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3-32.

Jasmin rolled a ball down a very steep ramp and got the distance function $s(t) = 2.3t^2$, where $t$ is measured in seconds and $s(t)$ is measured in feet. Sketch a graph of her distance function on your paper. Then, carefully approximate the speed of the ball at $t = 3$ seconds.

The velocity/speed at $t = 3$ should be the same as the IROC (or slope of the tangent line) at $t = 3$.

$v(3) = 13.8$ ____________ don't forget units!

Find the slope function of $s(t)$ by finding the IROC at $x = 3$:

$\text{IROC}_{t=3}=\lim\limits_{h\rightarrow 0}\frac{f(3+h)-f(3)}{h}$

1. Evaluate when $f(x) = s(t) = 2.3t^3$.
2. Expand and simplify the numerator.
3. Factor out an $h$ from the numerator.
4. 'Cancel out' the $h$ from the fraction.$5.\text{ Evaluate the }\lim\limits_{h\rightarrow 0}.\text{ This is the velocity at }t=3.$

$v(3)$ can also be found with Power Rule:
$s(t) = 2.3t^2$
$v(t) = 4.6t$
$v(3) =$ _____________

Use the eTool below to help solve the problem.
Click on the link to the right to view the full version of the eTool. Calc 3-32 HW eTool