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Additional resources for Calculus of a Single Variable
Y ϭ x 2π 18. s͑t͒ ϭ 1 t2 Ϫ 2 x 2 1 π 24. f ͑x͒ ϭ x͑3x Ϫ 9͒3 2 2π y = sin 2 x three thirteen. y ϭ Ί 6x 2 ϩ 1 x π y (b) y = sin 3x 1 10. f ͑t͒ ϭ ͑9t ϩ 2͒2͞3 four 12. g͑x͒ ϭ Ί9 Ϫ 4x 1 tϪ3 π 2 2π −2 2 eleven. f ͑t͒ ϭ Ί5 Ϫ t 19. f ͑t͒ ϭ π y forty four. (a) eight. y ϭ 2͑6 Ϫ x 2͒5 nine. g͑x͒ ϭ 3͑4 Ϫ 9x͒ 17. y ϭ x −2 In routines 7– 36, locate the by-product of the functionality. y = sin 2x 2 y = sin x 1 four. y ϭ three tan͑ x 2͒ y (b) 2 three. y ϭ Ίx3 Ϫ 7 6. y ϭ sin 137 The Chain Rule In workouts 67–74, review the by-product of the functionality on the given aspect. Use a graphing software to make sure your outcome. sixty seven. s͑t͒ ϭ Ίt2 ϩ 6t Ϫ 2, ͑3, five͒ ͑2, 2͒ five 1 f ͑x͒ ϭ three Ϫ2, Ϫ , x Ϫ2 2 1 1 f ͑x͒ ϭ 2 , four, ͑x Ϫ 3x͒2 sixteen 3t ϩ 2 f ͑t͒ ϭ , ͑0, Ϫ2͒ tϪ1 xϩ1 f ͑x͒ ϭ , ͑2, three͒ 2x Ϫ three y ϭ 26 Ϫ sec three 4x, ͑0, 25͒ 2 1 y ϭ ϩ Ίcos x, , x 2 five 3x three ϩ 4x, sixty eight. y ϭ Ί sixty nine. 70. seventy one. seventy two. seventy three. seventy four. 138 bankruptcy 2 Differentiation In routines seventy five – eighty two, (a) locate an equation of the tangent line to the graph of f on the given element, (b) use a graphing application to graph the functionality and its tangent line on the aspect, and (c) use the by-product function of the graphing application to substantiate your effects. functionality seventy six. f ͑x͒ ϭ 1 2 three xΊx seventy seven. y ϭ ͑ 4x3 ͑4, five͒ ͑2, 2͒ ͑Ϫ1, 1͒ ͑1, four͒ ͑, zero͒ Ί2 ,Ϫ 2 four ,1 four ,2 four ϩ5 ϩ three͒ 2 seventy eight. f ͑x͒ ϭ ͑9 Ϫ x2͒2͞3 seventy nine. f ͑x͒ ϭ sin 2x eighty one. f ͑x͒ ϭ tan 2 x eighty two. y ϭ 2 tan3 x 3t2 Ίt2 ϩ 2t Ϫ 1 eighty four. f ͑x͒ ϭ Ίx ͑2 Ϫ x͒2, ninety nine. f ͑x͒ ϭ cos eighty five. s ͑t͒ ϭ three 86. y ϭ ͑t2 Ϫ 9͒Ίt ϩ 2, , 0, 34 25 − x 2 f (x) = x −3 −2 −1 y 104. four three 2 three x x −2 −2 −3 −4 four In workouts a hundred and five and 106, the connection among f and g is given. clarify the connection among fЈ and gЈ. a hundred and five. g͑x͒ ϭ f ͑3x͒ 2 106. g͑x͒ ϭ f ͑x 2͒ (1, 1) 1 6 1 2 three four y 103. ⎪x⎪ 2 − x2 three (3, four) four x three 88. Bullet-nose curve 2 −4 four three 2 three four 2 y 102. −2 −3 eight − 6 −4 − 2 y one hundred and one. x y four In routines 101–104, the graphs of a functionality f and its spinoff fЈ are proven. Label the graphs as f or fЈ and write a brief paragraph declaring the factors you utilized in making your choice. To print an enlarged replica of the graph, visit the web site www. mathgraphs. com. −2 y 6 three 2 recognized Curves In routines 87 and 88, locate an equation of the tangent line to the graph on the given aspect. Then use a graphing software to graph the functionality and its tangent line within the related viewing window. f (x) = ͑0, 1͒ , Ί3 6 WRITING approximately ideas ͑2, Ϫ10͒ 87. best half circle x2, a hundred. g͑t͒ ϭ tan 2t, ͑4, eight͒ ͑4 Ϫ 2t͒Ί1 ϩ t four ͑x ϩ 2͒3 ninety six. f ͑x͒ ϭ sec 2 x ninety four. f ͑x͒ ϭ 21, 32 , 1 xϪ6 1 sixty four ninety seven. h͑x͒ ϭ nine ͑3x ϩ 1͒3, ͑1, nine ͒ 1 1 , zero, ninety eight. f ͑x͒ ϭ Ίx ϩ four 2 In routines eighty three– 86, (a) use a graphing application to discover the spinoff of the functionality on the given aspect, (b) locate an equation of the tangent line to the graph of the functionality on the given element, and (c) use the application to graph the functionality and its tangent line within the similar viewing window. eighty three. g͑t͒ ϭ ninety three. f ͑x͒ ϭ ninety two. f ͑x͒ ϭ 4͑x 2 Ϫ 2͒3 In routines 97–100, assessment the second one by-product of the functionality on the given element.