Section 3.3 Increasing and Decreasing Functions and the First Derivative Test 3 Section 3.3 Increasing and Decreasing Functions and the First Derivative Test. f 8 3. 3, Decreasing on:, 3 3 3,,, Decreasing on:, 5. f 7. f 3 Discontinuit: 0 < 0 < < Sign of f: Conclusion: Increasing Decreasing Increasing on, 0 Decreasing on 0, f f g 8 g Critical number: Sign of g: < < g, Decreasing on:, < < g 9. Domain:, 8 ± < < < < < < Sign of : Decreasing Increasing on, Decreasing on,,,. f 3. f 0 f 3 f 0 Critical number: 3 Critical number: < < 3 3 < < Sign of f: f f 3, Decreasing on:, 3 Relative minimum: 3, 9 < < < < Sign of f: f f Conclusion: Increasing Decreasing, Decreasing on:, Relative maimum:, 5
Chapter 3 Applications of Differentiation 5. f 3 3 f 0, < < < < < < Sign of f: f f f Conclusion: Increasing Decreasing Increasing,,, Decreasing on:, Relative maimum:, 0 Relative minimum:, 7 7. f 3 3 3 f 3 3 0, < 0 < < < < Sign of f: f f f Decreasing 0, Decreasing on:, 0,, Relative maimum:, Relative minimum: 0, 0 9. f 5 5 5 f, < < < < < < Sign of f: Conclusion: Increasing Decreasing Increasing,,, Decreasing on:, Relative maimum:, 5 Relative minimum:, 5 f f f
Section 3.3 Increasing and Decreasing Functions and the First Derivative Test 5. f 3 3. f 3 3 3 3 Critical number: 0 < 0 < < Sign of f: Conclusion: Increasing Increasing, No relative etrema f f f 3 f 3 3 Critical number: < < < < Sign of f:, Decreasing on:, Relative minimum:, 0 f f 5. f 5 5 f 5 5, Critical number: 5, < 5 > 5 < < 5 5 < < Sign of f: f f Conclusion: Increasing Decreasing Decreasing on:, 5 5, Relative maimum: 5, 5 7. f f, Discontinuit: 0 < < < 0 < < < < Sign of f: f f f f Conclusion: Increasing Decreasing Decreasing Increasing,,, Decreasing on:, 0, 0, Relative maimum:, Relative minimum:,
Chapter 3 Applications of Differentiation 9. f 9 f 9 9 8 9 Critical number: Discontinuities: 0 3, 3 < < 3 3 < 0 < < 3 3 < < Sign of f: f f f f Conclusion: Increasing Increasing Decreasing Decreasing, 3, 3, 0 Decreasing on: 0, 3, 3, Relative maimum: 0, 0 3. f f 3 3 3, Discontinuit: < < 3 3 < < < < < < Sign of f: f f f f Conclusion: Increasing Decreasing Decreasing Increasing, 3,, Decreasing on: 3,,, Relative maimum: 3, 8 Relative minimum:, 0 33. f cos, 0 < < f sin 0, 5 Sign of f: f 0 < < f f Conclusion: Increasing Decreasing Increasing < < 5 5 < < 0, Decreasing on:, 5, 5, Relative maimum: Relative minimum: 5, 5 3 3,
Section 3.3 Increasing and Decreasing Functions and the First Derivative Test 7 35. f sin sin, 0 < < f sin cos cos cos sin 0, 7, 3, Sign of f: f 0 < < < < 7 7 < < 3 f f f f Conclusion: Increasing Decreasing Increasing Decreasing Increasing 3 < < < < Decreasing on: 0, Relative minima: Relative maima:, 7, 7 7,,,, 3, 3,, 3,,,, 0 37. f 9, 3, 3 (a) f 9 9 (c) 9 9 0 0 f 8 ± 3 (d) Intervals: 3, 3 ±3 3, 3 3, 3 8 0 f f f Decreasing Increasing Decreasing f is increasing when when is negative. f f is positive and decreasing 39. f t t sin t, 0, (a) ft t cos t t sin t (c) tt cos t sin t 0 tt cos t sin t t 0 or t tan t 0 30 0 0 0 0 π f π t t cot t t.889, 5.0870 (graphing utilit) t.889, t 5.0870 (d) Intervals: 0,.889.889, 5.0870 5.0870, ft ft ft Increasing Decreasing Increasing f is increasing when is negative. f f is positive and decreasing when
8 Chapter 3 Applications of Differentiation. f 5 3 3 f g 3 3 for all ±. f 3 3 3, ± f smmetric about origin zeros of f: 0, 0, ±3, 0 3 3 3 3, ± f 0 3 5 (, ) 3 5 No relative etrema Holes at, and, (, ) 3 5 3 3. f c is constant f 0 5. f is quadratic f is a line. 7. f has positive, but decreasing slope In Eercises 9 53, f on,, f on, and f on,. 9. g f 5 5. g f 53. g f 0 g f g f g f 0 g0 f0 g f g0 f0 55. f, undefined,, < f is increasing on,. > f is decreasing on,. Two possibilities for f are given below. (a) 3 5 8 3
Section 3.3 Increasing and Decreasing Functions and the First Derivative Test 387 Section 3.3 Increasing and Decreasing Functions and the First Derivative Test.. f,, 0,, Decreasing on:, Decreasing on:,, 0,. 0, Discontinuit: < < < < < 0 < < Sign of f: Conclusion: Increasing Decreasing Decreasing Increasing Increasing on,, 0, Decreasing on,,, 0 8. h 7 3 h 7 3 33 3 h 0 ±3 < < 3 3 < < 3 3 < < Sign of h: h h h Decreasing Increasing on 3, 3 Decreasing on, 3, 3, 0. ± Discontinuit: 0 < < < 0 < < < < Sign of : Conclusion: Increasing Decreasing Decreasing Increasing Increasing:,,, Decreasing:, 0, 0,
