THE CHAIN RULE AND PARAMETRIC EQUATIONS

CHAPTER 2 ADDITIONAL AND ADVANCED EXERCISES

3.5 THE CHAIN RULE AND PARAMETRIC EQUATIONS

19. pœÈ3 œt (3t)^"Î# Ê œ ^"(3t)^"Î# (3 œ t) ^"(3t)^"Î#œ ^"

# #

dp

dt dt

d

2 3 t

† _È

20. qœÈ2r œr^# a2rr^#b^"Î# Ê ^dq_dr œ^"_#a2rr^#b^"Î# _dr^d a2rr^#bœ^"_#a2rr^#b^"Î#(22r)œ ^{" r}

2r r

† _{È #}

21. sœ ₃⁴₁ sin 3t₅⁴₁ cos 5t Ê ^ds_dt œ ₃⁴₁ cos 3t†_dt^d (3t)₅⁴₁( sin 5t) †_dt^d (5t)œ ⁴₁ cos 3t⁴₁ sin 5t (cos 3t sin 5t)

œ ⁴₁

22. sœsinˆ^{3 t}_#¹‰cosˆ^{3 t}_#¹‰ Ê ^ds_dt œcosˆ^{3 t}_#¹‰†_dt^d ˆ^{3 t}_#¹‰sinˆ^{3 t}_#¹‰†_dt^d ˆ^{3 t}_#¹‰œ³₂¹ cosˆ^{3 t}_#¹‰³₂¹ sinˆ^{3 t}_#¹‰ cos œ ³₂¹ˆ ^{3 t}_#¹ sin ^{3 t1}_# ‰

23. rœ(csc cot ) ))^" Ê ^dr_d) œ (csc cot ) ))^# _d^d₎(csc cot ) ))œ ^csc _{(csc cot} ^)cot ₎^)csc₎₎#^#) œ^{csc (cot} _{(csc cot} ⁾ ₎^{)csc )}₎₎#⁾

œ _{csc cot} ^csc ₎⁾ ₎

24. rœ (sec tan ) ))^" (sec tan Ê _d^dr₎ œ ) ))^# _d^d₎(sec tan ) ))œ ^sec _{(sec tan} ^)tan ₎^)sec₎₎#^#) œ^{sec (tan} _{(sec tan} ⁾ ₎^{)sec )}₎₎#⁾

œ _{sec tan} ^sec ₎⁾ ₎

25. yœx sin x^{# %} x cos^#x Ê ^dy_dx œx ^# _dx^d asin x^% bsin x^% †_dx^d a bx^# x _dx^d acos^#xbcos^#x†_dx^d (x) x 4 sin x (sin x) 2x sin x x 2 cos x (cos x) cos x

œ ^#ˆ ^$ _dx^d ‰ ^% ˆ ^$ †_dx^d ‰ ^# x 4 sin x cos x 2x sin x x 2 cos x ( sin x) cos x œ ^#a ^$ b ^% aa ^$ b b ^#

4x sin x cos x 2x sin x 2x sin x cos x cos x

œ ^{# $ % $ #}

26. yœ^"_x sin^&x₃^x cos x ^$ Ê y^wœ ^"_{x dx}^d asin^&xbsin^&x†_dx^d ˆ ‰^"_{x 3 dx}^{x d} acos x^$ bcos x^$ †_dx^d ˆ ‰₃^x

5 sin x cos x sin x 3 cos x ( sin x) cos x

œ ^"_xa ^' ba ^& bˆ_x^"#‰₃^xaa ^# b ba ^$ bˆ ‰^"₃

sin x cos x sin x x cos x sin x cos x

œ ⁵_x ^' _x^"# ^{& #} ₃^{" $}

27. yœ₂₁^" (3x2)⁽ˆ4_#^"_x#‰^" Ê _dx^dyœ ₂₁⁷ (3x2)^'†_dx^d (3x 2) ( 1) 4ˆ _#^"_x#‰^#†_dx^d ˆ4_#^"_x#‰ (3xœ ₂₁⁷ 2) 3 ( 1) 4 _{x x} œ(3x2)

x 4

' " " ' "

# #

† ˆ _#‰ ˆ ‰_$

$ #

Š _{# #}^" ‹ x

28. yœ(52x)^$"₈ˆ_x²1 ‰^% Ê _dx^dyœ 3(52x)^%( 2) ₈⁴ˆ_x²1‰ ˆ^$ _x²_#‰œ6(52x)^%ˆ ‰ ˆ_x^"_{# x}²1‰^$ œ ₍₅⁶_2x)% _x¹

$

#

Š²_x ‹

29. yœ(4x3) (x^% 1)^$ Ê ^dy_dx œ(4x3) ( 3)(x^% 1)^%†_dx^d (x 1) (x 1)^$(4)(4x3)^$†_dx^d (4x3) (4xœ 3) ( 3)(x^% 1)^%(1) (x 1)^$(4)(4x3) (4)^$ œ 3(4x3) (x^% 1)^%16(4x3) (x^$ 1)^$ 3(4xœ ^(4x_(x1)³⁾%^$c 3) 16(x1)dœ^(4x_(x^{3) (4x$}₁₎%⁷⁾

30. yœ(2x5)^"ax^#5x b^' Ê ^dy_dxœ(2x5)^"(6) xa ^#5x (2xb^& 5) ax^#5x ( 1)(2xb^' 5)^#(2) 6 xœ a ^#5xb^{&2 x}_(2x^{a #}₅₎^5x#^b'

31. h(x)œx tan 2ˆ Èx‰ Ê7 h (x)^w œx _dx^d ˆtan 2xˆ ^"Î#‰‰tan 2xˆ ^"Î#‰†_dx^d (x)0

x sec 2x 2x tan 2x x sec 2 x tan 2 x x sec 2 x tan 2 x

œ ^#ˆ ^"Î#‰†_dx^d ˆ ^"Î#‰ ˆ ^"Î#‰œ ^#ˆ È ‰†_È^"_x ˆ È ‰œÈ ^#ˆ È ‰ ˆ È ‰

32. k(x)œx sec^# ˆ ‰^"_x Ê k (x)^w œx ^# _dx^d ˆsec ^"_x‰secˆ ‰^"_x †_dx^d a bx ^# œx sec^# ˆ ‰^"_x tanˆ ‰^"_x †_dx^d ˆ ‰^"_x 2x secˆ ‰^"_x

x sec tan 2x sec 2x sec sec tan

œ ^# ˆ ‰^"_x ˆ ‰ ˆ^"_x † _x^"#‰ ˆ ‰^"_x œ ˆ ‰^"_x ˆ ‰^"_x ˆ ‰^"_x

33. f( )) œˆ₁^{sin sin} _cos ) ‰ Ê f ( )) œ2ˆ_{1 cos} ) ‰ _d^d ˆ₁^sin _cos ) ‰œ₁^{2 sin} _cos ) ⁽¹ cos )(cos )_{(1 cos} (sin )( sin )₎

