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- A ROCK SHOT UPWARD FROM A VOLCANO ON A DIFFERENT PLANET
WITH A VELOCITY OF 80 METERS PER SECOND.
THE HEIGHT AFTER T SECONDS IS GIVEN BY THE FUNCTION
H OF T = 80 + 48T - 0.5T SQUARED.
WE WANT TO FIND THE AVERAGE VELOCITY
OVER THE GIVEN TIME INTERVALS HERE ON THE LEFT
AND THEN PREDICT THE INSTANTANEOUS VELOCITY
AFTER 30 SECONDS.
WELL, THE AVERAGE VELOCITY OVER THE GIVEN INTERVALS
WILL BE THE SLOPE OF THE SECANT LINES
PASSING THROUGH THE TWO END POINTS OF EACH INTERVAL
AND THE INSTANTANEOUS VELOCITY WILL BE
THE SLOPE OF THE TANGENT LINE AT T = 30.
SO IF WE CONSIDER THE TWO END POINTS
FOR EACH OF THESE INTERVALS
NOTICE HOW THE TWO POINTS ARE GETTING
CLOSER AND CLOSER TOGETHER APPROACHING T = 30.
SO WE CAN USE THESE AVERAGE VELOCITIES TO MAKE A PREDICTION
ABOUT THE INSTANTANEOUS VELOCITY.
AND TO FIND THE AVERAGE VELOCITIES
WE CAN USE THE SLOPE FORMULA OR THIS FORMULA HERE
WHERE WE'LL FIND THE CHANGE IN THE HEIGHT
AND DIVIDE BY THE CHANGE IN TIME.
BUT BEFORE WE DO THIS LET'S TAKE A LOOK AT AN ANIMATION
TO GET A BETTER IDEA OF HOW THIS IS GOING TO WORK.
LOOKING AT THIS BLACK FUNCTION HERE
THE SLOPE OF THIS BLUE TANGENT LINE AT THIS BLUE POINT
WOULD BE THE INSTANTANEOUS RATE OF CHANGE.
AND NOW IF WE CONSIDER THE SLOPE OF THE RED SECANT LINE
PASSING THROUGH THE BLUE POINT AND RED POINT
NOTICE AS THE TWO POINTS GET CLOSER AND CLOSER TOGETHER
THE SLOPE OF THE SECANT LINE IS GOING TO APPROACH
THE SLOPE OF THE TANGENT LINE AS WE SEE HERE.
SO USING THE SLOPES OF THE SECANT LINE
WE'RE GONNA MAKE A PREDICTION
ABOUT THE SLOPE OF THE TANGENT LINE.
AS YOU CAN SEE AS THE TWO POINTS GET CLOSER AND CLOSER TOGETHER,
THE SLOPE OF THE SECANT LINE
WILL APPROACH THE SLOPE OF THE TANGENT LINE.
SO TO FIND THE SLOPE OF THE SECANT LINES
USING THE END POINTS OF EACH OF THESE INTERVALS
WILL TAKE QUITE A BIT OF WORK
AND BECAUSE OF THIS I'VE ALREADY SET MOST OF IT UP.
NOTICE HOW AFTER THIS FIRST INTERVAL T SUB 1 = 30,
T SUB 2 IS = TO 31.
IN THE SECOND INTERVAL T SUB 1 = 30,
T SUB 2 = 30.5, T SUB 1 = 30, T SUB 2 = 30.1,
T SUB 1 = 30 AND T SUB 2 = 30.01.
SO THE AVERAGE VELOCITY OF EACH OF THESE INTERVALS
WILL BE THE CHANGE IN THE HEIGHT
OR THE CHANGE IN THE FUNCTION VALUE
DIVIDED BY THE CHANGE IN TIME.
AND I DO WANT TO SHOW HOW WE CAN SAVE QUITE A BIT OF TIME
BY PERFORMING THESE CALCULATIONS ON A GRAPHING CALCULATOR.
WE FIRST WANT TO ENTER THE HEIGHT FUNCTION,
SO PRESS Y = AND WE'LL TYPE IN OUR FUNCTION.
INSTEAD OF USING T WE'LL USE X,
SO WE'LL HAVE 80 + 48X - 0.5X SQUARED
AND NOW WE'LL GO BACK TO THE HOME SCREEN
TO DETERMINE THESE AVERAGE VELOCITIES.
TO FIND THE CHANGE IN HEIGHT
BECAUSE OUR FUNCTION IS STORED IN Y1
WE'RE GOING TO PRESS VARS, RIGHT ARROW, ENTER, ENTER,
THIS BRINGS UP Y1,
SO TO FIND H OF 31 WE CAN ENTER Y1 OF 31
AND NOW WE'LL SUBTRACT Y1 OF 30,
SO AGAIN VARS, RIGHT ARROW, ENTER, ENTER,
IN PARENTHESIS "30,"
PRESS ENTER.
HERE IS THE CHANGE IN THE HEIGHT OVER THIS FIRST INTERVAL
AND WE'LL DIVIDE BY THE CHANGE IN TIME.
31 - 30 = 1 GIVING US AN AVERAGE VELOCITY
ON THIS FIRST INTERVAL OF 17.5 METERS PER SECOND.
LET'S GO AND DO ONE MORE OF THESE INTERVALS
THEN YOU CAN TRY THE LAST TWO ON YOUR OWN.
SO WE WANT Y1 OF 30.5 - Y1 OF 30
AND WE'LL DIVIDE THIS BY THE CHANGE IN TIME
OR DIVIDE BY 0.5.
SO WE'RE GONNA PRESS VARS, RIGHT ARROW, ENTER, ENTER,
Y1 OF 30.5 THIS TIME AND THEN WE'LL SUBTRACT Y1 OF 30.
SO HERE'S THE CHANGE IN HEIGHT.
WE'LL DIVIDE BY 0.5 WHICH IS THE CHANGE IN TIME
WHICH GIVES US AN AVERAGE VELOCITY
ON THIS INTERVAL OF 17.75 METERS PER SECOND.
SO NOTICE AS THE TWO END POINTS OF EACH INTERVAL
GET CLOSER AND CLOSER TOGETHER APPROACHING T = 30,
THE AVERAGE VELOCITY APPEARS TO BE APPROACHING
18 METERS PER SECOND AT T = 30 WHICH WOULD BE OUR PREDICTION
FOR THE INSTANTANEOUS VELOCITY AT T = 30.
I HOPE YOU FOUND THIS EXAMPLE HELPFUL.