Your roommate arrives home and says, "i am so hungry. i would give up my iphone for a bowl of chili right now." you're famished, too, but you say, "here is my chili-let's trade." yor have a time preference than your roommate, because o weaker: you're willing to make trades proposed by others o weaker: you're willing to give up an immediate benefit for a long-term one o stronger: you consider communicating more important than eating o weaker: your roommate is getting what he wants from you o stronger: you are able to think more rationally about your choices

The frequency of vibrations of a vibrating violin string is given by f = 1 2l t ρ where l is the length of the string, t is its tension, and ρ is its linear density.† (a) find the rate of change of the frequency with respect to the following. (i) the length (when t and ρ are constant) (ii) the tension (when l and ρ are constant) (iii) the linear density (when l and t are constant) (b) the pitch of a note (how high or low the note sounds) is determined by the frequency f. (the higher the frequency, the higher the pitch.) use the signs of the derivatives in part (a) to determine what happens to the pitch of a note for the following. (i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates df dl 0 and l is ⇒ f is ⇒ (ii) when the tension is increased by turning a tuning peg df dt 0 and t is ⇒ f is ⇒ (iii) when the linear density is increased by switching to another string df dρ 0 and ρ is ⇒ f is ⇒