Day 18 in Mat129

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Announcements

Today: hand in pp. 137, #52-54 (it was due Thursday, 2/18!;)

New assignment:

Thu

2/25

Section 3.6 and the Chain rule

Read 3.9: Related Rates
Begin on-line homework for sections 3.7 (called 3.7a, due Fri, 3/5).
Problems to hand in: p. 154, #82-85 (due Thu, 3/4)

Section 3.6: Trigonometric Derivatives
- Derivative of cosine from the definition, and trig identities: we required a couple of limits:
  - ${\lim_{h\rightarrow{0}}\frac{\sin(h)}{h}=1}$
  - ${\lim_{h\rightarrow{0}}\frac{\cos(h)-1}{h}=0}$
- Problems:
  - Derivative of tangent
  - Definition of secant, cosecant
    - ${sec(x)=\frac{1}{\cos(x)}}$
    - ${csc(x)=\frac{1}{\sin(x)}}$
  - #23, p. 145
  - #27, p. 145
  - #47, p. 146
Section 3.7: The Chain Rule
- The chain rule is about the derivative of composition of two functions. Many important functions -- perhaps even most important functions -- can be thought of as compositions.
- Compositions are like a succession of trips. Here's a nice explanation of the Chain Rule you might want to check out.
- Motivation from the limit definition of the derivative:
  
  ${\frac{d}{dx}\left[f\left(g(x)\right)\right]}$ ${=\lim_{h\rightarrow{0}}\frac{f(g(x+h))-f(g(x))}{h}}$
  
  ${\approx\lim_{h\rightarrow{0}}\frac{f(g(x)+hg'(x))-f(g(x))}{h}}$
  
  ${=\hspace{.3cm}\lim_{hg'(x)\rightarrow{0}}g'(x)\left[\frac{f(g(x)+hg'(x))-f(g(x))}{hg'(x)}\right]}$
  
  ${=g'(x)f'(g(x))}$
  
  ${=(stuff)'f'(stuff)}$
- Examples:
  - Preliminary questions 1-4, p. 152
  - #21-50, inner and outer, p. 153
    - #29
    - #37
    - #41
  - #68, p. 153

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