43 Derivative as Instantaneous Rate of Cange Consider a function tat describes te position of a racecar moving in a straigt line away from some starting point Let y s t suc tat t represents te time in ours since te race began and st represents te distance in miles te car as driven from its starting position In tis scenario, te point (0,0) indicates tat te car is zero miles from te starting point after zero ours, and te point (1,40) indicates tat te car is 40 miles from te starting point after one our Suc a function is a position function Quite naturally, te driver of te racecar is interested in is/er speed, wic is te rate of cange of te car's position Indeed, car speeds in miles per our equal an average, namely, te distance traveled divided by te elapsed interval of time If te racecar is 40 miles from te starting point after one our of driving, ten te car's average rate of speed must ave been 40 miles per our In general, consider te time interval from t = a to t = a + Over tis interval, s a s a, and te average velocity is given by te cange in position is cange in position time elapsed s a s a s a s a a a At any given point during te race, owever, te driver is less interested in is average speed and more interested in is speed at tat moment, is/er instantaneous velocity We can approac instantaneous velocity by computing te average velocities over sorter and sorter intervals of time tat is, as approaces zero; tereby, arriving at te instantaneous velocity by taking te limit as defined below Given a position function va, is given is given by Comparing te definition above to te definition of te slope of a tangent line, we see tat instantaneous velocity at time a (te instantaneous rate of cange) is equivalent to te slope of a line tangent to te position function at time a, wic we defined as te derivative If a is some v a is actually a function Replacing a wit te variable t, we ave te point in general, ten definition for a velocity function, te derivative of a position function Given a position function y s t v a y s t, te instantaneous velocity at point a, denoted s a s a lim, te instantaneous velocity of te moving object is a function of time given by te derivative of s st st vt s ' t lim In te preceding discussion, we found te instantaneous velocity, wic was really te instantaneous rate at wic te position canged In general ten, for any given function y f x, te instantaneous rate of cange of te y-values equals te limit of te ratio of cange
44 in y to cange in x as te cange in x approaces zero In symbols, we write lim y x Allowing to represent te cange in x, we ave: f x f x instantaneous rate of cange lim Note tat te instantaneous rate of cange equals te derivative of f, so we ave anoter interpretation of te derivative Geometrically, te derivative is a function f ' tat gives te slope of a line tangent to te function f at any given point Also, te derivative is a function f ' tat gives te instantaneous rate of cange of te function f at any given point If f is a position function, f ' is te velocity function Just as te velocity function is te derivative of te position function, te acceleration function is te derivative of te velocity function In general, if y s t gives te position of an object, vt gives te velocity of te object, and vt s ' t and at v' t x 0 at gives te acceleration of te object, ten
45 Example Exercise S t 5t 3t 10 describes te position of an object in meters after t seconds Find te velocity of te object after tree seconds Te velocity equals te derivative of te position function S t t t 5 3 10 3 3 S S v3 S' 3 lim 53 33 10 53 33 10 v3 lim 5 9 6 9 3 10 44 v3 lim v 3 lim 33 5 v3 lim v 3 lim v 45 30 5 3 45 33 5 3lim33 5 v 3 33 Te velocity equals 33 meters per second
46 Practice Problems in Calculus: Concepts and Contexts by James Stewart 1st ed problem set: Section 6 #13 17 odd, Section 7 #3 nd ed problem set: Section 6 #13 17 odd, Section 7 #5 3rd ed problem set: Section 6 #15 19 odd, Section 7 #5 Practice Problems in Calculus: Early Transcendentals by Briggs and Cocran 1st ed problem set: Section 31 #57, Section 3 #66a, #66b s t 49t Given te position function velocity function Answer: vt vt s t s t lim Possible Exam Problem 49 t 49t lim 49 t t 49t vt lim 98t 49 vt lim v t lim 98t 49 v t 98t, use te definition of instantaneous velocity to find te Application Exercise Te function A 58t 083t gives te eigt of a projectile wit an initial velocity of 58 meters per second Wat is te velocity of te projectile after one second?
47 Practice Problems #1 A particle is moving along a orizontal line according to s3t 1 were s meters is te directed distance from some starting point at t seconds Wat is te instantaneous velocity in meters per second wen 3 seconds ave elapsed? # A stone is dropped from a eigt of 64 feet and s16t 64 gives te eigt of te stone in feet t seconds after being dropped Wat is te velocity of te stone te moment tat it strikes te ground? #3 Te function s 560t 16t gives te eigt of a rocket in feet t seconds after launc Wat is te velocity of te rocket two seconds after launc? #1 18 meters per second #3 496 feet per second