warmup: yes calculator 1) 2). warmup find k such that the line is tangent to the graph of the...
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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