Worksheet: Derivatives of Vector-Valued Functions

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

Q1:

Calculate 𝑓(𝑠), and find the vector form of the equation of the tangent line 𝐿 at 𝑓(0) for 𝑓(𝑠)=(2𝑠,2𝑠,𝑠)cossin.

  • A𝑓(𝑠)=(2𝑠,2𝑠,1)sincos, 𝐿(0,1,1)+𝑡(1,0,0):
  • B𝑓(𝑠)=(22𝑠,22𝑠,1)sincos, 𝐿(1,0,0)+𝑡(0,2,1):
  • C𝑓(𝑠)=(2𝑠,2𝑠,1)sincos, 𝐿(1,0,0)+𝑡(0,1,1):
  • D𝑓(𝑠)=(22𝑠,22𝑠,1)sincos, 𝐿(0,2,1)+𝑡(1,0,0):
  • E𝑓(𝑠)=(22𝑠,22𝑠,1)sincos, 𝐿(1,0,0)+𝑡(0,2,1):

Q2:

Calculate 𝑓(𝑠), and find the vector form of the equation of the tangent line at 𝑓(0) for 𝑓(𝑠)=𝑠+1,𝑠+1,𝑠+1.

  • A𝑓(𝑠)=1,2𝑠,3𝑠𝐿(1,0,0)+𝑡(1,1,1),:
  • B𝑓(𝑠)=2,2𝑠+1,3𝑠+1𝐿(2,0,0)+𝑡(2,1,1),:
  • C𝑓(𝑠)=(1,2𝑠,3𝑠)𝐿(1,1,1)+𝑡(1,0,0),:
  • D𝑓(𝑠)=1,2𝑠,3𝑠𝐿(1,1,1)+𝑡(1,0,0),:
  • E𝑓(𝑠)=2,2𝑠+1,3𝑠+1𝐿(2,1,1)+𝑡(2,0,0),:

Q3:

Calculate f(𝑠), and find the vector form of the equation of the tangent line at f(0) for f(𝑠)=(𝑒+1,𝑒+1,𝑒+1).

  • Af(𝑠)=𝑒,2𝑒,2𝑠𝑒, 𝐿(2,2,2)+𝑡(1,2,0):
  • Bf(𝑠)=𝑒,𝑒,𝑒, 𝐿(2,2,2)+𝑡(1,1,1):
  • Cf(𝑠)=𝑒,2𝑒,2𝑠𝑒, 𝐿(1,2,0)+𝑡(2,2,2):
  • Df(𝑠)=𝑒+1,2𝑒+1,2𝑠𝑒+1, 𝐿(2,3,1)+𝑡(2,2,2):
  • Ef(𝑠)=𝑒+1,2𝑒+1,2𝑠𝑒+1, 𝐿(2,2,2)+𝑡(2,3,1):

Q4:

Consider the curve r(𝑠)=(2𝑠,2𝑠,2𝑠)sinsincos. Determine r(𝑠) and find the tangent 𝐿 to the curve when 𝑠=0.

  • Ar(𝑠)=(22𝑠,22𝑠,2𝑠)cossinsin, 𝐿(2,0,0)+𝑡(0,0,2):
  • Br(𝑠)=(2𝑠,2𝑠,2𝑠)coscossin, 𝐿(0,0,2)+𝑡(2,1,0):
  • Cr(𝑠)=(22𝑠,4𝑠,2𝑠)coscossin, 𝐿(2,2,0)+𝑡(0,0,2):
  • Dr(𝑠)=(22𝑠,2𝑠𝑠,2𝑠)cossincossin, 𝐿(0,0,2)+𝑡(2,2,0):
  • Er(𝑠)=(22𝑠,4𝑠𝑠,2𝑠)cossincossin, 𝐿(0,0,2)+𝑡(2,0,0):

Q5:

Given that rijk(𝑡)=𝑎𝑡+𝑡𝑒+𝑐𝑡sincos, where 𝑎 and 𝑏 are constants, find r(𝑡).

