Lesson Worksheet: Derivatives of Vector-Valued Functions Mathematics • Higher Education

In this worksheet, we will practice determining the derivatives of vector-valued functions and finding unit tangent vectors.

Q1:

Given that rijk(𝑑)=π‘Žπ‘‘+𝑑𝑒+𝑐𝑑sincos, where π‘Ž and 𝑏 are constants, find rβ€²(𝑑).

  • Aβˆ’2π‘Žπ‘Žπ‘‘π‘Žπ‘‘+𝑒(1+𝑏𝑑)+2𝑐𝑐𝑑𝑐𝑑sincoscossinijk
  • Bπ‘Žπ‘Žπ‘‘π‘Žπ‘‘+𝑒(1+𝑏𝑑)βˆ’π‘π‘π‘‘π‘π‘‘sincoscossinijk
  • C2π‘Žπ‘‘π‘Žπ‘‘+𝑒(1+𝑑)βˆ’2𝑐𝑑𝑐𝑑sincoscossinijk
  • Dβˆ’π‘Žπ‘Žπ‘‘π‘Žπ‘‘+𝑒(1+𝑑)+𝑐𝑐𝑑𝑐𝑑sincoscossinijk
  • E2π‘Žπ‘Žπ‘‘π‘Žπ‘‘+𝑒(1+𝑏𝑑)βˆ’2𝑐𝑐𝑑𝑐𝑑sincoscossinijk

Q2:

Find the derivative of a vector-valued function π‘Ÿ(𝑑)=(2𝑑)π‘–βˆ’(𝑑)𝑗+π‘’π‘˜sincos.

  • A2(𝑑)𝑖+(𝑑)𝑗+π‘’π‘˜cossin
  • Bcossin(2𝑑)π‘–βˆ’(𝑑)𝑗+π‘’π‘˜ο
  • C2(2𝑑)π‘–βˆ’(𝑑)𝑗+π‘’π‘˜cossin
  • Dcossin(2𝑑)𝑖+(𝑑)𝑗+π‘’π‘˜ο
  • E2(2𝑑)𝑖+(𝑑)𝑗+π‘’π‘˜cossin

Q3:

Find the derivative of the vector-valued function rijk(𝑑)=𝑒+𝑒+3.

  • Aπ‘’βˆ’π‘’+3ijk
  • B𝑒+π‘’οοŠ±οij
  • Cπ‘’βˆ’π‘’οοŠ±οij
  • Dπ‘’βˆ’π‘’+ijk
  • Eπ‘’βˆ’π‘’οοij

Q4:

Find the derivative of a vector-valued function rijk(𝑑)=𝑑++.

  • A𝑑+2
  • B1
  • Cijk++
  • Di
  • E3

Q5:

Find the derivative of a vector-valued function rij(𝑑)=(3βˆ’2𝑑)+ο€Ή2𝑑+3π‘‘βˆ’2ο…οŠ¨.

  • Aβˆ’2βˆ’(4𝑑+3)ij
  • B4𝑑+1
  • Cβˆ’2+(4𝑑+3)ij
  • D2+(4𝑑+3)ij
  • Eβˆ’2+(4𝑑)ij

Q6:

Find the derivative of a vector-valued function rijk(𝑑)=ο€Ή1+𝑑+ο€Ή5𝑑+1+𝑑+2ο…οŠ©οŠ¨οŠ©.

  • Aο€Ή3𝑑+(10𝑑)+ο€Ή3π‘‘ο…οŠ¨οŠ¨ijk
  • Bο€Ή1+3𝑑+(10𝑑)+ο€Ή3π‘‘ο…οŠ¨οŠ¨ijk
  • C6𝑑+10π‘‘οŠ¨
  • D3𝑑+10π‘‘οŠ¨
  • E(3𝑑)+(10𝑑)+ο€Ή3𝑑ijk

Q7:

Find the derivative of the following vector-valued function: r(𝑑)=5𝑑+3𝑑2𝑒5(𝑑).cos

  • A12𝑒10𝑑+3βˆ’5(𝑑)sin
  • B10𝑑+32𝑒5(𝑑)cos
  • C10𝑑+312𝑒5(𝑑)sin
  • D10𝑑+312π‘’βˆ’5(𝑑)sin
  • E10𝑑+32π‘’βˆ’5(𝑑)sin

Q8:

Find the derivative of the vector-valued function rij(𝑑)=ο€»βˆš3π‘‘ο‡βˆ’ο€Ήπ‘’ο…οŠ―ο.

  • Aο€Ώ13√3π‘‘ο‹βˆ’ο€Ύπ‘’9ij
  • Bο€Ώ32√3π‘‘ο‹βˆ’ο€Ή9𝑒ij
  • Cο€Ώ1√3π‘‘ο‹βˆ’ο€Ήπ‘’ο…ij
  • Dο€Ώ32√3𝑑+ο€Ή9𝑒ij
  • Eο€Ώ1√3π‘‘ο‹βˆ’ο€Ύπ‘’9ij

Q9:

Find the derivative of the vector-valued function rijk(𝑑)=ο€Ή3𝑑+𝑑+(5(𝑑))βˆ’ο€Ή5𝑑+3π‘‘βˆ’5ο…οŠ©οŠ¨ln.

  • Aο€Ή9𝑑+1+5(𝑑)βˆ’(10𝑑+3)ijkln
  • Bο€Ή9𝑑+(5𝑑)βˆ’(10𝑑)ijk
  • Cο€Ή9𝑑+1ο…βˆ’ο€Ό5π‘‘οˆ+(10𝑑+3)ijk
  • Dο€Ή9𝑑+1+(5𝑑)βˆ’(10𝑑+3)ijk
  • Eο€Ή9𝑑+1+ο€Ό5π‘‘οˆβˆ’(10𝑑+3)ijk

Q10:

For the curve defined by the vector-valued equation r(𝑑)=3π‘‘βˆ’5π‘‘βˆš5π‘‘ο£οŠ¨, find the value of rβ€²(2).

  • A7√104
  • B12√104
  • C1√104
  • D7√1020
  • Eοšβˆ’7√104

This lesson includes 2 additional questions and 54 additional question variations for subscribers.

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