- Exam 1 is Thursday (Feb. 25)
- Material since start of semester (e.g., 3D coordinates and surfaces, vectors and their products, lines, planes, vector-valued functions, their derivatives, limits, integrals, arc length, muPad, etc.)
- Whole class period
- 4 - 6 short-answer questions
- Open references (book, notes, computer), closed person
- Can use calculators, CASes, etc.
- But tell me where you do
- Unit circle (something you can double-check)
- r(t) = ⟨ cos t, sin t, 0 ⟩
- Length of circle from t=0 to t=2π came out to be 2π, as you would expect from c = 2πr
- What about length of arc from t = 0 to t = 1?
- This is why radians are so popular: they are the unit of angle measure for which the length of an arc covering Θ radians is Θ times the circle’s radius
- 4-dimensional curve
- r(t) = ⟨ sin t, cos t, 2t, t ⟩, 0 ≤ t ≤ π
- r(t) = ⟨ sin t, cos t, 2t, t ⟩, 0 ≤ t ≤ π
- The computer-generated snake (at least the body)
- r(t) = ⟨ t, sin t, 0 ⟩, 0 ≤ t ≤ 4π
- Example of an arc length integral that isn’t symbolically integrable
- Try integrating with muPad
- Unit tangent vector for snake?
- r(t) = ⟨ t, sin t, 0 ⟩, 0 ≤ t ≤ 4π
- Key idea: unit tangent vector is of constant length, so the only way it can change is to change its direction
- Its derivative therefore measures how fast a curve is curving: larger magnitude of derivative indicates fast bending, smaller magnitude indicates more gradual curve
- Furthermore, direction indicates the direction the curve is bending towards
- Example: curvature for a simple helix
- Finish curvature
- Read section 13.4