M408M Learning Module Pages
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Chapter 10: Parametric Equations and Polar Coordinates

Chapter 12: Vectors and the Geometry of Space

Chapter 13: Vector Functions

Learning module LM 13.1/2: Vector valued functions

Learning module LM 13.3: Velocity, speed and arc length:

      Position, velocity and acceleration
      Speed and arc length
      Worked problems

Learning module LM 13.4: Acceleration and curvature:

Chapter 14: Partial Derivatives

Chapter 15: Multiple Integrals

Worked problems

Worked problems

In this video, we work two example problems:

Find the length of the curve ${\bf r}(t) = \ln(t) {\bf i} + \frac{t^2}{2} {\bf j} + \sqrt{2} \, t {\bf k}$ from $t=1$ to $t=4$ .
Find the time at which the speed of a particle with position ${\bf r}(t) = \langle t^2, 5t, t^2 - 16t \rangle$ is minimized.