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2nd Midterm Review

2nd Midterm Review. Friday, April 3 No notes/calculators Coverage: Through Section 6.1 (Except for Sections 5.6 and 5.7) Solutions will be posted on web. First and second derivatives. Sign of 1st derivative implies increasing/decreasing Sign of 2nd derivative implies concave up/down

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2nd Midterm Review

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  1. 2nd Midterm Review • Friday, April 3 • No notes/calculators • Coverage: Through Section 6.1 (Except for Sections 5.6 and 5.7) • Solutions will be posted on web

  2. First and second derivatives • Sign of 1st derivative implies increasing/decreasing • Sign of 2nd derivative implies concave up/down • Stationery point: 1st derivative vanishes • Inflection point: Change of concavity

  3. 1st and 2nd derivative test • 1st derivative test: Relative max. (resp. min.) if 1st derivative changes from + to - (resp. - to +) • Second Derivative test: If f (x)=0 and f (x) < 0 then rel. max.; If f (x)=0 and f (x) > 0 then rel. min. • Relative extrema must occur at critical points

  4. Sketching graphs • Determine critical points • Determine intervals of increase/decrease • Determine places where 2nd derivative vanishes • Determine inflection points and intervals of concave up/down • Model problems: Section 5.3 HW

  5. Absolute Max/Min • Continuous functions on closed bounded intervals have absolute max. & min. • These must occur at critical points • Functions with exactly one relative extremum must have an absolute extremum at that point

  6. Applied Max/Min Problems • Draw picture • Find formula for quantity to be minimized • Eliminate variables using conditions in problem • Find interval of remaining variable • Find relevant max/min

  7. Rolle’s Theorem & Mean Value Theorem • What goes up must come down • Average velocity must be attained • Model problem: x3+4x+5=0 cannot have more than one root.

  8. L’Hôpital’s Rule • Various indeterminate cases: 0/0, / , 0  , 1, ...

  9. Integration

  10. Integration formulae

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