) ) ) ) ) ) ) ) ) )

# w † † #

œ (2 sin ) cos cos sin œ œ

(1 cos ) (1 cos ) (1 cos )

(2 sin ) (cos 1) 2 sin

) ) ) )

) ) )) ))

a b

^{# #}

$ $ #

34. g(t)œˆ¹_{sin t}^{cos t}‰" Ê g (t)w œ ˆ¹_{sin t}^{cos t}‰# _dt^d ˆ¹_{sin t}^{cos t}‰œ ₍₁^{sin t}_{cos t)} (sin t)( sin t)_{(sin t)}( cos t)(cos t)

"

† ^# #† #

œ ^{a sin t}₍₁^#_{cos t)}^{cos t}# ^{cos t#b} œ ₁^"_{cos t}

35. rœsina b)^# cos (2 ) ) Ê _d^dr₎ œsina b)^# ( sin 2 ) ) _d^d₎(2 )) cos (2 ) cos) a a b)^# b†_d^d₎a b)^# sin ( sin 2 )(2) (cos 2 ) cos (2 ) 2 sin sin ( ) 2 cos (2 ) cos œ a b)^# ) ) a a b)^# b ) œ a b)^# # ) ) ) a b)^#

36. rœŠsec È)‹tanˆ ‰^"₎ Ê ^dr_d) œŠsec È)‹ˆ sec ^{# "}₎‰ ˆ₎^"#‰tanˆ ‰^"₎ sec Š È) tan È)‹ Š_#È^"₎‹

sec sec tan sec tan sec

œ ₎^"# È) ^#ˆ ‰^"_{) #È}^"₎ ˆ ‰^"₎ È) È)œŠ È)‹ ”^{tan È}_È^)tan₎ ^{ˆ ‰"}) ^sec₎^##^{ˆ ‰"}) •

#

37. qœsinŠ _t^t ₁‹ Ê ^dq_dt œcosŠ _t^t ₁‹ _dt^d Š _t^t ₁‹œcosŠ _t^t ₁‹ ^{t 1 (1) t t 1}

t 1

È È È È

È ˆÈ ‰

ˆÈ ‰

† † ^†dtd #

cosœ Š_Èt^t₁‹†^{Èt 1}_t1^{2È tt 1} œcosŠ_Èt^t₁‹ Š^2(t_2(t¹⁾₁₎$Î#^t‹œ Š_2(t^t₁₎²$Î#‹cosŠ_Èt^t₁‹

38. qœcotˆ^{sin t}_t ‰ Ê ^dq_dt œ csc^#ˆ^{sin t}_t ‰†_dt^d ˆ^{sin t}_t ‰œ ˆ csc^#ˆ^{sin t}_t ‰‰ ˆ^{t cos t}_t# ^{sin t}‰

39. yœsin ( t^# 1 2) Ê ^dy_dt œ2 sin ( t1 2)†_dt^d sin ( t1 2)œ2 sin ( t1 2) cos ( t† 1 2)†_dt^d ( t1 2) 2 sin ( t 2) cos ( t 2)

œ 1 1 1

40. yœsec^#1t Ê ^dy_dt œ(2 sec t)1 †_dt^d (sec t)1 œ(2 sec t)(sec t 1 1 tan t)1 †_dt^d ( t)1 œ2 1sec^#1t tan t1

41. yœ(1cos 2t)^% Ê ^dy_dt œ 4(1cos 2t)^&†_dt^d (1cos 2t)œ 4(1cos 2t)^&( sin 2t) †_dt^d (2t)œ ₍₁^{8 sin 2t}_{cos 2t)}&

42. yœˆ1cotˆ ‰_#^t ‰^# Ê ^dy_dt œ 2 1ˆ cotˆ ‰_#^t ‰^$†_dt^dˆ1cotˆ ‰_#^t ‰œ 2 1ˆ cotˆ ‰_#^t ‰^$†ˆcsc^#ˆ ‰_#^t ‰†_dt^dˆ ‰_#^t œ ^csc

cot

# #

#

ˆ ‰ ˆ ‰

t ˆ1 t ‰^$

43. yœsin cos (2ta 5) b Ê ^dy_dt œcos (cos (2t5))†_dt^d cos (2t5)œcos (cos (2t5)) ( sin (2t† 5))†_dt^d (2t5) 2 œ cos (cos (2t5))(sin (2t5))

44. yœcos 5 sinˆ ˆ ‰₃^t ‰ Ê ^dy_dt œ sin 5 sinˆ ˆ ‰₃^t ‰†_dt^d ˆ5 sinˆ ‰₃^t ‰œ sin 5 sinˆ ˆ ‰₃^t ‰ ˆ5 cosˆ ‰₃^t ‰†_dt^dˆ ‰₃^t sin 5 sin cos

œ ⁵₃ ˆ ˆ ‰₃^t ‰ ˆ ˆ ‰₃^t ‰

45. yœ<1tan^%ˆ ‰₁^t_# ‘^$ Ê ^dy_dt œ3 1< tan^%ˆ ‰₁^t_# ‘^#†_dt^d <1tan^%ˆ ‰₁^t_# ‘œ3 1< tan^%ˆ ‰₁^t_# ‘ <^# 4 tan^$ˆ ‰₁^t_# †_dt^d tanˆ ‰₁^t_# ‘

12 1 tan tan sec 1 tan tan sec

œ < ^%ˆ ‰₁^t_# ‘ <^{# $}ˆ ‰₁^t_# ^#ˆ ‰₁^t_# †₁^"_#‘œ< ^%ˆ ‰₁^t_# ‘ <^{# $}ˆ ‰₁^t_# ^#ˆ ‰₁^t_# ‘

46. yœ^"₆c1cos (7t) ^# d^$ Ê ^dy_dt œ ₆³c1cos (7t)^# d^#†2 cos (7t)( sin (7t))(7) œ 7 1c cos (7t) (cos (7t) sin (7t))^# d^#

