Lesson Worksheet: Second Derivatives of Parametric Equations Mathematics • Higher Education

In this worksheet, we will practice finding second derivatives and higher-order derivatives of parametric equations by applying the chain rule.

Q1:

Given that ๐‘ฅ=๐‘ก+5๏Šฉ and ๐‘ฆ=๐‘กโˆ’3๐‘ก๏Šจ, find dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ.

  • A2(3โˆ’๐‘ก)๐‘ก
  • B2(3โˆ’๐‘ก)9๐‘ก๏Šซ
  • C๐‘ก2(3โˆ’๐‘ก)
  • D2(3โˆ’๐‘ก)3๐‘ก(2๐‘กโˆ’3)๏Šฉ
  • E9๐‘ก2(3โˆ’๐‘ก)๏Šซ

Q2:

Given that ๐‘ฅ=2๐‘’๏Šจ๏ and ๐‘ฆ=๐‘ก๐‘’๏Šฑ๏Šจ๏, find dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ.

  • A8๐‘’4๐‘กโˆ’3๏Šฌ๏
  • B4๐‘กโˆ’38๐‘’๏Šฌ๏
  • C2(4๐‘กโˆ’3)
  • D3โˆ’4๐‘ก8๐‘’๏Šฌ๏
  • E2(3โˆ’4๐‘ก)

Q3:

Given that ๐‘ฅ=๐‘ก+1๏Šจ and ๐‘ฆ=๐‘’โˆ’1๏, find dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ.

  • A๐‘’(๐‘กโˆ’1)๐‘ก๏
  • B๐‘’(๐‘กโˆ’1)2๐‘ก๏๏Šฉ
  • C๐‘กโˆ’12๐‘ก๏Šจ
  • D4๐‘ก๐‘’(๐‘กโˆ’1)๏Šฉ๏
  • E๐‘’(๐‘กโˆ’1)4๐‘ก๏๏Šฉ

Q4:

Given that dd๐‘ง๐‘ฅ=5๐‘ฅโˆ’6 and dd๐‘ฆ๐‘ฅ=2๐‘ฅโˆ’1๏Šจ, determine dd๏Šจ๏Šจ๐‘ง๐‘ฆ at ๐‘ฅ=1.

Q5:

Find dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ if ๐‘ฅ=โˆ’๐‘’๏Šช๏Š and ๐‘ฆ=โˆ’2๐‘›๏Šช.

  • A๐‘’๐‘›(โˆ’8๐‘›+6)๏Šฑ๏Šช๏Š๏Šจ
  • Bโˆ’๐‘’๐‘›๏Šฑ๏Šช๏Š๏Šจ
  • C๐‘›2๐‘’(4๐‘›โˆ’3)๏Šจ๏Šฑ๏Šฎ๏Š
  • D2๐‘’๐‘›๏Šฑ๏Šช๏Š๏Šฉ

Q6:

If ๐‘ฅ=25๐‘งsec and โˆš3๐‘ฆ=5๐‘งtan, find dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ.

  • A112
  • B13
  • C16
  • D12

Q7:

If ๐‘ฆ=(๐‘ฅ+4)๏€นโˆ’4๐‘ฅโˆ’1๏…๏Šจ and ๐‘ง=(๐‘ฅโˆ’5)(๐‘ฅ+4), find (2๐‘ฅโˆ’1)๐‘ฆ๐‘ง๏Šฉ๏Šจ๏Šจdd.

  • Aโˆ’72๐‘ฅโˆ’104๐‘ฅ+30๏Šจ
  • Bโˆ’48๐‘ฅ+72๐‘ฅ+44๐‘ฅโˆ’34๏Šฉ๏Šจ
  • Cโˆ’24๐‘ฅ+24๐‘ฅ+34๏Šจ
  • Dโˆ’96๐‘ฅ+16๐‘ฅ+120๐‘ฅโˆ’84๐‘ฅ+16๏Šช๏Šฉ๏Šจ

Q8:

Determine dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ, given that ๐‘ฅ=6๐‘›ln๏Šซ and ๐‘ฆ=โˆ’8๐‘›๏Šฉ.

