RATE OF CHANGE AND INSTANTANEOUS VELOCITY

RATE OF CHANGE AND INSTANTANEOUS VELOCITY Section 2.2A Calculus AP/Dual, Revised 2017 viet.dang@humbleisd.net 7/30/2018 1:34 AM 2.2A: Rates of Change 1

AVERAGE VELOCITY A. Rates of change play a role whenever we study the relationship between two changing quantities. A familiar example is velocity, which is the rate of change of position with respect to time. B. If an object is traveling in a straight line, the average velocity over a given time interval from point A to point B C. We cannot define instantaneous velocity as a ratio because we would have to divide by the length of the time interval, which is zero. However we can estimate the instantaneous velocity by computing the average velocity over successively smaller intervals. 7/30/2018 1:34 AM 2.2A: Rates of Change 2

AVERAGE RATE OF CHANGE A. The average rate of change describes the constant rate at which a function would have to change over an interval if the function were to achieve the same vertical displacement over the length of the interval investigated. B. Average Velocity Formula: ഥx s t f b f a b a Change in Position Time Elapsed 7/30/2018 1:34 AM 2.2A: Rates of Change 3

STEPS IN AVERAGE RATE OF CHANGE A. Plug your a and b in to get the corresponding f a and f b. B. Plug into the formula and reduce. C. Average Rate of Change uses only endpoints of the interval. It does not reflect any fluctuations within the interval. D. Label. 7/30/2018 1:34 AM 2.2A: Rates of Change 4

EXAMPLE 1 A car travels along a straight road and the driver observes the mile markers along the trip as shown in the table. What are the units for the average speed over the 24 minutes? Explain your answer. Time (in min.) 0 8 20 24 Mile Marker 14 27 42 50 The average speed is Miles f b f a y b a x Minute f b f a b a 24 0 24 2min 50 14 36 3 miles 3 2 miles per minute over 24 minutes 7/30/2018 1:34 AM 2.2A: Rates of Change 5

EXAMPLE 2 Water is pumped into a tank at a rate modeled by W t 2000e t2 /20 liters per hour for 0 t 8, where t is measured in hours. Using the table below, estimate R 2 and show all work that leads to the answer. Explain your answer. t (hours) 0 1 3 6 8 R t (Liters/Hr) R '2 1340 1190 950 740 700 f ( a) f b b R '2 a 7/30/2018 1:34 AM 2.2A: Rates of Change 6 R '2 liters liters 950 1190 hour hour 3 hours 1 hour f ( 3) f ( 1) 3 1

EXAMPLE 2 Water is pumped into a tank at a rate modeled by W t 2000e t2 /20 liters per hour for 0 t 8, where t is measured in hours. Using the table below, estimate R 2 and show all work that leads to the answer. Explain your answer. R '2 t (hours) 0 1 3 6 8 R t (Liters/Hr) liters liters 950 1190 hour hour 3 hours 1 hour 1340 1190 950 740 700 liters 240 hour liters R ' 2 120 2 hours hour 2 liters The rate of change is 120 ( 120 liters per hour per hour ) at t 2. 2 hour 7/30/2018 1:34 AM 2.2A: Rates of Change 7

YOUR TURN Jo Ann jogs along a straight path. For 0 t 40, Jo s speed is given by a differential function v t where t is measured in minutes and v t is measured in meters per minute. Using the table below, estimate v 16 and show all work that leads to the answer. Explain your answer and label accordingly. t (Min) 0 12 20 24 40 v t (Met/Min) 0 200 240 220 150 meters The rate of change is 5 or 5 meters per min per min at t 16. 2 min 7/30/2018 1:34 AM 2.2A: Rates of Change 8

EXAMPLE 3 A tennis ball is dropped from a height of 100 feet, its height s at time t is given by the position function, d t 16t 2 + 100 where d is measured in feet and t is measured in seconds. Find the average velocity over the time interval of 1, 2. f ( a) f b d b a 2 d 1 2 1 36 84 48 2 1 1 48 ft./ sec 7/30/2018 1:34 AM 2.2A: Rates of Change 9

EXAMPLE 3 A tennis ball is dropped from a height of 100 feet, its height s at time t is given by the position function, d t 16t 2 + 100 where d is measured in feet and t is measured in seconds. Find the average velocity over the time interval of 1, 2. 7/30/2018 1:34 AM 2.2A: Rates of Change 10

EXAMPLE 4 If an object is dropped from a tall building, then the distance it has fallen after t seconds is given by the function d t 16t 2. Find its average speed (average rate of change) between a and a + h seconds. 16 7/30/2018 1:34 AM 2.2A: Rates of Change 11

YOUR TURN Suppose you are driving to a friend s house. After 30 minutes, you are 15 miles from your house. After 3 hours, you are 60 miles from your house. What is your average rate of change between these times (in hours)? Label accordingly. s ' t 18 miles / hour 7/30/2018 1:34 AM 2.2A: Rates of Change 12

INSTANTANEOUS VELOCITY A. If we look at a graph of the position of the object, a secant line can be drawn through the points t 1, f a and t 2, f b. The average velocity of the object would be the slope of the secant line through t 1, f a and t 2, f b. B. If the secant line from t 2 is moved closer to t 1, the secant line gets closer and closer to becoming a tangent line to the graph at t 1, f a. The slope of the tangent line is the instantaneous velocity of the object when the time is t 1 C. The instantaneous rate of change at is the limit of the average rates of change as gets closer and closer to We must consider values of on both the left side of and on the right side of as we take the limit. D. An expression such as s quotient. t f b f a b a is called a difference 7/30/2018 1:34 AM 2.2A: Rates of Change 13

