Question

The position of an object at time t is given by s(t) = -4 - 2t. Find the instantaneous velocity at t = 6 by finding the derivative. MUST USE LIMIT DEFINITION

Answer 1

I know the derivative is -2 and is constant but I have to show work using limit definition

Answer 2

Alright so we have \(\large s(t) = -4 - 2t\)

The definition of the derivative states \(\large \lim_{h \rightarrow 0}\frac{f(t + h) - f(t)}{h}\)

\(\large f(t) = -4 - 2t\)
\(\large f(t + h) = -4 - 2(t + h)\)

So we have

\[\large \lim_{h \rightarrow 0}\frac{(-4 - 2(t + h)) - (-4 - 2t)}{h}\]
\[\large \lim_{h \rightarrow 0}\frac{(-4 - 2t -2h) - (-4 - 2t)}{h}\]
\[\large \lim_{h \rightarrow 0}\frac{-4 - 2t -2h +4 + 2t}{h}\]

Can you simplify from there?

Answer 3

\[\lim_{t \rightarrow 0} \frac{ -4-2(t+h)-(-4-2t) }{ h } = \frac{ -4-2t+2h+4+2t }{ h }\]

Answer 4

Careful distributing @Photon336 that should be a -2h :)

Answer 5

thanks so much for the reply, combine like terms on top? so -2h/h?

Answer 6

yeah lol take your time when expanding, don't rush it like me

Answer 7

@johnweldon1993

Answer 8

Usually you end up with a t term and can plug in 6, so did I simplify wrong? @johnweldon1993 @Photon336

Answer 9

ohh nvrmind, the 2 h's cancel and your left with -2, thanks

Answer 10

Yeah there ya go! *sorry for the late reply, have to eat at some point lol*