47. yœa1cos ta b^# b^"Î# Ê ^dy_dt œ^"_#a1cos ta b^# b^"Î#†_dt^da1cos ta b^# bœ ^"_#a1cos ta b^# b^"Î#ˆsin ta b^# †_dt^d a bt^# ‰ 1œ ^"_#a cos ta b^# b^"Î#asin ta b^# b†2tœ _È1^{t sin t}_{cos t}^{a b#}_{a b#}

48. yœ4 sinŒÉ1Èt 9 Ê ^dyœ4 cosŒÉ1Èt9 ŒÉ1Èt9œ4 cosŒÉ1Èt9 ˆ1Èt‰

dt dt dt

d d

1 t

† † ^" †

#É È

œ ^{2 cos 1 t} œ ^{cos 1 t}

1 t 2 t t t

ŒÉ È ŒÉ È

É È È É È

†

49. yœˆ1^"_x‰^$ Ê y^wœ3 1ˆ ^"_x‰ ˆ^# _x^"_#‰œ _x³_#ˆ1^"_x‰^# Ê y^wwœ ˆ _x³_#‰†_dx^d ˆ1^"_x‰^#ˆ1^"_x‰^#†_dx^d ˆ ‰_x³_#

2 1 1 1 1 1 1

œ ˆ _x³_#‰ ˆ ˆ _x^"‰ ˆ_x^"_#‰‰ˆ ‰ ˆ_x⁶_{$ x}^"‰^#œ _x⁶_%ˆ _x^"‰_x⁶_$ˆ _x^"‰^#œ _x⁶_$ˆ _x^"‰ ˆ_x^" _x^"‰

1 1

œ _x⁶$ˆ _x^"‰ ˆ ²_x‰

50. yœˆ1Èx‰^" Ê y^wœ ˆ1Èx‰^#ˆ^"_#x^"Î#‰œ ^"_#ˆ1Èx‰^#x^"Î#

Ê y^wwœ ^"_#’ˆ1Èx‰^#ˆ^"_#x^$Î#‰x^"Î#( 2) 1 ˆ Èx‰^$ˆ^"_#x^"Î#‰“

x 1 x x 1 x x 1 x x 1 x 1

œ " " œ " "

# # $Î# # " $ # " $ # "Î#

’ ˆ È ‰ ˆ È ‰ “ ˆ È ‰ < ˆ È ‰ ‘

1 x 1 1 x

œ _#^"_xˆ È ‰^$Š_#È^"_x ^"_# ‹œ _#^"_xˆ È ‰^$Š_#³_#È^"_x‹

51. yœ^"₉ cot (3x1) Ê y^wœ ^"₉ csc (3x^# 1)(3)œ ^"₃ csc (3x^# 1) Ê y^wwœ ˆ ₃²‰(csc (3x1)†_dx^d csc (3x1)) csc (3x 1)( csc (3x 1) cot (3x 1) (3x 1)) 2 csc (3x 1) cot (3x 1)

œ ²₃ †_dx^d œ ^#

52. yœ9 tanˆ ‰^x₃ Ê y^wœ9 secˆ ^#ˆ ‰^x₃ ‰ ˆ ‰₃^" œ3 sec^#ˆ ‰^x₃ Ê y^wwœ3 2 sec† ˆ ‰ ˆ^x₃ secˆ ‰^x₃ tanˆ ‰^x₃ ‰ ˆ ‰₃^" œ2 sec^#ˆ ‰^x₃ tanˆ ‰^x₃ 53. g(x)œÈx Ê g (x)^w œ _#Èx^" Ê g(1)œ1 and g (1)^w œ ^"_#; f(u)œu^& Ê1 f (u)^w œ5u ^% Ê f (g(1))^w œf (1)^w œ5;

therefore, (f‰g) (1)^w œf (g(1)) g (1)^w † ^w œ5†^"_# œ _#⁵

54. g(x)œ(1x)^" Ê g (x)^w œ (1 x)^#( 1) œ _{(1 x)}^" # Ê g( 1) œ ^"_# and g ( 1)^w œ ^"₄; f(u)œ 1 ^"_u f (u) f (g( 1)) f 4; therefore, (f g) ( 1) f (g( 1))g ( 1) 4 1

Ê ^w œ _u^"# Ê ^w œ ^wˆ ‰^"_# œ ‰ ^w œ ^{w w} œ †^"₄ œ

55. g(x)œ5Èx Ê g (x)^w œ Ê g(1)œ5 and g (1)^w œ ; f(u)œcotˆ ‰ Ê f (u)^w œ csc^#ˆ ‰ ˆ ‰

#5 #5 u u

x 10 10 10

È 1 1 1

csc f (g(1)) f (5) csc ; therefore, (f g) (1) f (g(1))g (1)

œ ₁₀^{1 #}ˆ ‰¹₁₀^u Ê ^w œ ^w œ ₁₀^{1 #}ˆ ‰_#¹ œ ₁₀¹ ‰ ^w œ ^{w w} œ ₁₀¹ †_#⁵ = ¹₄

56. g(x)œ1x Ê g (x)^w œ1 Ê gˆ ‰^"₄ œ ₄¹ and g^wˆ ‰^"₄ œ1; f(u)œ u sec u ^# Ê f (u)^w œ 1 2 sec u sec u tan u†

1 2 sec u tan u f g f 1 2 sec tan 5; therefore, (f g) f g g 5

œ ^# Ê ^wˆ ˆ ‰^"₄ ‰œ ^wˆ ‰₄¹ œ ^{# 1}_{4 4}¹ œ ‰ ^wˆ ‰^"₄ œ ^wˆ ˆ ‰^"₄ ‰ ˆ ‰^{w "}₄ œ 1

57. g(x)œ10x^# Êx 1 g (x)^w œ20x Ê1 g(0)œ1 and g (0)^w œ1; f(u)œ_u#^2u₁ Ê f (u)^w œ ^{u 1 (2)}_{u 1}^(2u)(2u)

#

# #

a b a b

f (g(0)) f (1) 0; therefore, (f g) (0) f (g(0))g (0) 0 1 0 œ a_u^2u_#^#₁b²# Ê ^w œ ^w œ ‰ ^w œ ^{w w} œ † œ

58. g(x)œ _x^"# Ê1 g (x)^w œ _x²$ Ê g( 1) œ0 and g ( 1)^w œ2; f(u)œˆ_u^u₁¹‰^# Ê f (u)^w œ2ˆ_u^u₁¹‰ _du^d ˆ_u^u₁¹‰

