8.2: Calculus of Vector-Valued Functions
( \newcommand{\kernel}{\mathrm{null}\,}\)
- Compute the derivative of r(t)=2t2i−1tj+√tk.
- Compute the derivative of r(t)=(cost,sint,t).
- Compute the derivative of r(t)=t2eti+ln(sect)j.
- Compute the derivative of r(t)=tantt2+t+1i−j+arctan(t2)k.
- Find the tangent and the unit tangent vectors to the curve r(t)=(ln(2t),t2+3t) at t=1.
- Find the tangent and unit tangent vectors to the curve r(t)=(cos(2t),2t,sin(3t)) at t=π2.
- Find the tangent and the unit tangent vectors to the curve r(t)=eti−e3t2j+4e−2tk at t=ln(2).
- Find the equation of the tangent line to the curve r(t)=(sint,cost) at t=π4.
- Find the equation of the tangent line to the curve r(t)=t2i+tj+tk at t=−1.
- The position of a particle at time t is given by the function r(t)=(cost,sin3t). Find the velocity, speed, and acceleration of the particle at t=π4.
- The position of a particle at time t is given by the function r(t)=ti−t2j+t3k. Find the velocity, speed, and acceleration of the particle at t=1.
- Consider a scalar function f(t) and a vector-valued function r(t). Suppose f(0)=4, f′(0)=−2, r(0)=(5,3,4), and r′(0)=(0,−2,−4). Find ddt(f(t)r(t))|t=0.
- Consider vector-valued functions r(t) and s(t). Suppose r(5)=i−3j, r′(5)=−i+4j+5k, s(5)=−4i+3j+k, and s′(5)=−5i+2j−2k. Find ddt(r(t)⋅s(t))|t=5.
- Consider vector-valued functions r(t) and s(t). Suppose r(1)=(−2,2,4), r′(1)=(4,0,4), s(1)=(1,2,−2), and s′(1)=(4,0,3). Find ddt(r(t)×s(t))|t=1.
- Evaluate the integral ∫e11ti+t2j dt.
- Evaluate the integral ∫10(3√t,1t+1,e−t) dt.
- The acceleration function of a particle at time t is given by a(t)=−5costi−5sintj. If the particle has an initial velocity of v(0)=9i+2j and starts at position r(0)=5i, find the velocity function v(t) and position function r(t) of the particle.