5.1B ITC Notes Date: ____________________________ Antiderivatives and Indefinite Integration Tree Growth: An evergreen nursery usually sells a certain tree after 8 years of growth and shaping. The dh 1.6t 4 where t is the time in years and h is growth rate during those 8 years is approximately by dt the height in centimeters. The seedlings are 10 cm tall when planted (t=0). a) Find the height after t years. b) How tall are the trees when they are sold (after 8 years)? dP of a population of bacteria is proportional to the square root of dt t, where P is the population size and t is the time in days (0 t 10) dP k t dt The initial size of the population is 500. After one day the population has grown to 600. Estimate the population after 7 days. Population Growth: The rate of growth Vertical Motion: The ball is thrown upwards from a height of 80 feet with an initial velocity of 64 feet per second. (Use -32 feet per second as the acceleration due to gravity) a) Find the position function s(t) b) When does the ball hit the ground and with what velocity? Vertical Motion: The Grand Canyon is 1600 meters deep at its deepest point. A rock is dropped from the rim above this point. (Use -9.8 meters per second as the acceleration due to gravity) a) Find the position function s(t) b) How long will it take to hit the canyon floor? c) What was its velocity at impact? Deceleration: A car is traveling at 60mph when the brakes are fully applied, producing a constant deceleration of 20 feet per second. What is the distance traveled before the car comes to a complete stop? Acceleration: The maker of a certain automobile advertises that it takes 10 seconds to accelerate from 0 kilometers per hour to 90 kilometers per hour. Assuming constant acceleration, compute the following: a) The acceleration in meters per second b) The distance the car travels during the 10 seconds. Rectilinear Motion: A particle moves with acceleration function x ''(t) 12t 2 6t 8 . Its initial velocity is 18 meters per second and its initial displacement is 40 meters. Find the position function after t seconds.