Question

The position of an object at time t is given by s(t) = -2 - 6t. Find the instantaneous velocity at t = 2 by finding the derivative.

Answer 1

@bibby

Answer 2

as I remember it, the velocity is s'(t)
what is the first derivative of -2-6t

Answer 3

(d/dx)(-2-6t)?

Answer 4

well you just rephrased my question

Answer 5

it would be d/dt (-2 - 6t)

which is (term by term
\[ \frac{d}{dt} (-2) -6 \frac{d}{dt}t \]

Answer 6

constants don't change with time. in other words d/dt (-2) is 0
that leaves
\[ -6 \frac{d}{dt} t \]

Answer 7

okay, so what do I do with the derivative?

Answer 8

@bibby @phi

Answer 9

take the derivative of t

Answer 10

-12t?

Answer 11

after you get s'(t) , which in this case is
s'(t)= -6 (a constant)
you evaluate it at t=2
you (of course) get -6

Answer 12

or just -12?

Answer 13

if you don't know how to find the derivative of d/dt t
then you need to review the intro...

Answer 14

\(\frac{d}{dt}(-2-6t)=\frac{d}{dt})-2_-\frac{d}{dt}6t\)

Answer 15

see
http://www.khanacademy.org/math/differential-calculus/taking-derivatives/power_rule_tutorial/v/derivative-properties-and-polynomial-derivatives

Answer 16

I thought I had to plug in 2 for t though

Answer 17

you do, after you find s'(t)

the steps are:
s(t) = -2 - 6t

you take the derivative with respect to "t"
d/dt (-2) is 0
d/dt (-6t ) is -6
you get

s'(t)= -6

now you find
s'(2)
but s'(2) = -6
(in other words, the "function" s'(t) is a constant value, always equal to -6)

Answer 18

because s'(t) does not have a "t" in it... you just get -6 no matter what t is

Answer 19

so s'(t)=-6 with or without the t?

Answer 20

I would say:
so s'(t)=-6

there is no "t"
it sometimes happens. But it's ok, (it's like y= -6, a straight line, with y always = -6 no matter what x is... same idea here)

Answer 21

so what's the instantaneous velocity then?

Answer 22

s'(t) evaluated at t=2

Answer 23

and because s'(t) is constant , always equal to -6
s'(2) is -6
i.e. instantaneous velocity is -6

Answer 24

okay, and just for my own record because this wasn't in my lesson at all, what exactly is an instantaneous velocity? I know velocity is speed with direction but idk about the instantaneous velocity...

Answer 25

velocity is change in distance divided by change in time
(sounds vaguely like the definition of slope, if distance is y and time is x)

if the velocity is always changing (speeding up or slowing down)
then to find velocity we have to look at the change in distance over a *very small change in time*

if we plot distance over time, and velocity is changing, the curve would not be a straight line. The slope (velocity) can only be approximated as the slope of a tangent line to the curve.

Answer 26

Khan explains it pretty well
http://www.khanacademy.org/math/differential-calculus/taking-derivatives/secant-line-slope-tangent/v/slope-of-a-line-secant-to-a-curve

Answer 27

Okay, thanks :D

Answer 28

If you have time, watch Khan's videos.