AI Tutor
Back to Calculus of Vector-Valued Functions of One Variable

Exercise 35

Avg difficulty 2.5

Compute the derivative of each of the following functions in two different ways: (1) use the rules provided in the theorem stated just after Activity, and (2) rewrite each given function so that it is stated as a single function (either a scalar function or a vector-valued function with three components), and differentiate component-wise. Compare your answers to ensure that they are the same.

  1. r⃗(t)=sin⁡(t)⟨2t,t2,arctan⁡(t)⟩\vr(t) = \sin(t) \langle 2t, t^2, \arctan(t) \rangle

  2. s⃗(t)=r⃗(2t)\vs(t) = \vr(2^t), where r⃗(t)=⟨t+2,ln⁡(t),1⟩\vr(t) = \langle t+2, \ln(t), 1 \rangle

  3. f(t)=⟨cos⁡(t),sin⁡(t),t⟩⋅⟨−sin⁡(t),cos⁡(t),1⟩f(t) = \langle \cos(t), \sin(t), t \rangle \cdot \langle -\sin(t), \cos(t), 1 \rangle

  4. r⃗(t)=⟨cos⁡(t),sin⁡(t),t⟩×⟨−sin⁡(t),cos⁡(t),1⟩\vr(t) = \langle \cos(t), \sin(t), t \rangle \times \langle -\sin(t), \cos(t), 1 \rangle

Long, Keller & Schlicker, Active Calculus Multivariable, §10.3 Calculus of Vector-Valued Functions of One Variable, Exercise 35

AI Tutor

Adapted from Active Calculus Multivariable © 2013–2026 Nicholas Long, Mitchel T. Keller, and Steven Schlicker. Licensed under CC BY-SA 4.0. Source: Active Calculus Multivariable. Imported at revision 692c03c.