Solve Derivative Problems

Solve Derivative Problems-30
Notice that if we zoom in close enough to a curve, it begins to look like a straight line.We can find a very good approximation to the slope of the curve at the point `t = 1` (it will be the slope of the tangent to the curve, marked in pink) by observing the points that the curve passes through near `t = 1`.

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It also supports computing the first, second and third derivatives, up to 10.

During the "up" phase, the ball has negative acceleration and as it falls, the acceleration is positive.

This idea of "zooming in" on the graph and getting closer and closer to get a better approximation for the slope of the curve (thus giving us the rate of change) was the breakthrough that led to the development of differentiation.

Up until the time of Newton and Leibniz, there was no reliable way to describe or predict this constantly changing velocity.

We have found the rate of change by looking at the slope.

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Clearly, if we were to zoom in closer, our curve would look even more straight and we could get an even better approximation for the slope of the curve.

It goes from positive (when the ball is going up), slows down to zero, then becomes negative (as the ball is coming down).

All the time during this motion the velocity is changing.

We will return to this problem later and see how to do it in the Applications of Differentiation chapter.

The approach we follow here is the same as that discovered historically: You can skip the first few sections if you just need the differentiation rules, but that would be a shame because you won't see why it works the way it does.

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