Need help understand graphs!!

Question

I also dont understand how you can determine what the derivatives graph would look like? and is the answer it a) B??

b) Find the velocity at time t

c) What is the velocity after 4 seconds

d) At what time t is the particle at rest

e) Find the total distance travelled during the first 6 seconds

Answer 1

1(a).

The function x = f(t) = 2t / (1 + t²).

Make the table :

t	x = f(t) = 2t / (1 + t2)	(t, x)
0	x = 2(0) / [1 + (0)2] = 0 / 1 = 0	(0, 0)
1	x = 2(1) / [1 + (1)2] = 2 / 2 = 1	(1, 1)
2	x = 2(2) / [1 + (2)2] = 4/5	(2, 4/5)
3	x = 2(3) / [1 + (3)2] = 6/10 = 3/5	(3, 3/5)
4	x = 2(4) / [1 + (4)2] = 8/17	(4, 8/17)
5	x = 2(5) / [1 + (5)2] = 10/26 = 5/13	(5, 5/13)

Graph the function x = f(t) = 2t / (1 + t²) by using the above the table values.

The above graph matching with given graph B.

Answer 2

1(b).

Velocity = dx/dt = (d/dt)[ 2t / (1 + t²) ]

V = [ (1 + t²) · (d/dt)(2t) - (2t) · (d/dt)(1 + t²) ] / ((1 + t²)²

V = [ 2(1 + t²) - 4t² ] / ((1 + t²)²

V = (2 - 2t²) / (1 + t²)²

1(c).

Velocity V = (2 - 2t²) / (1 + t²)²

Substitute the value of t = 4 in the above function.

V = [2 - 2(4)²] / [1 + (4)²]²

V = [2 - 32] / (17)²]

V = - (30/289) = - 0.104 m / s.

1(d).

Velocity V = 0.

2(1 - t²) / (1 + t²)² = 0

1 - t² = 0

t² = 1

t = ± 1 = 1, since time is positive.

At 1 second the velocity is zero.

1(e).

Distance : x = f(t) = 2t / (1 + t²).

Substitute the value of t = 6 in the above function.

x = f(6) = 2(6) / (1 + 6²)

x = 12 / 37 meters.