + 2)2 [x3/2 (ex sin(x) + √ 22. f (x) = (x + 1)(x + 2)(x + 3) Answer: f 0 (x) = (x + 1)0 (x + 2)(x + 3) + (x + 1)(x + 2)0 (x + 3) + (x + 1)(x + 2)(x + 3)0 = 1(x + 2)(x + 3) + (x + 1)1(x + 3) + (x + 1)(x + 2)1 = (x + 2)(x + 3) + (x + 1)(x + 3) + (x + 1)(x + 2) = (x + 2 + x + 1)(x + 3) + x2 + 3x + 2 = (2x + 3)(x + 3) + x2 + 3x + 2 = 2x2 + 9x + 9 + x2 + 3x + 2 = 3x2 + 12x + 11 9 23. f (x) = (2x + 1) sin(x)ex Answer: f 0 (x) = (2x + 1)0 sin(x)ex + (2x + 1)(sin(x))0 ex + (2x + 1) sin(x)(ex )0 = 2 sin(x)ex + (2x + 1) cos(x)ex + (2x + 1) sin(x)ex = ex [(2x + 3) sin(x) + (2x + 1) cos(x)] 24. f (x) = e3x + sin(3x) Answer: 10 f (x) = ex ex ex + sin(x) cos(2x) + cos(x) sin(2x) = ex ex ex + sin(x)(cos(x) cos(x) − sin(x) sin(x)) + cos(x)2 sin(x) cos(x) = ex ex ex + sin(x) cos(x) cos(x) − sin(x) sin(x) sin(x)) + 2 sin(x) cos(x) cos(x) f 0 (x) = (ex )0 ex ex + ex (ex )0 ex + ex ex (ex )0 + (sin(x))0 cos(x) cos(x) + sin(x)(cos(x))0 cos(x) + sin(x) cos(x)(cos(x))0 − (sin(x))0 sin(x) sin(x) − sin(x)(sin(x))0 sin(x) − sin(x) sin(x)(sin(x))0 + 2(sin(x))0 cos(x) cos(x) + 2 sin(x)(cos(x))0 cos(x) + 2 sin(x) cos(x)(cos(x))0 = 3(ex )0 ex ex + (sin(x))0 cos(x) cos(x) + 2 sin(x)(cos(x))0 cos(x) − 3(sin(x))0 sin(x) sin(x) + 2(sin(x))0 cos(x) cos(x) + 4 sin(x)(cos(x))0 cos(x) = 3ex ex ex + cos(x) cos(x) cos(x) − 2 sin(x) sin(x) cos(x) − 3 cos(x)) sin(x) sin(x) + 2 cos(x) cos(x) cos(x) − 4 sin(x) sin(x) cos(x) = 3e3x + 3 cos(x) cos(x) cos(x) − 9 sin(x) sin(x) cos(x) = 3e3x + 3 cos(x) cos(x) cos(x) − 3 sin(x) sin(x) cos(x) − 6 sin(x) sin(x) cos(x) = 3e3x + 3 cos(x)(cos2 (x) − sin(x)2 ) − 3 sin(x)(2 sin(x) cos(x)) = 3e3x + 3 cos(x)(cos(2x)) − 3 sin(x)(sin(2x)) = 3e3x + 3 cos(x + 2x) = 3e3x + 3 cos(3x) 1 − cos2 (x) sin(x) Answer: 25. f (x) = 11 sin2 (x) sin(x) = sin(x) f (x) = f 0 (x) = cos(x) √ 26. f (x) = x sin(x) − x2 πex − tan(x) Answer: √ √ ( x sin(x) − x2 )0 (πex − tan(x)) − ( x sin(x) − x2 )(πex − tan(x))0 f (x) = (πex − tan(x))2 √ √ x √ + ( sin(x) x cos(x) − 2x)(πe − tan(x)) − ( x sin(x) − x2 )(πex − sec2 (x)) 2 x = (πex − tan(x))2 0 27. f (x) = (x sin(x))2 Answer: f (x) = x sin(x)x sin(x) = x2 sin(x) sin(x) f 0 (x) = (x2 )0 sin(x) sin(x) + x2 (sin(x))0 sin(x) + x2 sin(x)(sin(x))0 = 2x sin(x) sin(x) + 2x2 cos(x) sin(x) = 2x sin(x) (sin(x) + x cos(x)) II. Find an equation to the tangent line to the curve at the given point. 1. y = xex , (0, 0) Answer: 12 y 0 = ex + xex m = y 0 (0) = e0 + 0 = 1 Tangent line: y = x 2. y = x2 + sin(x), (π, π 2 ) Answer: y 0 = 2x + cos(x) m = y 0 (π) = 2π + cos(π) = 2π − 1 Tangent line: (y − π 2 ) = (2π − 1)(x − π) 3. y = sin(3x), (π/6, 1) Answer: y 0 = 3 cos(3x) m = y 0 (π/6) = 3 cos(3π/6) = 3(0) = 0 Tangent line: (y − 1) = 0(x − π/6) =⇒ y = 1 4. y = Cex , (0, C) Answer: 13 y 0 = Cex m = y 0 (0) = Ce0 = C Tangent line: (y − C) = C(x − 0) 3+x , (2, 1) 1 + x2 Answer: 5. y = (3 + x)0 (1 + x2 ) − (3 + x)(1 + x2 )0 y = (1 + x2 )2 (1 + x2 ) − (3 + x)2x = (1 + x2 )2 1 − 6x − x2 = (1 + x2 )2 0 m = y 0 (2) = 1 − 12 − 4 −15 −3 = = 2 5 25 5 Tangent line: (y − 1) = − 53 (x − 2) 14