13.2 Derivatives and Integrals of Vector Functions PDF

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13.2 Derivatives and Integrals of Vector Functions Thursday, March 18, 2021

2꞉07 PM

Derivatives of Vector Functions -

For any vector function, its derivative generates its tangent vectors. Product rule for vector functions uses dot/cross product depending on operation

Integrals of Vector Functions -

An integral from t=a to t=b represents the displacement between r(a) and r(b) Displacement - the shortest distance between the points Taken piece-by-peice

WebAssign 1.V

Watch It Player | Cengage Learning Master It (webassign.net) 𝑟 ! (𝑡 ) = ⟨6𝑡 cos 6𝑡 + sin 6𝑡 , 2𝑡, cos 5𝑡 − 5𝑡 sin 5𝑡 ⟩

2.V

Watch It Player | Cengage Learning Master It (webassign.net) 𝑟 ! (𝑡 ) = 33 (𝑒 "# − 𝑡𝑒 "# ),

6 , 2𝑒 # 8 $ 𝑥 +1

′(0) ⟨3  6 2 ⟩  |𝑟 ! (0 )| = √9 + 36 + 4 = √49 = 7 3 𝑇(0 ) = 3 6 2 8 7

3.xxxxV

𝑟 ! (𝑡 ) = B𝑡 & , 𝑒 # , 2𝑡𝑒 $# C

 𝑡' #  𝑡 ' # $# 𝑡 ' # $# 𝑒 $# $# $# ) D𝑟 𝑡 𝑑𝑡 = 3 , 𝑒 , D2𝑡𝑒 𝑑𝑡 8 = 3 , 𝑒 , 𝑡𝑒 − D𝑒 8 = 3 , 𝑒 , 𝑡𝑒 − 8 6 6 6 2    𝑢 = 2𝑡 𝑑𝑢 = 2 𝑒 $# 𝑣= 2 𝑑𝑣 = 𝑒 $# 𝑡 ' # $# 1 = 3 , 𝑒 , 𝑒 H𝑡 − I8 2 6 

!(

1 3 𝑡' 3 + 1, 𝑒 # , 𝑒 $# H𝑡 − I + 8 2 2 6

4.xV

Watch It Player | Cengage Learning 𝑟(𝑡 ) = ⟨𝑒 "(# cos 4𝑡 , 𝑒 "(# sin 4𝑡 , 𝑒 "(# ⟩ = ⟨1, 0, 1⟩ 𝑡=0 𝐿(𝑡) = ⟨1, 0, 1⟩ + 𝑡𝑟 ! (0) 𝑟 ! (𝑡 ) = ⟨−4𝑒 "(# cos 4𝑡 − 4𝑒 "(# sin 4𝑡 , −4𝑒 "(# sin 4𝑡 + 4𝑒 "(# cos 4𝑡 , −4𝑒 "(# ⟩ = ⟨−4𝑒 "(# (cos 4𝑡 + sin 4𝑡 ), 4𝑒 "(# (sin 4𝑡 + cos 4𝑡 ), −4𝑒 "(# ⟩ 𝑟 ! (0 ) = ⟨−4, 4, −4 ⟩ 𝐿(𝑡) = ⟨1,0,1⟩ + 𝑡⟨−4, 4, −4⟩ = ⟨1 − 4𝑡, 4𝑡, 1 − 4𝑡⟩...