2 f (g( 1)) f (0) 4; therefore,

œ ˆ^u_u¹₁‰†^(u1)(1)_(u1)^(u# ¹⁾⁽¹⁾ œ ^2(u_(u¹⁾⁽²⁾₁₎$ œ _(u^4(u₁₎¹⁾$ Ê ^w œ ^w œ (f‰g) ( 1)^w œf (g( 1))g ( 1)^{w w} œ ( 4)(2)œ 8

59. (a) yœ2f(x) 2f (x) Ê ^dy_dxœ ^w Ê ^dy_dx¹ œ2f (2)^w œ2ˆ ‰₃^" œ ₃²

x=2

(b) yœf(x)g(x) f (x)Ê ^dy_dxœ ^w g (x) ^w Ê ^dy_dx¹ œf (3)^w g (3)^w œ2 5

x=3 1

(c) yœf(x) g(x) f(x)g (x)† Ê ^dy_dxœ ^w g(x)f (x) ^w Ê ^dy_dx¹ œf(3)g (3)^w g(3)f (3)^w œ3 5† ( 4)(2 )œ158

x=3 1 1

(d) yœ_g(x)^f(x) Ê ^dy_dx œ^{g(x)f (x)w}_[g(x)]^{f(x)g (x)}# ^w Ê ^dy_dx œ ^{g(2)f (2)w}_[g(2)]^{f(2)g (2)}# ^w œ ⁽²⁾ #^{(8)( 3)} œ ³⁷₆

"

¹ # x=2

ˆ ‰₃

(e) yœf(g(x)) f (g(x))g (x) Ê ^dy_dx œ ^{w w} Ê ^dy_dx¹ œf (g(2))g (2)^{w w} œf (2)( 3)^w œ₃^"( 3) œ 1

x=2

(f) yœ(f(x))^"Î# Ê ^dy_dx œ^"_#(f(x))^"Î#†f (x)^w œ _#È^{f (x)w}_f(x) Ê ^dy_dx¹ œ _#È^{f (2)w}_f(2)œ _#^{ˆ ‰}_È^"₈ œ _6È^"₈ œ_1#^"_È2 œ^È₂₄²

x=2

3

(g) yœ(g(x))^# Ê ^dy_dx œ 2(g(x))^$†g (x) ^w Ê ^dy_dx¹ œ 2(g(3))^{$ w}g (3)œ 2( 4)^$†5œ ₃⁵_#

x=3

(h) yœa(f(x))^#(g(x))^#b^"Î# Ê ^dy_dx œ^"_#a(f(x))^#(g(x))^#b^"Î#a2f(x) f (x)† ^w 2g(x) g (x)† ^w b

Ê ^dy_dx¹ œ a(f(2)) (g(2)) b a2f(2)f (2)2g(2)g (2)bœ a8 2 b ˆ2 8 ₃2 2 ( 3) ‰

x=2

" " "

# # # "Î# w w # # # "Î# † † † †

œ _3È⁵₁₇

60. (a) yœ5f(x)g(x) 5f (x)Ê ^dy_dx œ ^w g (x) ^w Ê ^dy_dx¹ œ5f (1)^w g (1)^w œ5ˆ₃^"‰ˆ₃⁸‰œ1

x=1

(b) yœf(x)(g(x)) f(x) 3(g(x)) g (x)^$ Ê ^dy_dxœ a ^{# w} b(g(x)) f (x) ^{$ w} Ê ^dy_dx¹ œ $f(0)(g(0)) g (0)^{# w} (g(0)) f (0)^{$ w}

x=0

3(1)(1)œ ^#ˆ ‰^"₃ (1) (5)^$ œ6

(c) yœ_g(x)^f(x)₁ Ê ^dy_dx œ^(g(x)_(g(x)^{1)f (x)w} ₁₎^{f(x) g (x)}# ^w Ê ^dy_dx¹ œ ^(g(1)_(g(1)^{1)f (1)w} ₁₎^{f(1)g (1)}# ^w x=1

œ ^{( 4 " )ˆ}_{( 4 1)}^"3^‰ #^(3)ˆ38^‰œ1

(d) yœf(g(x)) f (g(x))g (x) Ê ^dy_dx œ ^{w w} Ê ^dy_dx¹ œf (g(0))g (0)^{w w} œf (1)^w ˆ ‰₃^" œ ˆ ₃^"‰ ˆ ‰₃^" œ ₉^"

x=0

(e) yœg(f(x)) g (f(x))f (x) Ê ^dy_dx œ ^{w w} Ê ^dy_dx¹ œg (f(0))f (0)^{w w} œg (1)(5)^w œ ˆ ₃⁸‰(5)œ ⁴⁰₃

x=0

(f) yœax^""f(x)b^# Ê ^dy_dx œ 2 xa ^""f(x)b^$a11x^"!f (x) ^w b Ê ^dy_dx¹ œ 2(1f(1))^$a11f (1)^w b

x=1

2(1œ 3)^$ˆ11^"₃‰œ ˆ ₄²_$‰ ˆ ‰³²₃ œ ^"₃

(g) yœf(xg(x)) f (xÊ ^dy_dxœ ^w g(x)) 1a g (x) ^w b Ê ^dy_dx¹ œf (0^w g(0)) 1a g (0)^w bœf (1) 1^w ˆ ₃^"‰

x=0

œ ˆ ^"₃‰ ˆ ‰₃⁴ œ ₉⁴

61. ^ds_dt œ ^ds_d)†^d_dt⁾: sœcos ) Ê _d^ds₎ œ sin ) Ê ^ds_d)¸₎₌₃ œ sinˆ ‰³¹ œ1 so that ^ds_dt œ ^ds_d)†^d_dt⁾ œ1 5† œ5

21 #

62. ^dy_dt œ ^dy_dx†^dx_dt: yœx^#7x Ê5 ^dy_dxœ2x Ê7 ^dy_dx¹ œ9 so that ^dy_dt œ ^dy_dx†^dx_dt œ9†₃^" œ3

x=1

63. With yœx, we should get ^dy_dxœ1 for both (a) and (b):

(a) yœ Ê^u₅ 7 _du^dyœ ₅^"; uœ5x35 Ê ^du_dx œ5; therefore, _dx^dy œ _du^dy†^du_dx œ₅^" †5œ1, as expected (b) yœ 1 ^"_u Ê _du^dyœ _u^"#; uœ(x1)^" Ê _dx^du œ (x 1)^#(1)œ _(x^"₁₎#; therefore _dx^dyœ _du^dy†_dx^du

(x 1) 1, again as expected œ ^"_u# †_(x^"₁₎# œ_a(x^"₁₎" #_b †_(x^"₁₎# œ ^#†_(x^"₁₎# œ

64. With yœx^$Î#, we should get ^dy_dx œ_#³x^"Î# for both (a) and (b):

(a) yœu 3u ; ^$ Ê ^dy_duœ ^# uœÈx Ê _dx^duœ _#È^"_x; therefore, ^dy_dx œ ^dy_du†_dx^du œ3u^#†_#È^"_x œ3ˆÈx‰^#†_#È^"_x œ _#³Èx, as expected.

