1. Analytically determine the equation of the line tangent to f at x = 2.

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1 Warm Up!! Given: 1. Analytically determine the equation of the line tangent to f at x = Using transformations to help you, sketch the graph of. 3. On the same grid, sketch the graph of. y x

2 Given: 4. Is f increasing or decreasing at x = 2? Justify analytically. 5. Write the equation of. 2

3 Big Ideas: * Displacement versus Position versus Distance Traveled * Velocity versus Speed * Speeding up and slowing down Goals * Given an equation for the displacement or position of a moving particle, write an equation for > the velocity of the object at any time, t > the acceleration of the object at any time, t * Analytically analyze the motion of the particle using the equations for the velocity and acceleration. 3

4 Displacement vs. Distance Traveled Displacement is the directed distance between a particle's terminal position and its initial position. May be positive or negative or zero The distance traveled is the total distance traveled from first point to second point. Always non negative Displacement: you only care where you end up. Distance: you care how you got there 4

5 The velocity is the instantaneous rate of change in the displacement with respect to time. That is, the velocity function is the derivative of the displacement function. Let be the displacement of a particle. The unit for f is a unit of length (ft, in, m, etc) Then is the velocity. The unit for is. 5

6 A velocity has two parts: the sign and the rate The sign of the velocity tells you the direction you are moving: * positive means displacement is increasing (so usually particle is moving up or to the right). * negative means the displacement is decreasing (so usually the particle is moving down or to the left). The rate tells you have fast the displacement is changing in the given direction. A velocity of zero means the particle is at rest. 6

7 Example: The height in meters of a falcon above the ground at x minutes is given by a. What is the displacement of the falcon at 1.5 minutes? b. What is the velocity of the falcon at min? Justify your answer using correct notation and include units. c. At t = 2 min, is the falcon flying up or down? How do you know? 7

8 Velocity vs. Speed Velocity describes the direction and rate that a particle is moving. Velocity is the derivative of displacement. Speed describes the rate that a particle is moving. Speed has no direction. Speed is the absolute value of velocity. 8

9 Acceleration is the instantaneous rate of change of velocity. Acceleration is the derivative of the velocity. The unit of acceleration is the unit of velocity divided by time. displacement velocity acceleration The sign of the acceleration describes whether the velocity function is increasing or decreasing since acceleration is the "slope" of velocity. (Note: This does not tell you whether a particle is speeding up! Remember "speed" and "velocity" are different.) 9

10 Remember our falcon? (minutes, meters) Is the falcon moving up or down at t = 1.8 min? Justify analytically. What is the falcon's speed at t = 1.8? Justify analytically. What is the falcon's acceleration at t = 1.8. Justify analytically. 10

11 Speeding up Slowing down Speed is increasing Speed is decreasing Velocity and acceleration have the same sign. Velocity and acceleration have the different signs. If the velocity is positive and the slope of the velocity is positive, then the velocity is getting further from zero. The speed, therefore, is getting further from zero. Similarly, if the velocity is negative and the slope of the velocity is positive, then the velocity is getting further from zero. Therefore, the speed is getting further from zero so is increasing. If the velocity is positive but the slope of the velocity is negative, then the velocity is getting closer to zero. The speed, therefore, is getting closer to zero. Similarly, if the velocity is negative but the slope of the velocity is positive, then the velocity is getting closer to zero. Therefore, the speed is getting closer to zero. 11

12 1. What is the difference between displacement of an object and the distance it travels while it is in motion? 2. What is the difference between the velocity of a moving object and the speed at which it is traveling? 12

13 3. True or False: "If the acceleration of a moving object is negative, then the object is slowing down." 13

14 4. A particle moves along the x axis with a displacement x feet from the origin given as a function of t seconds by. a. Write the equation of the velocity. b. At t = 2, what are the velocity and acceleration of the particle? c. Is the particle speeding up or slowing down at t = 2? At what rate? 14

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17 Attachments TI SmartView_TI 84.exe

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