388 Chapter 3 Applications of Differentiation. f 8 0. f 8 0 Critical number: < < < < Sign of f: f f, Decreasing on:, Relative minimum:, f 8 f 8 0 Critical number: Sign of f: < < < < f Conclusion: Increasing Decreasing, Decreasing on:, Relative maimum:, f. f 3 5 f 3 3 0, < 0 < < < < Sign of f: f f f Conclusion: Increasing Decreasing Increasing Increasing on, 0,, Decreasing on 0, Relative maimum: 0, 5 Relative minimum:, 7 8. f f 3, 0 < < < 0 < < Sign of f: f f f Conclusion: Increasing Decreasing Increasing,, 0, Decreasing on:, 0 Relative maimum:, 0 Relative minimum: 0,
Section 3.3 Increasing and Decreasing Functions and the First Derivative Test 389 0. f 3. f 3 f 3 3 3 8 Critical number: < < < < Sign of f: f f f 3 3 3 3 Critical number: Sign of f: 0 < f 0 < < f, Decreasing on:, Relative minimum:, 0, Decreasing on:, 0 Relative minimum: 0,. f 3. f 3 3 Critical number: < < < < Sign of f: Conclusion: Increasing Increasing, No relative etrema f f f 3 f 3 3, > 3, < 3 Critical number: 3 < < 3 3 < < Sign of f: f f 3, Decreasing on:, 3 Relative minimum: 3, 8. f f Discontinuit: < < < < Sign of f: f f Conclusion: Increasing Increasing,,, No relative etrema
390 Chapter 3 Applications of Differentiation 30. f 3 3 f 3 3 Critical number: Discontinuit: 0 < < < 0 < < Sign of f: Decreasing, 0 f Decreasing on:,, 0, Relative minimum:, f f 3. f 3 f 3 3 0 Discontinuit: < < < < Sign of f: f f Conclusion: Increasing Increasing,,, No relative etrema 3. f sin cos sin, 0 < < f cos 0, 3, 5, 7 Sign of f: 0 < < < < 7 < < 5 < < 3 < < Conclusion: Increasing Decreasing Increasing Decreasing Increasing Decreasing on: 0, Relative maima:, 3, 3,, 5 f, 5, 5, 7,, 7, Relative minima: 3,, 7, f 3 f 5 f 7 f
Section 3.3 Increasing and Decreasing Functions and the First Derivative Test 39 3. f sin cos, 0 < < f cos sin cos 0, 3 Sign of f: f 0 < < Conclusion: Increasing Decreasing Increasing < < 3 f 3 < < f 0, Relative maimum: Decreasing on: Relative minimum: 3, 3,, 3,, 38. f 05 3, 0, 5 5 3 (a) f 3 5 f 3 3 3 (c) 5 3 3 0 Critical number: 3 (d) Intervals: 0, 3 f 3, 5 f Increasing Decreasing f is increasing when when is negative. f f is positive and decreasing 0. f cos, 0, (a) f sin 8 f π π 3π π (c) sin 0 (d) Intervals: sin 0, f Critical number: Increasing f is increasing when, f Increasing f is positive.
39 Chapter 3 Applications of Differentiation. f t cos t sin t sin t gt, < t <. f is a line of slope f. ft sin t cos t sin t f smmetric with respect to -ais zeros of f: ± Relative maimum: 0, Relative minimum:,,, 3 3. f is a th degree polnomial f is a cubic polnomial. 8. f has positive slope 3 3 3 In Eercises 50 5, f on,, f on, and f on,. 50. g 3f 3 5. g f 5. g f 0 g 3f g f g f 0 g5 3f5 g0 f0 g8 f 5. Critical number: 5 58. f.5 f is decreasing at. f 3 f is increasing at. 5, f 5 is a relative minimum. st.9sin t (a) vt 9.8sin t speed If, vt 9.8t. the speed is maimum, 9.8 sin t 0. C (a) 3t 7 t 3, t 0 t 0 0.5.5.5 3 Ct 0 0.055 0.07 0.8 0.7 0.7 0.7 The concentration seems greater near t.5 hours. 0.5 (c) C 7 t 3 3 3t3t 7 t 3 37 t 3 7 t 3 0 3 0 The concentration is greatest when t.38 hours. C 0 when t 3 3.38 hours. B the First Derivative Test, this is a maimum.
Section 3.3 Increasing and Decreasing Functions and the First Derivative Test 393. P. 5000, 0 35,000 0 < <,00,00 < < 35,000 0,000 Sign of P: P P.,00 0,000 0 Increasing when 0 < <,00 hamburgers. Decreasing when,00 < < 35,000 hamburgers. P. R 0.00T T 00 (a) R 0.00T 3 0.00T T 00 0 5 T 0, R 8.3 00 00 5 The minimum resistance is approimatel R 8.37 at T 0.. f sin3 cos3 The maimum value is approimatel.7. You could use calculus b finding f and then observing that the maimum value of f occurs at a point where f 0. For instance, f0.5 0, and f 0.5.7. 8. (a) Use a cubic polnomial f a 3 3 a a a 0. f 3a 3 a a 0, 0:, 000: (c) The solution is 0 a 0 f 0 0 0 a f0 0 000 a 3 a f 000 0 8a 3 8a f 0 a 0 a 0, a 375, a 5 3 f 5 3 375. (d) 00 (, 000) 3 8 (0, 0) 00