Calculus of vector-valued functions practice problems
Derivatives, tangent vectors, unit tangents and integrals of vector-valued functions. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Evaluate \( \displaystyle \int_0^{\frac{\pi}{2}} \left\langle \cos{\left(t \right)}, \sin{\left(t \right)}, t \right\rangle\, dt \).
Find the unit tangent vector \( \displaystyle \mathbf T \) of \( \displaystyle \mathbf r(t) = \left\langle t, t^{2} - 1, 2 t \right\rangle \) at \( \displaystyle t = 0 \).
Evaluate \( \displaystyle \int_0^{\pi} \left\langle t^{2}, 2 t, \cos{\left(t \right)} \right\rangle\, dt \).
Find the unit tangent vector \( \displaystyle \mathbf T \) of \( \displaystyle \mathbf r(t) = \left\langle t \cos{\left(t \right)}, t \sin{\left(t \right)}, t \right\rangle \) at \( \displaystyle t = 1 \).
For \( \displaystyle \mathbf r(t) = \left\langle t^{2}, t^{3}, t \right\rangle \), find \( \displaystyle \mathbf r'(t) \) and the tangent vector at \( \displaystyle t = \pi \).
For \( \displaystyle \mathbf r(t) = \left\langle 3 t, 4 t^{2}, 2 t^{3} \right\rangle \), find \( \displaystyle \mathbf r'(t) \) and the tangent vector at \( \displaystyle t = \pi \).
Find the unit tangent vector \( \displaystyle \mathbf T \) of \( \displaystyle \mathbf r(t) = \left\langle t \cos{\left(t \right)}, t \sin{\left(t \right)}, t \right\rangle \) at \( \displaystyle t = \frac{\pi}{2} \).
For \( \displaystyle \mathbf r(t) = \left\langle e^{t}, e^{- t}, t \right\rangle \), find \( \displaystyle \mathbf r'(t) \) and the tangent vector at \( \displaystyle t = 2 \).
For \( \displaystyle \mathbf r(t) = \left\langle t \cos{\left(t \right)}, t \sin{\left(t \right)}, t \right\rangle \), find \( \displaystyle \mathbf r'(t) \) and the tangent vector at \( \displaystyle t = 2 \).
Find the unit tangent vector \( \displaystyle \mathbf T \) of \( \displaystyle \mathbf r(t) = \left\langle t, t^{2} - 1, 2 t \right\rangle \) at \( \displaystyle t = 1 \).