The following table gives the position $s\(\left\)(t\(\right\))$ of an object moving along a line at time $t$. Determine the average velocities over the time intervals $\(\left\[\lbrack\)1,1.01\(\right\]\rbrack\)$, $\(\left\[\lbrack\)1,1.001\(\right\]\rbrack\)$, and $\(\left\]\lbrack\)1,1.0001^{}\(\right\).]$. Then make a conjecture about the value of the instantaneous velocity at $t=1$. <IMAGE>

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→3^− f(x)

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→3^+ f(x)

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→3 f(x)

Given the function $f\(\left\)(x\(\right\))=-16x^2+64x$, complete the following. <IMAGE>

Find the slopes of the secant lines that pass though the points $\(\left\)(x,f\(\left\)(x\(\right\))\(\right\))$ and $\(\left\)(2,f\(\left\)(2\(\right\))\(\right\))$, for $x=1.5,1.9,1.99,1.999,$ and $1.9999$ (see figure).

Given the function $f\(\left\)(x\(\right\))=-16x^2+64x$, complete the following. <IMAGE>

Make a conjecture about the value of the limit of the slopes of the secant lines that pass through $\(\left\)(x,f\(\left\)(x\(\right\))\(\right\))$ and $\(\left\)(2,f\(\left\)(2\(\right\))\(\right\))$ as $x$ approaches $2$.

The position of an object moving vertically along a line is given by the function $s\(\left\)(t\(\right\))=-16t^2+128t$. Find the average velocity of the object over the following intervals.

$\(\left\[\lbrack\)1,4\(\right\]\rbrack\)$

The position of an object moving vertically along a line is given by the function $s\(\left\)(t\(\right\))=-16t^2+128t$. Find the average velocity of the object over the following intervals.

$\(\left\[\lbrack\)1,2\(\right\]\rbrack\)$