Applications Of The Derivative

Applications of the Derivative to the Natural and Social Sciences 1. A well-established formula from physics says that when an object is thrown into the air with an initial velocity of v0 and from a height of s0 feet above the ground, its height (s, in feet above the ground) after it’s been in the air for t seconds is given by: s(t) = −16t2 + v0 t + s0 We will do a demonstration of throwing a bouncy ball into the air (fun!) in which you will measure the height above the ground when the ball is released and how long it takes for the ball to come back to Earth. From these observations, answer the following questions. (a) What was the initial velocity of the ball? (b) What was the average velocity of the ball during the first two seconds of flight? (c) What was the instantaneous velocity of the ball at exactly 2 seconds into its flight? (d) What is the acceleration of the ball at t = 2? (e) How high did the ball go? (f) At what point(s) was the ball 2 feet above the ground? (g) How fast was the ball going when it hit the ground? Note that some of these questions may not involve calculus. 2. A sports clothing store sells all of its football jerseys for the same price, which we will denote by x (measured in dollars). The number of jerseys, q, that are sold in a day depends on the price, so we can say q = f (x). (a) Suppose we knew that f (100) = 150 and f 0 (100) = −20. What do these two quantities mean in terms of sales of jerseys? (b) The revenue earned by the store from the sale of jerseys is defined to be the number of jerseys sold times the price per jersey. In other words, if R represents revenue, then R = q · x, or said differently R = x · f (x). What is the revenue earned by the store when it prices jerseys at $100 each? What is the rate at which that revenue is changing when the price is $100? If the price were changed from $100 to $101 per jersey, what could the store owners expect in terms of revenue change?

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