INSTANTANEOUS VELOCITY A. Velocity of an object at time t is found by taking the DERIVATIVE of the position function. B. VELOCITY is a vector, which means it has direction. So, velocity can be positive, negative, or zero. C. SPEED is the absolute value of velocity, and cannot be negative. s t+ t s t D. Instantaneous Velocity Formula: lim t 0 t known as ds. dt s t v t. It is also 7/30/2018 1:34 AM 2.2A: Rates of Change 14

AVERAGE VELOCITY VS. INSTANTANEOUS VELOCITY Average Velocity Solve using Algebra Slope - (Not much Calculus needed) Instantaneous Velocity s t Take the Derivative 7/30/2018 1:34 AM 2.2A: Rates of Change 15

EXAMPLE 5 At time t 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s t 16t 2 + 16t + 32 where s is measured in feet and t is measured in seconds. (a) Graph the function on the graphing calculator and sketch the result below. (b) Find the average velocity (average rate of change) at which the diver is moving for the time interval 1, 1. 5. (c) Use your answers to (b) to estimate the instantaneous rate of change (instantaneous velocity) at which the diver is moving at time t 1 second. (d) When does the diver hit the water? (e) What is the diver s velocity at impact? What is their speed? (f) When does the diver stop moving upward and start their descent? 7/30/2018 1:34 AM 2.2A: Rates of Change 16

EXAMPLE 5A At time t 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s t 16t 2 + 16t + 32 where s is measured in feet and t is measured in seconds. (a) Graph the function on the graphing calculator and sketch the result below. 7/30/2018 1:34 AM 2.2A: Rates of Change 17

EXAMPLE 5B At time t 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s t 16t 2 + 16t + 32 where s is measured in feet and t is measured in seconds. (b) Find the average velocity (average rate of change) at which the diver is moving for the time interval 1, 1. 5. f ( a) f b s b a 1.5 s 1 1.5 1 20 32 1.5 1 12 0.5 24 ft./ sec 7/30/2018 1:34 AM 2.2A: Rates of Change 18

EXAMPLE 5C At time t 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s t 16t 2 + 16t + 32 where s is measured in feet and t is measured in seconds. (c) Use your answers to (b) to estimate the instantaneous rate of change (instantaneous velocity) at which the diver is moving at time t 1 second. 2 s t 16t + 16t + 32 ds 2 v t 16t 16t 32 dt + + ds 2 ds ds v t 16t + 16t + 32 dt dt dt v t t + 32 16 7/30/2018 1:34 AM 2.2A: Rates of Change 19

EXAMPLE 5C At time t 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s t 16t 2 + 16t + 32 where s is measured in feet and t is measured in seconds. (c) Use your answers to (b) to estimate the instantaneous rate of change (instantaneous velocity) at which the diver is moving at time t 1 second. v t 32t + 16 v t 32t + 16 v( t ) 32( 1) + 16 v t 16 ft. / sec. 7/30/2018 1:34 AM 2.2A: Rates of Change 20

EXAMPLE 5D At time t 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s t 16t 2 + 16t + 32 where s is measured in feet and t is measured in seconds. (d) When does the diver hit the water? s t 16t + 16t + 32 2 2 0 16t + 16t+ 32 ( 2 t t ) 0 16 2 0 16 t 2 t+ 1 t 2sec. 7/30/2018 1:34 AM 2.2A: Rates of Change 21

EXAMPLE 5E At time t 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s t 16t 2 + 16t + 32 where s is measured in feet and t is measured in seconds. (e) What is the diver s velocity at impact? What is their speed? v t 32t + 16 v ( 2) 32( 2) + 16 speed 48 ft. / sec 7/30/2018 1:34 AM 2.2A: Rates of Change 22

EXAMPLE 5F At time t 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s t 16t 2 + 16t + 32 where s is measured in feet and t is measured in seconds. (f) When does the diver stop moving upward and start their descent? v t 32t + 16 0 32t + 16 32t 16 t 1 sec. 2 7/30/2018 1:34 AM 2.2A: Rates of Change 23

EXAMPLE 6 The temperature T, in degrees Fahrenheit, of a cold potato placed in a hot oven is given by T f t, where t is the time in minutes since the potato was put in the oven. What is the meaning of the statement f 20 2? f ( t) T f '( 20) 2 At time t 20 minutes, the temperature of the potato is increasing at a rate of 2 F per minute. 7/30/2018 1:34 AM 2.2A: Rates of Change 24

AP FREE RESPONSE QUESTION 1 (NON-CALCULATOR) Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for selected points in the closed interval, 2, 13. Estimate f 4. Show the work that leads to the answer. f f ' x '4 f b f f a b a 5 f 3 5 3 x 2 3 5 8 13 f(x) 1 4 2 3 6 f f '4 '4 f '( 4) 3 2 4 ( 5) ( 3) 6 2 7/30/2018 1:34 AM 2.2A: Rates of Change 25

AP FREE RESPONSE QUESTION 2 (NON-CALCULATOR) W 8 221 gallons At 8 hours, there are 221 gallons of water W 1 257 12 At 12 hours, there are 257 gallons of water 7/30/2018 1:34 AM 2.2A: Rates of Change 26

AP FREE RESPONSE QUESTION 2 (NON-CALCULATOR) W ' 15 f 16 f 12 16 12 W ' 15 294 257 16 12 W ' 15 9.25 gallons / hour At t 15 hours, the rate of change of water flowing is 9.25 gallons per hour 7/30/2018 1:34 AM 2.2A: Rates of Change 27

ASSIGNMENT Page 114 93-97 all, 105 7/30/2018 1:34 AM 2.2A: Rates of Change 28