Download presentation
Presentation is loading. Please wait.
1
We’ll need the product rule.
Part (a) u v We’ll need the product rule. They want us to find acceleration, which is the derivative of velocity. u’ = -1 v’ = t cos (t2/2) a(t) = v’(t) = (-1)(sin (t2/2)) + (-t-1)(t cos (t2/2)) a(2) = = -sin cos 2 1.587 or 1.588
2
“Is the speed of the particle increasing at t=2 ? Why or why not ?”
v(2) = -(2+1) sin(4/2) v(2) = -3 sin 2 Since a(2) was positive, but v(2) is negative, the SPEED of the particle is DECREASING at t=2.
3
The velocity changes direction when v(t) = 0.
Part (b) The velocity changes direction when v(t) = 0. -(t+1) sin (t2/2) = 0 t could be -1, since that would make the first parenthesis equal to zero. Unfortunately, -1 is not in the interval 0 < t < 3.
4
The velocity changes direction when v(t) = 0.
-(t+1) sin (t2/2) = 0 t could also be 0, since the sine of 0 is equal to zero. Unfortunately, 0 is outside the interval 0 < t < 3.
5
The velocity changes direction when v(t) = 0.
-(t+1) sin (t2/2) = 0 NOTE: t = - 2p is not in the interval. Since sin (t2/2) must be equal to 0, t2/2 has to be equal to p. This works because sin p = 0. t2/2 = p t2 = 2p t = 2p
6
Using the graphing calculator...
Part (c) TD = (t+1) sin (t2/2) dt 3 TD = (t+1) sin (t2/2) dt (t+1) sin (t2/2) dt 2p 3 Using the graphing calculator... 1.069 -(-3.265) + TD = or 4.334
7
Part (d) The greatest distance from the particle to the origin occurs when s(t) is a MAXIMUM. s(t) will be a maximum either at the endpoints (t = 0 & t =3), or when its derivative [v(t)] is zero. From Part (c), we know that v(t) = 0 at t = 2p. 2p -(t+1) sin (t2/2) dt = From Part (c)... 3 -(t+1) sin (t2/2) dt = 1.069 2p
8
Here’s what the actual movement of the particle would look like on the x-axis:
-(t+1) sin (t2/2) dt = From Part (c)... 3 -(t+1) sin (t2/2) dt = 1.069 2p
9
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 t = 0 x = 1 2p -(t+1) sin (t2/2) dt = 3 2p
10
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 t = 0 x = 1 2p -(t+1) sin (t2/2) dt = 3 2p
11
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 2p -(t+1) sin (t2/2) dt = 3 2p
12
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 2p -(t+1) sin (t2/2) dt = 3 2p
13
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 2p -(t+1) sin (t2/2) dt = 3 2p
14
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 2p -(t+1) sin (t2/2) dt = 3 2p
15
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 2p -(t+1) sin (t2/2) dt = 3 2p
16
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 2p -(t+1) sin (t2/2) dt = 3 2p
17
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 2p -(t+1) sin (t2/2) dt = 3 2p
18
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 2p -(t+1) sin (t2/2) dt = 3 2p
19
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 2p -(t+1) sin (t2/2) dt = 3 2p
20
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 2p -(t+1) sin (t2/2) dt = 3 2p
21
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 t = 2p x = 2p 3 -(t+1) sin (t2/2) dt =
22
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 3 2p
23
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 3 2p
24
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 3 2p
25
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 3 2p
26
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 3 2p
27
Here’s what the actual movement of the particle would look like on the x-axis:
-4 -2 2 4 t = 3 x = 3 2p
28
Therefore, the greatest distance from the origin would be 2.265 units.
As you could see, the greatest distance from the origin happened right here. -4 -2 2 4 Since the entire distance traveled by the particle was only 4.334, it will not make it all the way back to the origin.. t = 2p x = 3 2p
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.