- Return of 3.7 homework
- Units, units, units!
- Use the chain rule, folks. It's the chain rule section!
- Hand in homework: Section 3.9, p. 169, #40-45
- New assignment:
Mon 3/15 Section 4.2 - Read 4.3 (if you haven't already): Mean Value Theorem
- Begin on-line homework for 4.2 (due Wed. 3/24)
- 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:
- Definitions:
- Absolute maximum
- Absolute minimum
- They may not exist for a function (examples)
- Theorem 1: continuous on a closed interval implies max and min
- Example
- Definitions:
- local extrema
- critical point
- Theorem 2: if f(c) local extremum, then c is critical
- Finding maxes and mins
- Theorem 3: extrema on a closed interval are at critical points or on boundaries
- Theorem 4: Rolle's Theorem
- Examples:
- #1
- #2
- #3
- #10 (do some algebra!)
- #16
- #61