Integration by Parts Back to: ∫udv =uv−∫vdu ∫x ln(x)dx Choose u (to differentiate (“du”)) dv (to integrate (“v”)) u =ln(x) 1 du = dx x dv = xdx x2 v= 2 x2 ⎛ x 2⎞⎞ ⎛ 1 ⎞ ∫x ln(x)dx = 2 ln(x) −∫ ⎜⎝ 22⎟⎠⎟⎠dx⎜⎝ x ⎟⎠ dx x2 x2 = ln(x) − + C 2 4 This second integral is simpler than the first

Integration by Parts Evaluate: y 3 5 ∫y e ∫udv =uv−∫vdu dy Choose u (to differentiate (“du”)) dv (to integrate (“v”)) y 5 u =y 3 y 3 5 y 2 5 dv =ye dy 3y 2 5e 5 6y 25e 6 125e 0 625e y 5 y 5 y 5 y 5 y 5 y 5 5y e −75 y e + 750 ye −3750 e =e ( 5y 3 − 75y 2 + 750y − 3750 )

Overview Forming Beliefs about others’ strategies that are chosen simultaneously with yours depends on assumptions of each player’s rationality, and each player’s beliefs of the rationality of other players. Dominate Strategies for a player give better payoffs compared with any other strategy, no matter what other players choose for their strategies. Any rational player should choose a dominate strategy. Weakly Dominate Strategies give at least as good payoffs compared with any other strategy, no matter what other players choose, and better payoffs for at least one choice of the other players. Dominated Strategies for a player give worse payoffs compared with any other strategy, no matter what other players choose for their strategies. No rational player should choose a dominated strategy. Weakly Dominated Strategies give at least as bad payoffs compared with any other strategy, no matter what other players choose, and worse payoffs for at least one choice of the other players. Rationalizable Strategies are a best response to some belief about what other players choose, where the other players are, themselves, believed to choose rationalizable strategies. BA 445 Lesson B.4 Simultaneous Price Competition 4

Examples: Based on readiness/abilities, differentiate content in math • subtraction for some, multiplication for others, division for others Based on interests, differentiate product in Spanish • In Spanish, students choose to give play-by-play of baseball game, critical review of restaurant, critical review of movie, or news broadcast Based on learner profiles, differentiate process in science • students choose how they learn about mammals (Internet, text, video, audio, etc.)

Trig Functions: (also on Day 10) Recall that when a trig integral has ODD powers, we save one term for the “du” and replace the remaining even powers with: sin 2 x =1 −cos2 x cos 2 x =1 −sin2 x And when a trig integral has EVEN powers, we use the following trig identities to integrate. 1 sin 2 x = (1 −cos(2 x)) 2 1 cos 2 x = (1 + cos(2 x)) 2

EOC 29 An inverted “V” is made of uniform boards and weights 356 N. Each side has the same length and makes the angle shown with the vertical. Find the magnitude of the static frictional force that acts on the lower end of each leg of the “V”.

EOC 29 An inverted “V” is made of uniform boards and weighs 356 N. Each side has the same length and makes the angle shown with the vertical. Find the magnitude of the static frictional force that acts on the lower end of each leg of the “V”. Fb Fn Fg fs Since ΣF = 0 in both directions: Fn = Fg = 356N/2 Fb = fs Find: fs

Arson Investigation Point of Origin (POO). Defined as where the fire originated. Cause of fire may be near the POO. Fire usually burns longer at POO. If accelerants or ignition devices used, they may be present at the POO. Multiple POO’s MAY indicate arson. “V” patterns usually point to the POO. Extensive ceiling damages may be present above the POO. Point of Origin (POO). Interior Examination. Work backward in relation to fire travel and from least to most damage. In accidental fires, floor damage is limited in respect to the ceiling damage. “V” patterns may help locate POO.

Arson Investigation Point of Origin (POO). Defined as where the fire originated. Cause of fire may be near the POO. Fire usually burns longer at POO. If accelerants or ignition devices used, they may be present at the POO. Multiple POO’s MAY indicate arson. “V” patterns usually point to the POO. Extensive ceiling damages may be present above the POO. Point of Origin (POO). Interior Examination. Work backward in relation to fire travel and from least to most damage. Ceiling damage may lead to POO. In accidental fires, floor damage is limited in respect to the ceiling damage. “V” patterns may help locate POO.

Applying Optimal Characteristics ASI Individual work (five-minute drill) Using Using your your Organizational Organizational Matrix, Matrix, identify identify individuals individuals in in your your institution institution who who exhibit exhibit these these characteristics characteristics and and would would make good good leaders leaders for for guiding guiding the the process process for for each each standards. standards. Guiding Guiding questions: questions: Do Do you you choose choose to to have have aa few few or or many many people people overseeing overseeing the the report? report? Do Do you you choose choose to to have have one one coordinator coordinator with multiple multiple information providers? Do Do you you choose choose to to have have information information providers providers also also write? write? Do Do you you have have aa succession succession plan plan for for people people who who leave? leave? 20 Share Share your your challenges challenges and and your your results results with with your your