  • A2𝑎𝑎𝑡𝑎𝑡+𝑒(1+𝑏𝑡)+2𝑐𝑐𝑡𝑐𝑡sincoscossinijk
  • B𝑎𝑎𝑡𝑎𝑡+𝑒(1+𝑏𝑡)𝑐𝑐𝑡𝑐𝑡sincoscossinijk
  • C2𝑎𝑡𝑎𝑡+𝑒(1+𝑡)2𝑐𝑡𝑐𝑡sincoscossinijk
  • D𝑎𝑎𝑡𝑎𝑡+𝑒(1+𝑡)+𝑐𝑐𝑡𝑐𝑡sincoscossinijk
  • E2𝑎𝑎𝑡𝑎𝑡+𝑒(1+𝑏𝑡)2𝑐𝑐𝑡𝑐𝑡sincoscossinijk

Q6:

Find the derivative of a vector-valued function rijk(𝑡)=1+𝑡+5𝑡+1+𝑡+2.

  • A3𝑡+(10𝑡)+3𝑡ijk
  • B1+3𝑡+(10𝑡)+3𝑡ijk
  • C6𝑡+10𝑡
  • D3𝑡+10𝑡
  • E(3𝑡)+(10𝑡)+3𝑡ijk

Q7:

Find the derivative of a vector-valued function rij(𝑡)=(32𝑡)+2𝑡+3𝑡2.

  • A2(4𝑡+3)ij
  • B4𝑡+1
  • C2+(4𝑡+3)ij
  • D2+(4𝑡+3)ij
  • E2+(4𝑡)ij

Q8:

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

  • A𝑒𝑒+3ijk
  • B𝑒+𝑒ij
  • C𝑒𝑒ij
  • D𝑒𝑒+ijk
  • E𝑒𝑒ij

Q9:

Find the derivative of a vector-valued function 𝑟(𝑡)=(2𝑡)𝑖(𝑡)𝑗+𝑒𝑘sincos.

  • A2(𝑡)𝑖+(𝑡)𝑗+𝑒𝑘cossin
  • Bcossin(2𝑡)𝑖(𝑡)𝑗+𝑒𝑘
  • C2(2𝑡)𝑖(𝑡)𝑗+𝑒𝑘cossin
  • Dcossin(2𝑡)𝑖+(𝑡)𝑗+𝑒𝑘
  • E2(2𝑡)𝑖+(𝑡)𝑗+𝑒𝑘cossin

Q10:

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

  • A𝑡+2
  • B1
  • Cijk++
  • Di
  • E3

Q11:

Find the derivative of the following vector-valued function: r(𝑡)=5𝑡+3𝑡2𝑒5(𝑡).cos

  • A12𝑒10𝑡+35(𝑡)sin
  • B10𝑡+32𝑒5(𝑡)cos
  • C10𝑡+312𝑒5(𝑡)sin
  • D10𝑡+312𝑒5(𝑡)sin
  • E10𝑡+32𝑒5(𝑡)sin

Q12:

Find the derivative of the vector-valued function rij(𝑡)=3𝑡𝑒.

  • A133𝑡𝑒9ij
  • B323𝑡9𝑒ij
  • C13𝑡𝑒ij
  • D323𝑡+9𝑒ij
  • E13𝑡𝑒9ij

Q13:

Find the derivative of the vector-valued function rijk(𝑡)=3𝑡+𝑡+(5(𝑡))5𝑡+3𝑡5ln.

  • A9𝑡+1+5(𝑡)(10𝑡+3)ijkln
  • B9𝑡+(5𝑡)(10𝑡)ijk
  • C9𝑡+15𝑡+(10𝑡+3)ijk
  • D9𝑡+1+(5𝑡)(10𝑡+3)ijk
  • E9𝑡+1+5𝑡(10𝑡+3)ijk

Q14:

For the curve defined by the vector-valued equation r(𝑡)=3𝑡5𝑡5𝑡, find the value of r(2).

  • A7104
  • B12104
  • C1104
  • D71020
  • E7104

Q15:

For the curve defined by the vector-valued equation rij(𝑡)=𝜋𝑡3+2𝑡+3𝑡+𝑡sin, find the value of r(1).

  • A12+13ij
  • B𝜋6+13ij
  • C12+6ij
  • D12+13ij
  • E𝜋6+13ij

Q16:

Calculate dd𝑡((𝑡)(𝑡))rs for the vector-valued functions rij(𝑡)=𝑡+𝑡sincos and sij(𝑡)=𝑡+𝑡cossin.

  • A2(2𝑡)cos
  • B2(2𝑡)sin
  • C2(2𝑡)+2(2𝑡)cossinij
  • D2(2𝑡)cos
  • E2(2𝑡)sin

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