(b) yœÈu Ê ^dy_duœ ; uœx 3x ; Ê _dxœ therefore, ^dy_dx œ^dy_{du dx}œ 3x œ 3x œ x ,

u u

du du 3

x

" " "

# $ # # # # "Î#

# #

È † È † È _$ †

again as expected.

65. yœ2 tanˆ ‰¹₄^x Ê _dx^dyœˆ2 sec ^{# 1}₄^x‰ ˆ ‰¹₄ œ ¹_# sec^{# 1}₄^x

(a) ^dy_dx¹ sec ˆ ‰₄ slope of tangent is 2; thus, y(1) 2 tanˆ ‰₄ 2 and y (1) tangent line is

x=1œ ¹_# ^{# 1} œ1 Ê œ ¹ œ ^w œ1 Ê

given by y œ2 1(x1) Ê yœ1x 2 1

(b) y^wœ _#¹ sec^#ˆ ‰¹₄^x and the smallest value the secant function can have in # x 2 is 1 Ê the minimum value of y is and that occurs when ^w _#¹ _#¹ œ¹_# sec^#ˆ ‰¹₄^x Ê 1œsec^#ˆ ‰¹₄^x Ê „ œ1 secˆ ‰¹₄^x Ê xœ0.

66. (a) yœsin 2x Ê y^wœ2 cos 2x Ê y (0)^w œ2 cos (0)œ2 Ê tangent to yœsin 2x at the origin is yœ2x;

yœ sinˆ ‰^x_# Ê y^wœ _#^" cosˆ ‰^x_# Ê y (0)^w œ _#^" cos 0œ Ê_#^" tangent to yœ sinˆ ‰^x_# at the origin is yœ ^"_#x. The tangents are perpendicular to each other at the origin since the product of their slopes is

1.

(b) yœsin (mx) Ê y^wœm cos (mx) Ê y (0)^w œm cos 0œm; yœ sinˆ ‰_m^x Ê y^wœ _m^" cosˆ ‰_m^x y (0) cos (0) . Since m 1, the tangent lines are perpendicular at the origin.

Ê ^w œ _m^" œ _m^" †ˆ_m^"‰œ

(c) yœsin (mx) Ê y^wœm cos (mx). The largest value cos (mx) can attain is 1 at xœ0 Ê the largest value y can attain is m because y^w k k k k^w œkm cos (mx)kœk k km cos mxkŸk km †1œk km . Also, yœ sinˆ ‰_m^x

y cos y cos cos the largest value y can attain is .

Ê ^wœ _m^" ˆ ‰_m^x Ê k k^w œ¸^"_m ˆ ‰_m^x ¸Ÿ¸ ¸ ¸_m^" ˆ ‰_m^x ¸Ÿ _{k km}^" Ê ^w ¸ ¸_m^"

(d) yœsin (mx) Ê y^wœm cos (mx) Ê y (0)^w œm Ê slope of curve at the origin is m. Also, sin (mx) completes m periods on [0 2 ]. Therefore the slope of the curve yß 1 œsin (mx) at the origin is the same as the number of periods it completes on [0 2 ]. In particular, for large m, we can think of “compressing" the graph ofß 1 yœsin x horizontally which gives more periods completed on [0 2 ], but also increases the slope of theß 1 graph at the origin.

67. xœcos 2t, yœsin 2t, 0Ÿ Ÿt 1 68. xœcos (1t), yœsin (1t), 0Ÿ Ÿt 1

cos 2t sin 2t 1 x y 1 cos ( t) sin ( t) 1

Ê ^{# #} œ Ê ^{# #} œ Ê ^# 1 ^# 1 œ

x y 1, y

Ê ^{# #}œ   !

69. xœ4 cos t, yœ2 sin t, 0Ÿ Ÿt 2 1 70. xœ4 sin t, yœ5 cos t, 0Ÿ Ÿt 21

1 1 1 1

Ê ^{16 cos t}₁₆^{# 4 sin t}₄^# œ Ê ^x₁₆^{# y}₄^# œ Ê ^{16 sin t}₁₆^{# 25 cos t}₂₅^# œ Ê ^x₁₆^# _#^y₅^# œ

71. xœ3t, yœ9t , ^# _ _ Êt yœx ^# 72. xœ Èt , yœt, t 0 Ê xœ Èy or yœx , x^# Ÿ0

73. xœ 2t 5, yœ _ _4t 7, t 74. xœ 3 3t, yœ2t, 0Ÿ Ÿt 1 Ê ^y_# œt

x 5 2t 2(x 5) 4t x 3 3 2x 6 3y

Ê œ Ê œ Ê œ ˆ ‰^y_# Ê œ

y 2(x 5) 7 y 2x 3 y 2 x, x

Ê œ Ê œ Ê œ ²₃ ! Ÿ Ÿ $

75. xœt, yœÈ1t , 1^# Ÿ Ÿt 0 76. xœÈt1, yœÈt, t 0

y 1 x y t x y 1, y 0

Ê œÈ ^# Ê ^# œ Ê œÈ ^#  

77. xœsec t^# 1, yœtan t, ¹_# t ¹_# 78. xœ sec t, yœtan t, _#¹ t _#¹

sec t 1 tan t x y sec t tan t 1 x y 1

Ê ^# œ ^# Ê œ ^# Ê ^{# #} œ Ê ^{# #} œ

79. (a) xœa cos t, yœ a sin t, 0Ÿ Ÿt 2 1 80. (a) xœa sin t, yœb cos t, ¹_# Ÿ Ÿt ⁵_#¹ (b) xœa cos t, yœa sin t, 0Ÿ Ÿt 2 1 (b) xœa cos t, yœb sin t, 0Ÿ Ÿt 21 (c) xœa cos t, yœ a sin t, 0Ÿ Ÿt 4 1 (c) xœa sin t, yœb cos t, ¹_# Ÿ Ÿt ⁹_#¹ (d) xœa cos t, yœa sin t, 0Ÿ Ÿt 4 1 (d) xœa cos t, yœb sin t, 0Ÿ Ÿt 41

81. Using a"ß $b we create the parametric equations xœ " at and y œ $ bt, representing a line which goes through a"ß $b at tœ !. We determine a and b so that the line goes through a%ß "b when tœ ".