  • Aโˆ’2๐‘›25๏Šฉ
  • Bโˆ’4๐‘›5๏Šฉ
  • Cโˆ’8๐‘›5๏Šฉ
  • Dโˆ’12๐‘›5๏Šจ

Q9:

Given that ๐‘ฅ=3๐‘ก+1๏Šฉ and ๐‘ฆ=3๐‘กโˆ’๐‘ก๏Šจ, find dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ.

  • A2(1โˆ’3๐‘ก)๐‘ก
  • B2(1โˆ’3๐‘ก)81๐‘ก๏Šซ
  • C๐‘ก2(1โˆ’3๐‘ก)
  • D2(1โˆ’3๐‘ก)9๐‘ก(6๐‘กโˆ’1)๏Šฉ
  • E81๐‘ก2(1โˆ’3๐‘ก)๏Šซ

Q10:

Given that ๐‘ฅ=3๐‘ก+1๏Šฉ and ๐‘ฆ=5๐‘กโˆ’๐‘ก๏Šจ, find dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ.

  • A2(1โˆ’5๐‘ก)๐‘ก
  • B2(1โˆ’5๐‘ก)81๐‘ก๏Šซ
  • C๐‘ก2(1โˆ’5๐‘ก)
  • D2(1โˆ’5๐‘ก)9๐‘ก(10๐‘กโˆ’1)๏Šฉ
  • E81๐‘ก2(1โˆ’5๐‘ก)๏Šซ

Q11:

Given that ๐‘ฅ=๐‘’๏Šช๏ and ๐‘ฆ=๐‘ก๐‘’๏Šฑ๏Šช๏, find dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ.

  • A4๐‘’8๐‘กโˆ’3๏Šง๏Šจ๏
  • B8๐‘กโˆ’34๐‘’๏Šง๏Šจ๏
  • C4(8๐‘กโˆ’3)
  • D3โˆ’8๐‘ก4๐‘’๏Šง๏Šจ๏
  • E4(3โˆ’8๐‘ก)

Q12:

Given that ๐‘ฅ=๐‘ก+4๏Šจ and ๐‘ฆ=3๐‘’โˆ’5๏, find dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ.

  • A3๐‘’(๐‘กโˆ’1)๐‘ก๏
  • B3๐‘’(๐‘กโˆ’1)2๐‘ก๏๏Šฉ
  • C๐‘กโˆ’12๐‘ก๏Šจ
  • D4๐‘ก3๐‘’(๐‘กโˆ’1)๏Šฉ๏
  • E3๐‘’(๐‘กโˆ’1)4๐‘ก๏๏Šฉ

Q13:

Given that dd๐‘ง๐‘ฅ=โˆ’7๐‘ฅ+7 and dd๐‘ฆ๐‘ฅ=3๐‘ฅโˆ’1๏Šจ, determine dd๏Šจ๏Šจ๐‘ง๐‘ฆ at ๐‘ฅ=0.

Q14:

Given that dd๐‘ง๐‘ฅ=โˆ’3๐‘ฅโˆ’8 and dd๐‘ฆ๐‘ฅ=โˆ’2๐‘ฅ+1๏Šจ, determine dd๏Šจ๏Šจ๐‘ง๐‘ฆ at ๐‘ฅ=โˆ’1.

Q15:

If ๐‘ฅ=44๐‘งsec and โˆš5๐‘ฆ=4๐‘งtan, find dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ.

  • A180
  • B110
  • C140
  • D18

Q16:

If ๐‘ฅ=88๐‘งsec and โˆš5๐‘ฆ=68๐‘งtan, find dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ.

  • A980
  • B95
  • C940
  • D98

Q17:

If ๐‘ฆ=(โˆ’๐‘ฅ+2)๏€น3๐‘ฅ+2๏…๏Šจ and ๐‘ง=(โˆ’๐‘ฅ+2)(๐‘ฅ+3), find (โˆ’2๐‘ฅโˆ’1)๐‘ฆ๐‘ง๏Šฉ๏Šจ๏Šจdd.

