Warmup: YES calculator

1)

2)

Warmup

Find k such that the line is tangent to the graph of the function

9-4x y :line )( 2 kxxxf

-10k -3,x when , 2 k 3, x

3

9x

94)42(x

4-2xk

42

2

2

when

x

xxx

so

kx

3.4: Velocity, Speed, and Rates of Change

f(x) = position function

f ’(x) = velocity function

f ”(x) = acceleration function

Speed is the absolute value of velocity.

)()( xvxfspeed

Acceleration is the derivative of velocity.

)()()( xfxvxa

example:

Its position is in: feet

Velocity would be in:feet

sec

Acceleration would be in:ft

sec sec

2

ft

sec

23 34)( tttf if t is in seconds, and f(t) is in feet

Write the position, velocity, and acceleration functions with appropriate units

tttf 612)( 2

23 34)( tttf

624)( ttf

time

distance

acc posvel pos(speeding up)

acc zerovel pos(constant speed)

acc negvel pos(slowing down)

velocityzero

acc negvel neg (speeding up) acc zero

vel neg (constant speed)

acc posvel neg(slowing down)

acc zero,velocity zero(not moving)

It is important to understand the relationship between a position graph, velocity and acceleration:

This is a POSITION GRAPH f(x)

0?on accelerati theis where5)

0?on accelerati theis where4)

0?on accelerati theis where3)

0 (x)' f , 0 (x)' f , 0 (x)' f is where2)

?decreasing ?increasing f(x) is where)1

0?on accelerati theis where5)

0?on accelerati theis where4)

0?on accelerati theis where3)

0? velocity theis where)2

?decreasing ?increasing f(x) is where)1

0?on accelerati theis where4)

0?on accelerati theis where3)

0?on accelerati theis where2)

?decreasing ?increasing (x)g is where)1

Rectilinear motion (motion of an object along a straight line):

• Position is the location of an object and is given as a function of time. Conventional notation uses s(t).

Displacement is the difference between the final position and the initial position…

displacement = s(final time) – s(initial time).

Total distance traveled… Sum of each distance between turns. (turns may occur when velocity =0)

Velocity info:

Advancing (moving right)….. when velocity is positive. v>0

Retreating (moving left) … when velocity is negative. v<0

Acceleration Info:Accelerating… when acceleration is positive. a>0

Decelerating… when acceleration is negative. a<0

Both: Speeding up (going faster)…

when velocity and acceleration have the same signs. (+)(+) or (-)(-)

Slowing down (going slower) … when velocity and acceleration have

opposite signs. (+)(-) or (-)(+)

sec]. 10 sec, [0 interval time

over the meters 250404t s(t)by described as linestraight

a along is particle a ofmotion theassume example, following In the23 tt

Example 2: Find the displacement of the object over the interval [0 sec, 10sec].

Example 3: What is the total distance traveled?

Example 4: Describe the motion of the object in terms of advancing (forward) and/or retreating (backwards).

Example 5: Describe when the object is accelerating and/or decelerating.

Example 6: is the object speeding up or slowing down at 1 second? justify answer.

v(1)= negativea(1)= negative

Since both v(1)<0 and a(1)<0 , the object is speeding up at t = 1 second

In problems 14-19 assume an object is moving rectilinearly in time according to s(t) = 4t2 – 6t + 1 meters over the time interval [0, 4] seconds.

14. Find the velocity, speed, and acceleration as functions of time and give the appropriate units of each.

15. What is the velocity at t = 1 sec?

16. What is the acceleration at t = 1 sec?

17. On what time interval(s) is the particle advancing (moving to the right) and retreating? Justify your answers.

18. What is the total distance traveled?

14.

15.

16.

17.

18.

Ex. A particle is moving along a line with its position at time t given by

ft)in is s and secondsin is(t 0for t ,23)( 3 ttts

Find:

a) Find the velocity function

b) Find v(0) and v(2)

c) When is the velocity 0? where is the particle at that time?

d) Is the particle speeding up or slowing down at t = 5 seconds. Justify your answer.

Use your calculator

The Average Speed = time

Total Distance

The Average Velocity = time

positionntDisplacemetotal )(

time

Change yin velocit The Average acceleration =

e) What is Bugs speeding up or slowing down at 3 seconds? Justify your answer.

d) Is the shot speeding up or slowing down at 4 seconds? Justify your answer.

f) At what value or values of t does the particle change directions?

the end