Since % œ " Ê œ &a a . Since " œ $ Ê œ %b b . Therefore, one possible parameterization is xœ " &t, y œ $ %t, 0Ÿ Ÿ "t .

82. Using a"ß $b we create the parametric equations xœ " at and y œ $ bt, representing a line which goes through at t . We determine a and b so that the line goes through when t . Since a a . a"ß $b œ ! a$ß #b œ " $ œ " Ê œ % Since # œ $ Ê œ &b b . Therefore, one possible parameterization is xœ " %t, y œ $ &t, 0Ÿ Ÿ "t . 83. The lower half of the parabola is given by xœy^# " for yŸ !. Substituting t for y, we obtain one possible

parameterization xœ " œt^# , y t, tŸ Þ0

84. The vertex of the parabola is at a"ß "b, so the left half of the parabola is given by yœx^# #x for xŸ ". Substituting t for x, we obtain one possible parametrization: xœt, yœ #t^# t, tŸ ".

85. For simplicity, we assume that x and y are linear functions of t and that the point x, y starts at a b a#ß $b for tœ ! and passes through a"ß "b at tœ ". Then xœf t , where fa b a b! œ # and fa b" œ ".

Since slopeœ^?_?^x_t œ ^"#_"! œ $ œ, x f ta bœ $ # œ # $t t. Also, yœg t , where ga b a b! œ $ and ga b" œ ". Since slopeœ^?_?^y_t œ ^"_"!³ œ 4. yœg ta bœ % $ œ $ %t t.

One possible parameterization is: xœ # $t, yœ $ %t, t  !.

86. For simplicity, we assume that x and y are linear functions of t and that the point x, y starts at a b a"ß #b for tœ ! and passes through a!ß !b at tœ ". Then xœf t , where fa b a b! œ " and fa b" œ !.

Since slopeœ^?_?^x_t œ ^{! "}_"!^{a b} œ " œ, x f ta bœ " " œ " t a b t. Also, yœg t , where ga b a b! œ # and ga b" œ !. Since slopeœ^?_?^y_t œ ^{! #}_"! œ # œ. y g ta bœ # # œ # #t t.

One possible parameterization is: xœ " t, yœ # #t, t  !.

87. tœ ¹₄ Ê xœ2 cos ¹₄ œÈ2, yœ2 sin ¹₄ œÈ2; ^dx_dt œ 2 sin t, ^dy_dt œ2 cos t Ê _dx^dy œ _dx/dt^dy/dtœ ^{2 cos t}_{2 sin t}œ cot t

cot 1; tangent line is y 2 1 x 2 or y x 2 2 ; csc t Ê ^dy¹ œ œ È œ Š È ‹ œ È ^dy œ

dx _tœ1 4 dt

4

1 ^w #

Ê ^{d y}_dx^## œ^{dy /dt}_dx/dt^w œ ^{csc t}_{2 sin t} œ _{2 sin t}$ Ê ^{d y}_dx^## œ 2

#

" ¹ È

tœ¹₄

88. tœ ²₃¹ Ê xœcos ²₃¹ œ ^"_#, yœÈ3 cos ²₃¹ œ ^È_#³; ^dx_dt œ sin t, ^dy_dt œ È3 sin t Ê _dx^dyœ ^È_{sin t}^{3 sin t}œÈ3

3 ; tangent line is y 3 x or y 3 x; 0 0

Ê ^dy_dx¹ œÈ Š ³‹œÈ < ˆ ‰‘ œÈ ^dy_dt œ Ê ^{d y}_dx œ _{sin t}⁰ œ

tœ²₃¹

È

# #" ^{w #}

#

Ê ^{d y}_dx^##¹ œ0

tœ²31

89. tœ ¹₄ Ê xœ ¹₄, yœ^"_#; ^dx_dt œ1, ^dy_dt œ _#^"_t Ê _dx^dyœ _dx/dt^dy/dtœ ₂¹_t Ê _dx^dy œ ^" œ1; tangent line is

È È ¹ #É

tœ¹₄ ^"₄

y œ^"_# 1†ˆx^"₄‰or yœ x ^"₄; ^dy_dt^w œ ^"₄t^$Î# Ê _dx^{d y#}# œ^{dy /dt}_dx/dt^w œ ^"₄t^$Î# Ê _dx^{d y#}#¹ œ 2

tœ¹₄

90. tœ3 Ê xœ È3 œ 1 2, yœÈ3(3)œ3; ^dx_dt œ (t1) , ^dy_dt œ³(3t) Ê _dx^dyœ ^(3t)

(t 1)

"

# "Î# # "Î#

ˆ ‰ ˆ ‰

3# "Î#

"

# "Î#

2; tangent line is y 3 2[x ( 2)] or y 2x 1;

œ ^{3 t 1} œ œ ^{3 3 1} œ œ œ

3t dy

dx 3(3)

È È

È È

¹t 3œ

dy d y

dt 3t dx

3t (t 1) 3 t 1 (3t) 3 3

2t 3t t 1 t 3t

w #

# "Î# # "Î#

œ^{È 3 ‘ È 3 ‘} œ _{È È} Ê # œ^{Š ‹} œ _È

Š ‹

3 2t 3t t 1

1 2 t 1 È È

È b cb

Ê ^{d y}_dx^##¹ œ ₃

t 3œ

"

91. tœ Ê1 xœ5, yœ1; ^dx_dt œ4t, ^dy_dt œ4t ^$ Ê _dx^dyœ _dx/dt^dy/dt œ^4t_4t^$ œt ^# Ê _dx^dy¹ œ ( 1)^#œ1; tangent line is

tœ1

y œ1 1 (x† 5) or yœ x 4; ^dy_dt^w œ2t Ê ^{d y}_dx^## œ^{dy /dt}_dx/dt^w œ _4t^2t œ " Ê ^{d y}_dx^## œ "

# ¹ #

tœ1

92. tœ ¹₃ Ê xœ ¹₃ sin ¹₃ œ ¹₃ ^È_#³, yœ 1 cos ¹₃ œ œ1 _#^" _#^"; ^dx_dt œ 1 cos t, ^dy_dt œsin t Ê _dx^dyœ _dx/dt^dy/dt 3 ; tangent line is y 3 x

œ _{1 cos t}^{sin t} Ê _dx^dy¹ œ_{1 cos}^sin œ œÈ œ_#^" È Š _{3 #}³‹

tœ¹₃

ˆ ‰ ˆ ‰

Š ‹ ˆ ‰

1 È

31 3

È3

#

"

