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.

  • A 2 ( 3 𝑡 ) 𝑡
  • B 2 ( 3 𝑡 ) 9 𝑡
  • C 𝑡 2 ( 3 𝑡 )
  • D 2 ( 3 𝑡 ) 3 𝑡 ( 2 𝑡 3 )
  • E 9 𝑡 2 ( 3 𝑡 )

Q2:

Given that 𝑥=2𝑒 and 𝑦=𝑡𝑒, find dd𝑦𝑥.

  • A 8 𝑒 4 𝑡 3
  • B 4 𝑡 3 8 𝑒
  • C 2 ( 4 𝑡 3 )
  • D 3 4 𝑡 8 𝑒
  • E 2 ( 3 4 𝑡 )

Q3:

Given that 𝑥=𝑡+1 and 𝑦=𝑒1, find dd𝑦𝑥.

  • A 𝑒 ( 𝑡 1 ) 𝑡
  • B 𝑒 ( 𝑡 1 ) 2 𝑡
  • C 𝑡 1 2 𝑡
  • D 4 𝑡 𝑒 ( 𝑡 1 )
  • E 𝑒 ( 𝑡 1 ) 4 𝑡

Q4:

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

  • A 1 4
  • B 1
  • C14
  • D9

Q5:

Find dd𝑦𝑥 if 𝑥=𝑒 and 𝑦=2𝑛.

  • A 𝑒 𝑛 ( 8 𝑛 + 6 )
  • B 𝑒 𝑛
  • C 𝑛 2 𝑒 ( 4 𝑛 3 )
  • D 2 𝑒 𝑛

Q6:

If 𝑥=25𝑧sec and 3𝑦=5𝑧tan, find dd𝑦𝑥.

  • A 1 1 2
  • B 1 3
  • C 1 6
  • D 1 2

Q7:

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

  • A 2 4 𝑥 + 2 4 𝑥 + 3 4
  • B 7 2 𝑥 1 0 4 𝑥 + 3 0
  • C 9 6 𝑥 + 1 6 𝑥 + 1 2 0 𝑥 8 4 𝑥 + 1 6
  • D 4 8 𝑥 + 7 2 𝑥 + 4 4 𝑥 3 4

Q8:

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

  • A 2 𝑛 2 5
  • B 4 𝑛 5
  • C 8 𝑛 5
  • D 1 2 𝑛 5

Q9:

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

  • A 2 ( 1 3 𝑡 ) 𝑡
  • B 2 ( 1 3 𝑡 ) 8 1 𝑡
  • C 𝑡 2 ( 1 3 𝑡 )
  • D 2 ( 1 3 𝑡 ) 9 𝑡 ( 6 𝑡 1 )
  • E 8 1 𝑡 2 ( 1 3 𝑡 )

Q10:

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

  • A 2 ( 1 5 𝑡 ) 𝑡
  • B 2 ( 1 5 𝑡 ) 8 1 𝑡
  • C 𝑡 2 ( 1 5 𝑡 )
  • D 2 ( 1 5 𝑡 ) 9 𝑡 ( 1 0 𝑡 1 )
  • E 8 1 𝑡 2 ( 1 5 𝑡 )

Q11:

Given that 𝑥=𝑒 and 𝑦=𝑡𝑒, find dd𝑦𝑥.

  • A 4 𝑒 8 𝑡 3
  • B 8 𝑡 3 4 𝑒
  • C 4 ( 8 𝑡 3 )
  • D 3 8 𝑡 4 𝑒
  • E 4 ( 3 8 𝑡 )

Q12:

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

  • A 3 𝑒 ( 𝑡 1 ) 𝑡
  • B 3 𝑒 ( 𝑡 1 ) 2 𝑡
  • C 𝑡 1 2 𝑡
  • D 4 𝑡 3 𝑒 ( 𝑡 1 )
  • E 3 𝑒 ( 𝑡 1 ) 4 𝑡

Q13:

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

  • A 4 2
  • B 7
  • C42
  • D7

Q14:

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

  • A 2 6
  • B5
  • C26
  • D23

Q15:

If 𝑥=44𝑧sec and 5𝑦=4𝑧tan, find dd𝑦𝑥.

  • A 1 8 0
  • B 1 1 0
  • C 1 4 0
  • D 1 8

Q16:

If 𝑥=64𝑧sec and 𝑦=44𝑧tan, find dd𝑦𝑥.

  • A 4 9
  • B 2 9
  • C 1 6 3
  • D 8 9

Q17:

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

  • A 7 2 𝑥 6 0 𝑥 + 1 8 𝑥 + 2 7 𝑥 + 6
  • B 5 4 𝑥 3 0 𝑥 8
  • C 3 6 𝑥 5 4 𝑥 + 1 4 𝑥 + 1 6
  • D 1 8 𝑥 + 1 8 𝑥 1 6

Q18:

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

  • A 2 ( 2 𝑡 2 𝑡 + 𝑡 2 𝑡 ) 𝑡 s i n s i n c o s c o s s i n
  • B 2 ( 2 𝑡 2 𝑡 + 𝑡 2 𝑡 ) 𝑡 s i n s i n c o s c o s s i n
  • C 𝑡 2 ( 2 𝑡 2 𝑡 + 𝑡 2 𝑡 ) s i n s i n s i n c o s c o s
  • D 2 ( 2 𝑡 2 𝑡 + 𝑡 2 𝑡 ) 𝑡 s i n s i n c o s c o s s i n
  • E 2 𝑡 2 𝑡 + 𝑡 2 𝑡 𝑡 2 𝑡 s i n s i n c o s c o s s i n c o s

Q19:

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

  • A 5 𝑡
  • B 5 6 𝑡 ( 6 𝑡 + 5 )
  • C 5 3 6 𝑡
  • D 5 𝑡
  • E 5 3 6 𝑡

Q20:

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

  • A ( 𝑡 1 ) 2 𝑡
  • B 1 𝑡 ( 𝑡 1 )
  • C 2 𝑡 ( 𝑡 1 )
  • D 2 𝑡 ( 𝑡 1 )
  • E 2 ( 𝑡 1 )

Q21:

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

  • A 2 7 5
  • B 2 5
  • C 2 5
  • D 5 2
  • E 2 7 5

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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