Calculus
SOLUTIONS TO SUGGESTED PROBLEMS FROM THE TEXT PART 2
3.5 2 3 4 6 15 18 28 34 36 42 43 44 48 49 3.6 1 2 6 12 17 19 23 30 31 34 38 40 43a 45 51 52 1 4 7 8 10 14 17 19 20 21 22 26 r’(θ) = cosθ – sinθ 2 2 cos θ – sin θ = cos2θ z’= -4sin(4θ) -3cos(2 – 3x) 2 cos(tanθ)/cos θ f’(x) = [-sin(sinx)](cosx) -sinθ w’ = (-cosθ)e y’ = cos(cosx + sinx)(cosx – sinx) 2 T’(θ) = -1 / sin θ x q(x) = e / sin x F(x) = -(1/4)cos(4x)
(a) dy/dt = -(4.9π/6)sin(πt/6) (b) indicates the change in depth of water (a) Graph at end (b) Max on 1 July; 4500; yes; 1 Jan (c) pos 1 April; neg 1 Oct (d) 0 2 2 2 (a) a cosθ + √l – a sin θ (b) i: -2a cm/sec 2 2 2 ii: -a√2 – a / (√l – a /2 cm/sec
28 36 37 42 52a 52b 1 2 4b 5 8 13 17 26a 29 39 41 1 2 3 17 22 29 36 44a 46 49 2 5 8 10 14 16b 21 25 26 27 5.2 1 6 8 10 14
Sketch at end Sketch at the end x = 0: not max/min x = 3/7: local max x = 1: local min
4.2
-1/3 g decreasing near x = x0 g has local min at x1 Sketch at end Sketch at end x = 4; y = 57
Max: 20 at x = 1 Min: -2 at x = -1; x = 8 Max: 2 at x = 0; x = 3 Min: 16 at x = -1; x = 2 (a) f(1) local min; f(0), f(2) local max (b) f(1) global min; f(2) global max
Global min = 2 at x = 1, No global max D=C r = 3B/(2A) Sketch at end Sketch at end. x = L/2 x = 2a Min: -2amps; Max: 2 amps
(a) xy + πy /8 (b) x + y + πy/2 (c) x = 100/(4 + π); y = 200/(4 + π)
2
2t / (t + 1) 1 / (x – 1) cosα/sinα (lnx) + 1 e . 1/x 1 -sin (lnt) / t 2 2 / (√1 – 4t ) 1 / t lnt 2 1 / (1 + 2u + 2u ) 0.8 -1 ‹ x ‹ 1 1 / ((ln 10) x) y=x-1 (a) f’(x) = 0 (b) f is constant 225 -0.12
Critical points: x = -3and x = 2 F(-3) local max; f(2) local min Critical points: x = 0 and x = 2 No local max; f(2) local min
2
4.4
5.1
r = (4/π) ; h = 2(4/π) (±1,0), (±1, 1/2 ) (0.59, 0.35) (a + b)/2 x = 4/5 x = 1, P = (1,1) Upper = 408; Lower = 390
(a) Always same direction: speed up, then slow down. (b) Over = 15, Under = 6
1/3
1/3
4.1
82 meters 250 meters Total change = 15 cm to the right 14.5 miles 8 cm to the
3.5 2 3 4 6 15 18 28 34 36 42 43 44 48 49 3.6 1 2 6 12 17 19 23 30 31 34 38 40 43a 45 51 52 1 4 7 8 10 14 17 19 20 21 22 26 r’(θ) = cosθ – sinθ 2 2 cos θ – sin θ = cos2θ z’= -4sin(4θ) -3cos(2 – 3x) 2 cos(tanθ)/cos θ f’(x) = [-sin(sinx)](cosx) -sinθ w’ = (-cosθ)e y’ = cos(cosx + sinx)(cosx – sinx) 2 T’(θ) = -1 / sin θ x q(x) = e / sin x F(x) = -(1/4)cos(4x)
(a) dy/dt = -(4.9π/6)sin(πt/6) (b) indicates the change in depth of water (a) Graph at end (b) Max on 1 July; 4500; yes; 1 Jan (c) pos 1 April; neg 1 Oct (d) 0 2 2 2 (a) a cosθ + √l – a sin θ (b) i: -2a cm/sec 2 2 2 ii: -a√2 – a / (√l – a /2 cm/sec
28 36 37 42 52a 52b 1 2 4b 5 8 13 17 26a 29 39 41 1 2 3 17 22 29 36 44a 46 49 2 5 8 10 14 16b 21 25 26 27 5.2 1 6 8 10 14
Sketch at end Sketch at the end x = 0: not max/min x = 3/7: local max x = 1: local min
4.2
-1/3 g decreasing near x = x0 g has local min at x1 Sketch at end Sketch at end x = 4; y = 57
Max: 20 at x = 1 Min: -2 at x = -1; x = 8 Max: 2 at x = 0; x = 3 Min: 16 at x = -1; x = 2 (a) f(1) local min; f(0), f(2) local max (b) f(1) global min; f(2) global max
Global min = 2 at x = 1, No global max D=C r = 3B/(2A) Sketch at end Sketch at end. x = L/2 x = 2a Min: -2amps; Max: 2 amps
(a) xy + πy /8 (b) x + y + πy/2 (c) x = 100/(4 + π); y = 200/(4 + π)
2
2t / (t + 1) 1 / (x – 1) cosα/sinα (lnx) + 1 e . 1/x 1 -sin (lnt) / t 2 2 / (√1 – 4t ) 1 / t lnt 2 1 / (1 + 2u + 2u ) 0.8 -1 ‹ x ‹ 1 1 / ((ln 10) x) y=x-1 (a) f’(x) = 0 (b) f is constant 225 -0.12
Critical points: x = -3and x = 2 F(-3) local max; f(2) local min Critical points: x = 0 and x = 2 No local max; f(2) local min
2
4.4
5.1
r = (4/π) ; h = 2(4/π) (±1,0), (±1, 1/2 ) (0.59, 0.35) (a + b)/2 x = 4/5 x = 1, P = (1,1) Upper = 408; Lower = 390
(a) Always same direction: speed up, then slow down. (b) Over = 15, Under = 6
1/3
1/3
4.1
82 meters 250 meters Total change = 15 cm to the right 14.5 miles 8 cm to the