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Mathematics LibreTexts

8.2: Calculus of Vector-Valued Functions

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  1. Compute the derivative of r(t)=2t2i1tj+tk.
     
  2. Compute the derivative of r(t)=(cost,sint,t).
     
  3. Compute the derivative of r(t)=t2eti+ln(sect)j.
     
  4. Compute the derivative of r(t)=tantt2+t+1ij+arctan(t2)k.
     
  5. Find the tangent and the unit tangent vectors to the curve r(t)=(ln(2t),t2+3t) at t=1.
     
  6. Find the tangent and unit tangent vectors to the curve r(t)=(cos(2t),2t,sin(3t)) at t=π2.
     
  7. Find the tangent and the unit tangent vectors to the curve r(t)=etie3t2j+4e2tk at t=ln(2).
     
  8. Find the equation of the tangent line to the curve r(t)=(sint,cost) at t=π4.
     
  9. Find the equation of the tangent line to the curve r(t)=t2i+tj+tk at t=1.
     
  10. 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.
     
  11. The position of a particle at time t is given by the function r(t)=tit2j+t3k. Find the velocity, speed, and acceleration of the particle at t=1.
     
  12. 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.
     
  13. Consider vector-valued functions r(t) and s(t). Suppose r(5)=i3j, r(5)=i+4j+5k, s(5)=4i+3j+k, and s(5)=5i+2j2k. Find ddt(r(t)s(t))|t=5.
     
  14. 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.
     
  15. Evaluate the integral e11ti+t2j dt.
     
  16. Evaluate the integral 10(3t,1t+1,et) dt.
     
  17. The acceleration function of a particle at time t is given by a(t)=5costi5sintj. 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.

8.2: Calculus of Vector-Valued Functions is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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