By equation (1.1), 
, where the 
term in the sum inside the parenthesis is 
(starting at 
). So the first approximation of 
using this formula is 
, and the second approximation is 
. Continue like this until two consecutive approximations have 
as the first digit before the decimal point. How many terms in the sum did this require? Be careful with rounding off in the approximations.
Attempts.
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Summing without aim, I forgot my purpose. Where am I?
(discontinued)
(refreshed)
Please scroll up to the and approximation.
This required seven or eight terms in the sum for having as the first digit before the decimal point.
after time 
is the given function 
. Find the instantaneous velocity of the object at a general time 
. You should mimic the earlier example for the instantaneous velocity when 
.
![[t,t+\Delta t]](https://physicspupil.com/wp-content/ql-cache/quicklatex.com-e8ef7e317d6da1a6e5f0665c06779114_l3.svg "Rendered by QuickLaTeX.com")
is 
, so since 
:
get closer and closer to 
. Then the average velocity 
gets closer and closer to 
. Thus, the object has instantaneous velocity ![\begin{aligned} & \qquad \textrm{instantaneous velocity at }t \\ & = \textrm{limit of average velocity over }[t,t+\Delta t]\textrm{ as }\Delta t\textrm{ approaches to }0 \\ & = \lim_{\Delta t\to 0}\frac{\Delta s}{\Delta t} \\ & = \lim_{\Delta t\to 0}(2t+\Delta t) \\ & = 2t+(0) \\ & = 2t \end{aligned}](https://physicspupil.com/wp-content/ql-cache/quicklatex.com-5d7474b339cdb36360b54eaa22faad1a_l3.svg "Rendered by QuickLaTeX.com")