#

1

y 3x 2;

Ê œÈ ^1È3 œ œ Ê œ œ ^{ˆ ‰}

3 dt (1 cos t) 1 cos t dx dx/dt 1 cos t

dy^w (1 cos t)(cos t) (sin t)(sin t) 1 d y^# dy /dt^w

# #

1cos t¹

œ _{(1cos t)}¹ # Ê _dx^{d y}# œ 4

# ¹

tœ¹3

93. tœ ¹₂ Ê xœcos ¹₂ œ0, yœ 1 sin ¹₂ œ2; ^dx_dt œ sin t, ^dy_dt œcos t Ê _dx^dy œ^{cos t}_{sin t} œ cot t

cot 0; tangent line is y 2; csc t csc t 1

Ê ^dy_dx¹ œ œ œ ^dy_dt œ Ê ^{d y}_dx œ ^{csc t}_{sin t} œ Ê ^{d y}_dx ¹ œ

tœ¹2 tœ¹2

1# ^w # ^# $ ^#

# #

#

94. tœ ¹₄ Ê xœsec^#ˆ¹₄‰ œ1 1, yœtanˆ¹₄‰œ 1; ^dx_dt œ2 sec t tan t, ^{# dy}_dt œsec t^#

cot t cot ; tangent line is

Ê ^dy_dx œ 2 sec t tan t œ 2 tan t œ Ê ^dydx œ 4 œ

sec t^#

# " " " "

# ¹ # ˆ ‰ #

tœ¹₄

1

y œ ( 1) ^"_#(x1) or yœ ^"_#x^"_#; ^dy_dt œ ^"_# csc t ^# Ê ^{d y}_dx œ2 sec t tan t œ ^"4 cot t^$

csc t

w #

# #

"

# #

Ê ^{d y}_dx^##¹ œ₄

tœ¹4

"

95. sœA cos (2 bt) 1 Ê vœ ^ds_dt œ A sin (2 bt)(2 b)1 1 œ 2 bA sin (2 bt). If we replace b with 2b to double the1 1 frequency, the velocity formula gives vœ 4 bA sin (4 bt) 1 1 Ê doubling the frequency causes the velocity to double. Also vœ #1bA sin (2 bt) 1 Ê œ a ^dv_dt œ 41^{# #}b A cos (2 bt). If we replace b with 2b in the1 acceleration formula, we get aœ 161^{# #}b A cos (4 bt) 1 Ê doubling the frequency causes the acceleration to quadruple. Finally, aœ 41^{# #}b A cos (2 bt) 1 Ê œ j ^da_dt œ81^{$ $}b A sin (2 bt). If we replace b with 2b in the jerk1 formula, we get jœ641^{$ $}b A sin (4 bt) 1 Ê doubling the frequency multiplies the jerk by a factor of 8.

96. (a) yœ37 sin<₃₆₅²¹ (x101)‘25 Ê y^wœ37 cos<₃₆₅²¹ (x101)‘ ˆ₃₆₅²¹‰œ ⁷⁴₃₆₅¹ cos<₃₆₅²¹ (x101) .‘ The temperature is increasing the fastest when y is as large as possible. The largest value of^w cos<₃₆₅²¹ (x101) is 1 and occurs when ‘ ₃₆₅²¹ (x101)œ0 Ê xœ101 Ê on day 101 of the year (µApril 11), the temperature is increasing the fastest.

(b) y (101)^w œ⁷⁴₃₆₅¹ cos<₃₆₅²¹ (101101)‘œ⁷⁴₃₆₅¹ cos (0)œ ⁷⁴₃₆₅¹ ¸0.64 °F/day

97. sœ " ( 4t)^"Î# Ê vœ ^ds_dt œ ^"_#(14t)^"Î#(4)œ2(14t)^"Î# Ê v(6)œ " %2( †6)^"Î#œ ²₅ m/sec;

vœ " 2( 4t)^"Î# Ê aœ^dv_dt œ ^"_# †2(14t)^$Î#(4)œ 4(14t)^$Î# Ê a(6)œ 4(14 6)† ^$Î#œ _{1 5#}⁴ m/sec^#

98. We need to show aœ ^dv_dt is constant: aœ^dv_dt œ^dv_ds †^ds_dt and ^dv_ds œ _ds^d ˆkÈs‰œ _2È^k_s Ê œ a ^dv_ds †^ds_dt œ ^dv_ds †v k s which is a constant.

œ _2È^k_s† È œ^k_#^#

99. v proportional to _È^" v _È for some constant k . Thus, a v

s s

k dv k dv dv ds dv

ds 2s dt ds dt ds

Ê œ Ê œ $Î# œ œ † œ † acceleration is a constant times so a is inversely proportional to s . œ _2s^k$Î# ^k_s œ ^k _s Ê _s

#

# #

†_{È #} ˆ ‰^{" " #}

100. Let ^dx_dt œf(x). Then, aœ^dv_dt œ^dv_dx†^dx_dt œ ^dv_dx†f(x)œ _dx^d ˆ ‰^dx_dt †f(x)œ _dx^d (f(x)) f(x)† œf (x)f(x), as required.^w 101. Tœ21É^L_g Ê ^dT_dL œ21 _g œ œ . Therefore, ^dT_du œ ^dT_dL ^dL_du œ kLœ œ 2 k1 É^L_g

g gL gL

k L

† ^" †^" † † g ^" †

#É É È È È #

È

L L

g g

1 1 1 1

, as required.

œ ^kT₂

102. No. The chain rule says that when g is differentiable at 0 and f is differentiable at g(0), then f‰g is differentiable at 0. But the chain rule says nothing about what happens when g is not differentiable at 0 so there is no contradiction.

103. The graph of yœ ‰(f g)(x) has a horizontal tangent at xœ1 provided that (f‰g) (1)^w œ0 Ê f (g(1))g (1)^{w w} œ0 either f (g(1)) 0 or g (1) 0 (or both) either the graph of f has a horizontal tangent at u g(1), or the

Ê ^w œ ^w œ Ê œ

graph of g has a horizontal tangent at xœ1 (or both).