  • A54๐‘ฅโˆ’30๐‘ฅโˆ’8๏Šจ
  • Bโˆ’36๐‘ฅโˆ’54๐‘ฅ+14๐‘ฅ+16๏Šฉ๏Šจ
  • C18๐‘ฅ+18๐‘ฅโˆ’16๏Šจ
  • Dโˆ’72๐‘ฅโˆ’60๐‘ฅ+18๐‘ฅ+27๐‘ฅ+6๏Šช๏Šฉ๏Šจ

Q18:

Given that ๐‘ฅ=๐‘กcos and ๐‘ฆ=2๐‘กsin, find dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ.

  • Aโˆ’2(2๐‘ก2๐‘ก+๐‘ก2๐‘ก)๐‘กsinsincoscossin๏Šฉ
  • Bโˆ’2(2๐‘ก2๐‘ก+๐‘ก2๐‘ก)๐‘กsinsincoscossin
  • Cโˆ’๐‘ก2(2๐‘ก2๐‘ก+๐‘ก2๐‘ก)sinsinsincoscos๏Šฉ
  • D2(2๐‘ก2๐‘ก+๐‘ก2๐‘ก)๐‘กsinsincoscossin๏Šจ
  • E2๐‘ก2๐‘ก+๐‘ก2๐‘ก๐‘ก2๐‘กsinsincoscossincos๏Šจ

Q19:

Given that ๐‘ฅ=3๐‘ก+1๏Šจ and ๐‘ฆ=3๐‘ก+5๐‘ก๏Šจ, find dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ.

  • Aโˆ’5๐‘ก
  • Bโˆ’56๐‘ก(6๐‘ก+5)๏Šจ
  • C536๐‘ก๏Šฉ
  • D5๐‘ก
  • Eโˆ’536๐‘ก๏Šฉ

Q20:

Given that ๐‘ฅ=๐‘กโˆ’๐‘กln and ๐‘ฆ=๐‘ก+๐‘กln, find dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ.

  • Aโˆ’(๐‘กโˆ’1)2๐‘ก๏Šฉ
  • Bโˆ’1๐‘ก(๐‘กโˆ’1)
  • Cโˆ’2๐‘ก(๐‘กโˆ’1)๏Šฉ
  • Dโˆ’2๐‘ก(๐‘กโˆ’1)
  • Eโˆ’2(๐‘กโˆ’1)๏Šจ

Q21:

If ๐‘ฆ=โˆ’5๐‘ฅโˆ’7๏Šฉ and ๐‘ง=3๐‘ฅ+16๏Šจ, find ๐‘‘๐‘ง๐‘‘๐‘ฆ๏Šจ๏Šจ at ๐‘ฅ=1.

  • Aโˆ’275
  • B25
  • Cโˆ’25
  • Dโˆ’52
  • E275

Q22:

Consider the parameric curve ๐‘ฅ=1+๐œƒsec and ๐‘ฆ=1+๐œƒtan. Determine whether this curve is concave up, down, or neither at ๐œƒ=๐œ‹6.

  • Aneither
  • Bupward
  • Cdownward

Q23:

Consider the parameric curve ๐‘ฅ=๐œƒcos๏Šฉ and ๐‘ฆ=๐œƒsin๏Šฉ. Determine whether this curve is concave up, down, or neither at ๐œƒ=๐œ‹6.

  • Adownward
  • Bneither
  • Cupward

Q24:

Consider the parameric curve ๐‘ฅ=๐œƒcos and ๐‘ฆ=๐œƒsin. Determine whether this curve is concave up, down, or neither at ๐œƒ=๐œ‹6.

  • Aupward
  • Bdownward
  • Cneither

Q25:

Consider the parameric curve ๐‘ฅ=๐œƒcos and ๐‘ฆ=2๐œƒsin. Determine whether this curve is concave up, down, or neither at ๐œƒ=๐œ‹6.

  • Aupward
  • Bdownward
  • Cneither

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