equation x = 6*t2 -3*t -9*t3 What is the instantaneous velocity at t = 3 seconds?

Question

6*t^2-3*t-9*t^3

Answer 1

velocity (v) is the first derivative of space (x) so
v=dx/dt = 12t-3-27t^2
for t=3 we get
v(3)= - 210 m/s

Answer 2

Calculus is equipped in reachable right here. Derive x with know to t making use of the capacity rule. This gets you: dx/dt = 4.5t^2 dx/dt=distance/time=velocity Now in basic terms plug in 2 and 3 for t interior the equation above to get on the spot velocity at those cases.

Answer 3

Velocity V=12t - 3 - 27t^2
at t=3, V = 12*3 -3 - 27*9
= 36 - 3 - 243
= -210

It's moving backwards fast!

Answer 4

Some parts of Calculus allow you to use derivatives, but you might have to use limits.

Using limits:

The instantaneous velocity is the limit as {h ---> 0} of [f(t + h) - f(t) ] / [(t + h) - (t)]

f(t + h) = 6(t + h)^2 - 3(t + h) - 9(t + h)^3

=6(t^2 + 2th + h^2) - 3(t + h) - 9(t + h)(t^2 + 2th + h^2)

=6t^2 + 12th + 6h^2 - 3t - 3h - 9(t^3 + 2ht^2 + th^2 + ht^2 + 2th^2 + h^3)

=6t^2 + 12th + 6h^2 - 3t - 3h - 9t^3 - 18ht^2 - 9th^2 - 9ht^2 - 18th^2 - 9h^3

= -9t^3 + (6t^2 - 18ht^2 - 9ht^2) + (12th - 3t - 9th^2 - 18th^2) + (6h^2 -3h - 9h^3)

Next,
[f(t + h) - f(t)] =

[-9t^3 + (6t^2 - 18ht^2 - 9ht^2) + (12th - 3t - 9th^2 - 18th^2) + (6h^2 -3h - 9h^3) ] - [6t^2 - 3t - 9t^3]

=[ (-9t^3 + 9t^3) + ( -6t^2 + 6t^2 - 18ht^2 - 9ht^2) + (3t + 12th - 3t - 9th^2 - 18th^2) + (6h^2 -3h - 9h^3) ]

=[ (0) + (0 - 18ht^2 - 9ht^2) + (0 + 12th - 9th^2 - 18th^2) + (6h^2 -3h - 9h^3) ]

=[ h(-18t^2 - 9t^2) + h(12t - 9th - 18th) + h(6h -3 - 9h^2) ]

Finally,

[f(t + h) - f(t) ] / [(t + h) - (t)]

=[ h(-18t^2 - 9t^2) + h(12t - 9th - 18th) + h(6h -3 - 9h^2) ] / [(t + h) - (t)]

=[h]*[ (-18t^2 - 9t^2) + (12t - 9th - 18th) + (6h -3 - 9h^2) ] / [h]
=[ (-18t^2 - 9t^2) + (12t - 9th - 18th) + (6h -3 - 9h^2) ]

Now evaluate the limit as {h ---> 0}

[ (-18t^2 - 9t^2) + (12t - 9t(0) - 18t(0)) + (6(0) -3 - 9(0)^2) ]
Instantaneous Velocity =[ -27t^2 + 12t - 3 ]
At t = 3
V = [ -27(3)^2 + 12(3) - 3 ]
= -210 meters / second

Using derivatives:

The instantaneous velocity is the derivative of the position function:

x'(t) = 12t - 3 - 27t^2
x'(3) = 12(3) - 3 - 27(3)^2
= -210 meters/ second

Answer 5

velocity = dx/dt (i.e. first derivative of x with respect to time)
= 12t - 3 - 27t^2

At t = 3 sec =====> v = 12*3 - 3 - 27*3^2 = 36-3-27*9 = -210

Answer 6

54-9-27*9

=-198

Answer 7

= check ur math book ,, nerdy