1. Calculus 6.3 Rectilinear Motion

2. 8. Apply derivatives 4Solve a problem that involves applications and combinations of these skills. 3  Use implicit differentiation to find the derivative of an inverse function  Use derivatives to solve problems involving rates of change with velocity, speed and acceleration  Use derivatives to find absolute and relative extrema in real-life context (Unit 4 Ch 5-6)  Use derivatives to model rates of change (Unit 4 Ch 5-6) 2  Uses implicit differentiation to find the derivative of an inverse but makes errors in writing the inverses or in computing the derivative  Can state the relationship between position, velocity, speed and acceleration, and can write the velocity function given position, or acceleration function given the velocity function  Shows understanding of using the derivatives to find extrema, but makes mistakes, or finds local extrema when absolute are required.  Writes an equation to describe a situation but does not write a differential equation, or makes mistakes in evaluating the function. 1With help, partial success at 2.0 & 3.0 content 0Even with help, no success

3. Vocabulary Day 1: –Rectilinear Motion Position function –Velocity function Instantaneous velocity Day 2: –Speed function Instantaneous speed Day 3 –Acceleration Function Speeding up Slowing down

4. Rectilinear Motion Motion on a line Moving in a positive direction from the origin Moving in a negative direction from the origin

5. Position Function Horizontal axis: –time Vertical Axis: –position on a line Moving in a positive direction from the origin time position Moving in a negative direction from the origin Position function: s(t) s = position (sposition duh!) t = time s(t)= position changes as time changes Sketchpad Example

6. Use the position and time graph to describe how the puppy was moving time position

7. Velocity Rate –position change vs time change –Velocity can be positive or negative positive: going in a positive direction negative: going in a negative direction Velocity Position

8. Velocity function Velocity is the slope of the position function (change in position /change in time) velocity = – Technically this is instantaneous velocity PositionVelocityMeaning Positive SlopePositive y’smoving in a positive direction Negative slopeNegative y’sMoving in a negative direction

9. Velocity Rate at which a coordinate of a particle changes with time –Insanities velocity s(t) = position with respect to time Instantaneous velocity at time t is: – time position v(t) = positive – increasing slope – moving in a positive direction v(t) = negative – decreasing slope – moving in a negative direction

10. Practice Let s(t)= t 3 -6t 2 be the position function of a particle moving along an s-axis were s is in meters and t is in seconds. –Graph the position function –On a number line, trace the path that the particle took. –Where will the velocity be positive? Negative? –Graph the instantaneous velocity. –Identify on the velocity function when the particle was heading in a positive direction and when it was heading in a negative direction.

11. Velocity or Speed Speed change in position with respect to time in any direction Velocity is the change in position with respect to time in a particular direction –Thus – Speed cannot be negative – because going backwards or forwards is just a distance –Thus – Velocity can be negative – because we care if we go backwards

12. Speed Absolute Value of Velocity – example: if two particles are moving on the same coordinate line with velocity of v=5 m/s and v=-5 m/s, then they are going in opposite directions but both have a speed of |v|=5 m/s

13. Example - s(t)= t 3 -6t 2 position time velocity time speed

14. Practice Graph the velocity function What will the speed function look like? At what time(s) was the particle heading in a negative direction? Positive direction?

15. Acceleration the rate at which the velocity of a particle changes with respect to time. –If s(t) is the position function of a particle moving on a coordinate line, then the instantaneous acceleration of the particle at time t is or

16. Example Let s(t) = t 3 – 6t 2 be the position function of a particle moving along an s-axis where s is in meters and t is in seconds. Find the instantaneous acceleration a(t) and shows the graph of acceleration verses time

17. Day 3: Speeding Up & Slowing down Speeding up when slope of speed is positive Slowing down when slope of speed is negative

18. Example When is s(t) speeding up and slowing down? position time velocity speed acceleration

19. Velocity & Acceleration function Slowing down Velocity + Acceleration - Speeding up Velocity - Acceleration - Slowing down Velocity - Acceleration + Speeding up Velocity + Acceleration +

20. Analyzing Motion GraphicallyAlgebraicallyMeaning Position Velocity  Acceleration Positive “s” values Positive side of the number line Negative side of the number line Negative “s” values s  (t)=velocity. Look for Critical Pts Postive “v” values 0 “v” values (CP) Negative “v” values Moving in + direction Turning/stopped Moving in a – direction v  (t)=acceleration Look for Critical Pts + a, + v = speeding up - a, - v = speeding up + a, - v = slowing down - a, + v = slowing down

21. Example Suppose that the position function of a particle moving on a coordinate line is given by s(t) = 2t 3 -21t 2 +60t+3 Analyze the motion of the particle for t>0 GraphicallyAlgebraicallyMeaning  Position Velocity Acceleration Never 0 (t>0), always postive Always on postive side of number line 02 5 + - +00 0<t<2 going pos direction t=2 turning 2<t<5 going neg. direction t=5 turning t>5 going pos. direction t=0 t=2 t=5 +- - + 0 2 5 3.5 v a - - + + 0<t<2 slowing down 2<t<3.5 speeding up 3.5<t<5 slowing down 5<t speeding up

22. Example Suppose that the position function of a particle moving on a coordinate line is given by s(t) = 2t 3 -21t 2 +60t+3 Analyze the motion of the particle for t>0 position velocity Acceleration

23. position velocity Acceleration Position t Direction of motion 02 5 +++ --- +++ stop positive direction negative direction positive direction 02 5 v(t) +++ --- +++ 7/2 a(t) ----------- ++++++ slowing down speeding up slowing down speeding up

24. Day 4: Applications; Gravity s = position (height) s 0 = initial height v 0 = initial velocity t = time g= acceleration due to gravity –g=9.8 m/s 2 (meters and seconds) –g=32 ft/s 2 (feet and seconds) s0s0

25. Day 4: Applications; Gravity at time t= 0 an object at a height s 0 above the Earth’s surface is given an upward or downward velocity of v 0 and moves vertically (up or down) due to gravity. If the positive direction is up and the origin is the surface of the earth, then at any time t the height s=s(t) of the object is : – g= acceleration due to gravity –g=9.8 m/s 2 (meters and seconds) –g=32 ft/s 2 (feet and seconds) s axis s0s0

26. Example Nolan Ryan was capable of throwing a baseball at 150ft/s (more than 102 miles/hour). Could Nolan Ryan have hit the 208 ft ceiling of the Houston Astrodome if he were capable of giving the baseball an upward velocity of 100 ft/s from a height of 7 ft? the maximum height occurs when velocity = 0 t=100/32=25/8 seconds s(25/8)=163.25 feet