- Today: hand in pp. 137, #52-54 (it was due Thursday, 2/18)
- New assignment:
Tue 2/23 Section 3.5; Trig Derivatives; the Chain rule - Begin on-line homework for sections 3.5 (due Mon, 3/1) and 3.6 (due Tue, 3/2)
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- "Little fleas have littler fleas upon their backs to bite em / and littler fleas have littler fleas / and so ad infinitum."
- Functions that are differentiable have derivative functions, that are functions and may be differentiable and hence have derivative functions, and so ad infinitum.
- There's nothing especially novel about the calculation of higher
derivatives.
- Using the calculator to compute higher derivative
- They do provide something new in the way of interpretation of the
behavior of a curve.
- Degree one polynomials have zero second derivatives.
- Non-zero second derivatives give us concavity, and inflection
- What happens when the second derivative
- is zero?
- is positive?
- is negative?
- changes from negative to positive?
- changes from positive to negative?
- An important case: the quadratic
- The quadratic gives us total insight into the cases of bowls and umbrellas.
- A good test of your understanding of bowls versus umbrellas
- Galileo and the second derivative of the position function under gravity:
- #33, p. 141
- #35, p. 141
- #36, p. 141
- #37, p. 141
- Derivative of cosine from the definition, and trig
identities. We'll need a couple of limits:
-
- Problems:
- #23, p. 145
- #27, p. 145
- #47, p. 146
- It's about the derivative of composition of two functions. Many
important functions -- perhaps even most important functions --
can be thought of as compositions.
- The Chain Rule
- Motivation from the limit definition of the derivative:
- Examples:
- Preliminary questions 1-4, p. 152
- #21-50, inner and outer, p. 153
- #29
- #37
- #41
- #68, p. 153