Worksheet: Second Derivatives of Parametric Equations

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

Q1:

Given that 𝑥=𝑡+5 and 𝑦=𝑡3𝑡, find dd𝑦𝑥.

  • A2(3𝑡)𝑡
  • B2(3𝑡)9𝑡
  • C𝑡2(3𝑡)
  • D2(3𝑡)3𝑡(2𝑡3)
  • E9𝑡2(3𝑡)

Q2:

Given that 𝑥=2𝑒 and 𝑦=𝑡𝑒, find dd𝑦𝑥.

  • A8𝑒4𝑡3
  • B4𝑡38𝑒
  • C2(4𝑡3)
  • D34𝑡8𝑒
  • E2(34𝑡)

Q3:

Given that 𝑥=𝑡+1 and 𝑦=𝑒1, find dd𝑦𝑥.

  • A𝑒(𝑡1)𝑡
  • B𝑒(𝑡1)2𝑡
  • C𝑡12𝑡
  • D4𝑡𝑒(𝑡1)
  • E𝑒(𝑡1)4𝑡

Q4:

Given that dd𝑧𝑥=5𝑥6 and dd𝑦𝑥=2𝑥1, determine dd𝑧𝑦 at 𝑥=1.

Q5:

Find dd𝑦𝑥 if 𝑥=𝑒 and 𝑦=2𝑛.

  • A𝑒𝑛(8𝑛+6)
  • B𝑒𝑛
  • C𝑛2𝑒(4𝑛3)
  • D2𝑒𝑛

Q6:

If 𝑥=25𝑧sec and 3𝑦=5𝑧tan, find dd𝑦𝑥.

  • A112
  • B13
  • C16
  • D12

Q7:

If 𝑦=(𝑥+4)4𝑥1 and 𝑧=(𝑥5)(𝑥+4), find (2𝑥1)𝑦𝑧dd.

  • A72𝑥104𝑥+30
  • B48𝑥+72𝑥+44𝑥34
  • C24𝑥+24𝑥+34
  • D96𝑥+16𝑥+120𝑥84𝑥+16

Q8:

Determine dd𝑦𝑥, given that 𝑥=6𝑛ln and 𝑦=8𝑛.

  • A2𝑛25
  • B4𝑛5
  • C8𝑛5
  • D12𝑛5

Q9:

Given that 𝑥=3𝑡+1 and 𝑦=3𝑡𝑡, find dd𝑦𝑥.

  • A2(13𝑡)𝑡
  • B2(13𝑡)81𝑡
  • C𝑡2(13𝑡)
  • D2(13𝑡)9𝑡(6𝑡1)
  • E81𝑡2(13𝑡)

Q10:

Given that 𝑥=3𝑡+1 and 𝑦=5𝑡𝑡, find dd𝑦𝑥.

  • A2(15𝑡)𝑡
  • B2(15𝑡)81𝑡
  • C𝑡2(15𝑡)
  • D2(15𝑡)9𝑡(10𝑡1)
  • E81𝑡2(15𝑡)

Q11:

Given that 𝑥=𝑒 and 𝑦=𝑡𝑒, find dd𝑦𝑥.

  • A4𝑒8𝑡3
  • B8𝑡34𝑒
  • C4(8𝑡3)
  • D38𝑡4𝑒
  • E4(38𝑡)

Q12:

Given that 𝑥=𝑡+4 and 𝑦=3𝑒5, find dd𝑦𝑥.

  • A3𝑒(𝑡1)𝑡
  • B3𝑒(𝑡1)2𝑡
  • C𝑡12𝑡
  • D4𝑡3𝑒(𝑡1)
  • E3𝑒(𝑡1)4𝑡

Q13:

Given that dd𝑧𝑥=7𝑥+7 and dd𝑦𝑥=3𝑥1, determine dd𝑧𝑦 at 𝑥=0.

Q14:

Given that dd𝑧𝑥=3𝑥8 and dd𝑦𝑥=2𝑥+1, determine dd𝑧𝑦 at 𝑥=1.

Q15:

If 𝑥=44𝑧sec and 5𝑦=4𝑧tan, find dd𝑦𝑥.

  • A180
  • B110
  • C140
  • D18

Q16:

If 𝑥=64𝑧sec and 𝑦=44𝑧tan, find dd𝑦𝑥.

  • A49
  • B29
  • C163
  • D89

Q17:

If 𝑦=(𝑥+2)3𝑥+2 and 𝑧=(𝑥+2)(𝑥+3), find (2𝑥1)𝑦𝑧dd.

  • A54𝑥30𝑥8
  • B36𝑥54𝑥+14𝑥+16
  • C18𝑥+18𝑥16
  • D72𝑥60𝑥+18𝑥+27𝑥+6

Q18:

Given that 𝑥=𝑡cos and 𝑦=2𝑡sin, find dd𝑦𝑥.

  • A2(2𝑡2𝑡+𝑡2𝑡)𝑡sinsincoscossin
  • B2(2𝑡2𝑡+𝑡2𝑡)𝑡sinsincoscossin
  • C𝑡2(2𝑡2𝑡+𝑡2𝑡)sinsinsincoscos
  • D2(2𝑡2𝑡+𝑡2𝑡)𝑡sinsincoscossin
  • E2𝑡2𝑡+𝑡2𝑡𝑡2𝑡sinsincoscossincos

Q19:

Given that 𝑥=3𝑡+1 and 𝑦=3𝑡+5𝑡, find dd𝑦𝑥.

  • A5𝑡
  • B56𝑡(6𝑡+5)
  • C536𝑡
  • D5𝑡
  • E536𝑡

Q20:

Given that 𝑥=𝑡𝑡ln and 𝑦=𝑡+𝑡ln, find dd𝑦𝑥.

  • A(𝑡1)2𝑡
  • B1𝑡(𝑡1)
  • C2𝑡(𝑡1)
  • D2𝑡(𝑡1)
  • E2(𝑡1)

Q21:

If 𝑦=5𝑥7 and 𝑧=3𝑥+16, find 𝑑𝑧𝑑𝑦 at 𝑥=1.

  • A275
  • B25
  • C25
  • D52
  • E275

Q22:

Consider the parameric curve 𝑥=1+𝜃sec and 𝑦=1+𝜃tan. Determine whether this curve is concave up, down, or neither at 𝜃=𝜋6.

  • Aneither
  • Bupward
  • Cdownward

Q23:

Consider the parameric curve 𝑥=𝜃cos and 𝑦=𝜃sin. Determine whether this curve is concave up, down, or neither at 𝜃=𝜋6.

  • Adownward
  • Bneither
  • Cupward

Q24:

Consider the parameric curve 𝑥=𝜃cos and 𝑦=𝜃sin. Determine whether this curve is concave up, down, or neither at 𝜃=𝜋6.

  • Aupward
  • Bdownward
  • Cneither

Q25:

Consider the parameric curve 𝑥=𝜃cos and 𝑦=2𝜃sin. Determine whether this curve is concave up, down, or neither at 𝜃=𝜋6.

  • Aupward
  • Bdownward
  • Cneither

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