104. (f‰g) ( 5)^w 0 Ê f (g( 5)) g ( 5)^w † ^w 0 Ê f (g( 5)) and g ( 5) are both nonzero and have opposite signs.^{w w} That is, either f (g( 5))c ^w 0 and g ( 5)^w 0 or f (g( 5))d c ^w 0 and g ( 5)^w 0 .d

105. As h Ä 0, the graph of yœ sin 2(x h) sin 2x h

approaches the graph of yœ2 cos 2x because

lim (sin 2x) 2 cos 2x.

hÄ !

sin 2(x h) sin 2x

h dx

œ d œ

106. As h Ä 0, the graph of yœ ^{cos (xc h)}_h^{#dcos xa b#} approaches the graph of yœ 2x sin x becausea b^#

lim cos x 2x sin x .

hÄ !

cos (x h) cos x

h dx

c ^#d a b^# œ d c a b# dœ a b#

107. ^dx_dt œcos t and ^dy_dt œ2 cos 2t Ê _dx^dy œ_dx/dt^dy/dt œ^{2 cos 2t}_{cos t} œ ^{2 2 cos ta} _{cos t}^{# 1b}; then _dx^dy œ0 Ê ^{2 2 cos ta} _{cos t}^{# 1b} œ0

2 cos t 1 0 cos t t , , , . In the 1st quadrant: t x sin and

Ê ^# œ Ê œ „_È^"₂ Ê œ ¹₄ ³₄^{1 5}₄^{1 7}₄¹ œ ₄¹ Ê œ ₄¹ œ ^È_#²

yœsin 2ˆ ‰¹₄ œ1 Ê Š^È_#²ß1 is the point where the tangent line is horizontal. At the origin: x‹ œ0 and yœ0 sin t 0 t 0 or t and sin 2t 0 t 0, , , ; thus t 0 and t give the tangent lines at Ê œ Ê œ œ1 œ Ê œ ¹_# 1 ³_#¹ œ œ1

the origin. Tangents at origin: ^dy_dx¹ 2 y 2x and ^dy_dx¹ 2 y 2x

t 0œ œ Ê œ tœ œ Ê œ

1

108. ^dx_dt 2 cos 2t and _dt 3 cos 3t _{dx dx/dt} ^{3 cos 3t}_{2 cos 2t} 2 2 cos t 1

dy dy dy/dt 3(cos 2t cos t sin 2t sin t)

œ œ Ê œ œ œ _a #_b

œ ^{3 2 cos t 1 (cos t)} 2 sin t cos t sin t œ œ

2 2 cos t 1 2 2 cos t 1 2 2 cos t

(3 cos t) 2 cos t 1 2 sin t (3 cos t) 4 cos t 3

ca b d

a b a b a b

a b a b

#

# # #

# # #

1 ; then

0 0 3 cos t 0 or 4 cos t 3 0: 3 cos t 0 t , and

dy

dx 2 2 cos t 1

(3 cos t) 4 cos t 3 3

œ Ê _a ^a # ^# _b ^b œ Ê œ œ œ Ê œ

# #1 #1

4 cos t^# œ3 0 Ê cos tœ „^È_#³ Ê œ t ₆¹, ⁵₆¹, ⁷₆¹, ¹¹₆¹. In the 1st quadrant: tœ ¹₆ Ê xœsin 2ˆ ‰₆¹ œ ^È_#³ and yœsin 3ˆ ‰¹₆ œ1 Ê Š^È_#³ß1 is the point where the graph has a horizontal tangent. At the origin: x‹ œ0 and yœ0 Ê sin 2tœ0 and sin 3tœ0 Ê œ t 0, , , ¹_# 1 ³_#¹ and tœ0, , ¹₃ ²₃¹, , 1 ⁴₃¹, ⁵₃¹ Ê œ t 0 and tœ1 give the tangent lines at the origin. Tangents at the origin: ^dy_dx¹ _{2 cos 0}^{3 cos 0 3} y ³x, and ^dy_dx¹

t 0œ œ œ _# Ê œ _# tœ

1

y x

œ ^{3 cos (3 )}_{2 cos (2 )}¹₁ œ ʳ_# œ ³_#

109. From the power rule, with yœx^"Î%, we get ^dy_dx œ^"₄x^$Î%. From the chain rule, yœÉÈx

x x , in agreement.

Ê ^dy_dxœ _dx œ œ₄

x x

d

x

" " " "

# # # $Î%

ÉÈ † ˆÈ ‰ ÉÈ † È

110. From the power rule, with yœx^$Î%, we get ^dy_dx œ₄³x^"Î%. From the chain rule, yœÉ Èx x

x x x x x

Ê ^dy_dxœ _dx Ê ^dy_dx œ œ œ

x x x x x x 4 x x

d 3

x

3 x

" " " "

#É È #É È #È #É È # É È

† ˆ È ‰ †Š † È ‹ †ˆ È ‰ È

x , in agreement.

œ ^{3 x} œ

4 x x

3 4 È È ÉÈ

"Î%

111. (a)

(b) ^df_dt œ1.27324 sin 2t0.42444 sin 6t0.2546 sin 10t0.18186 sin 14t (c) The curve of yœ^df_dt approximates yœ^dg_dt

the best when t is not 1, ¹_#, 0, , nor .¹_# 1

112. (a)

(b) ^dh_dt œ2.5464 cos (2t)2.5464 cos (6t)2.5465 cos (10t)2.54646 cos (14t)2.54646 cos (18t) (c)

111-116. Example CAS commands:

: Maple

f := t -> 0.78540 - 0.63662*cos(2*t) - 0.07074*cos(6*t) - 0.02546*cos(10*t) - 0.01299*cos(14*t);

g := t -> piecewise( t<-Pi/2, t+Pi, t<0, -t, t<Pi/2, t, Pi-t );

plot( [f(t),g(t)], t=-Pi..Pi );

Df := D(f);

Dg := D(g);

plot( [Df(t),Dg(t)], t=-Pi..Pi );

: (functions, domains, and value for t0 may change):

Mathematica

To see the relationship between f[t] and f'[t] in 111 and h[t] in 112

Clear[t, f]

f[t_] = 0.785400.63662 Cos[2t]0.07074 Cos[6t]0.02546 Cos[10t]0.01299 Cos[14t]

f'[t]

Plot[{f[t], f'[t]},{t, 1 1, }]

For the parametric equations in 113 - 116, do the following. Do NOT use the colon when defining tanline.

Clear[x, y, t]

t0 = p/4;

x[t_]:=1 Cos[t]

y[t_]:=1Sin[t]

p1=ParametricPlot[{x[t], y[t]},{t, 1 1, }]

yp[t_]:=y'[t]/x'[t]

ypp[t_]:=yp'[t]/x'[t]

yp[t0]//N ypp[t0]//N

tanline[x_]=y[t0]yp[t0] (xx[t0]) p2=Plot[tanline[x], {x, 0, 1}]

Show[p1, p2]