VELOCITY AND OTHER RATES OF CHANGE 1.9. THINK ABOUT THIS YOU WORK AT WAL-MART AS A SALES ASSOCIATE. YOU ARE PAID $7.80 PER HOUR. WRITE A FUNCTION OF TIME.

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Presentation transcript:

VELOCITY AND OTHER RATES OF CHANGE 1.9

THINK ABOUT THIS YOU WORK AT WAL-MART AS A SALES ASSOCIATE. YOU ARE PAID $7.80 PER HOUR. WRITE A FUNCTION OF TIME WHICH REPRESENTS HOW MUCH MONEY YOU MAKE. HOW FAST IS THE AMOUNT OF MONEY THAT YOU HAVE CHANGING?

RATE OF CHANGE THE SLOPE IS THE RATE THAT A FUNCTION IS CHANGING. THE DERIVATIVE WILL TELL US THE “RATE OF CHANGE” IS THE DERIVATIVE OF Y WITH RESPECT TO X WE ALSO CALL IT THE RATE OF CHANGE OF Y WITH RESPECT TO X

EXAMPLE 1 FIND THE RATE OF CHANGE OF THE AREA OF A CIRCLE WITH RESPECT TO IT’S RADIUS. EVALUATE AT R = 5IN WHAT UNITS SHOULD WE USE?

EXAMPLE 2 FIND THE RATE OF CHANGE OF THE VOLUME OF A SPHERE WITH RESPECT TO ITS SURFACE AREA. EVALUATE FOR SA= ∏

VELOCITY GIVEN A POSITION FUNCTION (A FUNCTION THAT WILL TELL YOU WHERE SOMETHING IS AT ANY POINT IN TIME), THE RATE OF CHANGE OF POSITION WITH RESPECT TO TIME IS CALLED VELOCITY. WHAT IS THE RATE OF CHANGE OF VELOCITY WITH RESPECT TO TIME?

ACCELERATION THE DERIVATIVE OF VELOCITY IS ACCELERATION S’(T) = V(T) S’’(T) = V’(T) = A(T)

LINEAR MOTION THE POSITION OF A PARTICLE MOVING ALONG AN AXIS IS S(T) = T 3 – 6T 2 + 9T. FIND THE VELOCITY AT TIME T. WHAT IS THE VELOCITY AFTER 2 S? AFTER 4 S? WHEN IS THE PARTICLE AT REST? WHEN IS THE PARTICLE MOVING FORWARD? DRAW A DIAGRAM TO REPRESENT THE MOTION OF THE PARTICLE FIND THE TOTAL DISTANCE THAT THE PARTICLE TRAVELLED DURING THE FIRST 5 SECONDS. FIND THE ACCELORATION AT TIME T

LINEAR MOTION

EXAMPLE A DYNAMITE BLAST PROPELS A ROCK STRAIGHT UP IN THE AIR. THE ROCK’S HEIGHT IS REPRESENTED BY THE EQUATION S(T)= 160T - 16T 2 FT. FIND THE VELOCITY AND ACCELERATION FUNCTIONS. WHAT IS THE MAXIMUM HEIGHT OF THE ROCK? HOW FAST IS THE ROCK MOVING WHEN IT HITS THE GROUND?

EXAMPLE CONTINUED

HOMEWORK