Question

1. suppose that an object is at position s(t) = 1/t feet at time t seconds. a. find the average velocity of the object over a time interval from time 1 second to time t seconds. b. find the instantaneous velocity of the object at time 1 second by taking the limit of the average velocity in part a as t → 1.

Explanation:

Step1: Recall average - velocity formula

The average velocity $v_{avg}$ over the interval $[a,b]$ is given by $v_{avg}=\frac{s(b)-s(a)}{b - a}$. Here, $a = 1$, $b=t$, and $s(t)=\frac{1}{t}$, $s(1) = 1$.

Step2: Calculate average velocity for part A

$v_{avg}=\frac{s(t)-s(1)}{t - 1}=\frac{\frac{1}{t}-1}{t - 1}=\frac{\frac{1 - t}{t}}{t - 1}=\frac{1 - t}{t(t - 1)}=-\frac{1}{t}$ for $t
eq1$.

Step3: Calculate instantaneous velocity for part B

The instantaneous velocity $v(1)$ is the limit of the average velocity as $t
ightarrow1$. We have $\lim_{t
ightarrow1}v_{avg}=\lim_{t
ightarrow1}-\frac{1}{t}$.
Substitute $t = 1$ into $-\frac{1}{t}$.

Answer:

A. $-\frac{1}{t}$
B. $- 1$