- FYI: next exam is Monday, 11/15
- Last minute homework assignment 7.4's due
- Assignment 7.7 due today
- For example, the sine function has inverse function
arcsine (or asin -- note the bad notation below -- don't be confused! But our notation in mathematics is sometimes "context sensitive"):
and cosine function has inverse function arccos:
- Arctan and arccot are also sometimes useful:
- Derivatives: They're computed in the usual "sneaky" inverse function way:
Exercise 49, p. 398
- Examples:
- #50, p. 398
- Optimal Positioning of Fixed Solar Panels
- First of all, the big picture: integration by parts is just the
product rule backwards:
Example: Exercise #2, p. 428
- Integration by parts may need to be carried out multiple times:
sometimes the idea is to simplify the integral each time, until a
really simple one arises allowing us to calculate the final solution
(e.g. Example 4, p. 425). Sometimes it's something of a trick: we
compute the integral multiple times in order to return to the original
integral, allowing us to solve an equation for the original integral
(e.g. Example 5, p. 426).
Example: Exercise #18, p. 428
- Suggested strategy:
- Choose u so that u' is simpler than u itself.
- Choose v' so that its antiderivative
is known, or can be evaluated.
- Sometimes v'=1 is a good choice.
Example: Exercise #25, p. 428
- Of course, all this works with definite integrals:
Example: Exercise #49, p. 428