- Return of 3.5 homework
- 39: Use Occam's razor
- 40: discuss all graphs!
- 41: shouldn't the velocity curve be differentiable?
- 43: use your soul! Speak of traffic (also concavity!)
- 44: set
, and solve for terminal velocity (better notation:
(Check units!)
- Hand in homework
- New assignment:
Thu 3/4 Section 4.1 - Read 4.2 (if you haven't already): Max/Mins
- Read 4.7 to understand Newton's method
- Problems to hand in: Section 4.1, p. 180, #22, 27, 31, 32, 52, 53 (due Thu. 3/18)
- Begin on-line homework for 4.7 (due Wed. 3/17)
- The basic idea falls out of the limit definition of the derivative:
if we just forget about the limit and hope that h is small enough:
- Alternate forms:
- The linear approximation can be used to
- predict a value of f close to x0 where f and f' have known values:
This means that we're predicting f (at left) using an affine function (at right), which is the equation of the tangent line to f at x0. If there isn't much curvature in the graph of f, this should be a pretty good estimate; if, on the other hand, there is much curvature then the approximation will only be good for small values of h.
- Examples:
- #9, p. 179
- #11, p. 179
- #16, p. 179
- search for a value of x that produces a desired value of f (especially 0, i.e. Newton's method), starting from some nearby value:
or, solving for x at which we hope f(x)=c,
Newton's method is an iterative method for seeking a value of x such that f(x)=0 (see section 4.7):
- Examples:
- #5, p. 235
- #16, p. 235
- predict a value of f close to x0 where f and f' have known values: