Suppose that your distance from home on a Sunday afternoon drive was given by the formula $D(t)=-7.7t^3+3.1t^2+24.6t$ where $D$ is the distance in miles and $t$ is the number of hours driven. Determine your speed during your two hour trip.
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Suppose that your distance from home on a Sunday afternoon drive was given by the formula $D(t)=-7.7t^3+3.1t^2+24.6t$ where $D$ is the distance in miles and $t$ is the number of hours driven. Determine your speed during your two hour trip.
$\mbox{Speed}=D'(t)=-23.1t^2+6.2t+24.6$
SAMPLE SOLUTION
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From this lesson we know that speed is the derivative $D'(t)$. Now we know that $D(t)$ is a polynomial function and that the derivative of a polynomial is known. Therefore,
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That's it! <Sign in to see all the formulas> is the answer.
We can actually go a little bit further by graphing both the distance and the speed. Since we only drove for two hours we want our graph to go between $0$ and $2$ on the x-axis where we plot the number of hours driven. In the graph below we see that after we have traveled approximately 1.175 hours our speed (red line) becomes negative. This indicates that we have started moving back toward home and our distance from home becomes smaller and smaller. We also see that the furthest we got from home was about 20 miles.
