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Slope of the Tangent to a Curve

Determine the slope of the tangent to each of the following curves:
1) y=2/x^2
2) y=8x-2x^2
3) y=x^4-2/x

Solution Preview

The slope of a line is a measure of how "steep" it is. When you climb a steep hill, you're climbing a hill with great slope. Level ground has a zero slope, and something entirely vertical, like a wall, is said to have infinite slope. When we climb a hill, we are, in a sense, going forward as well as up. Slope measures how much we go up as we go forward. Zero slope means we don't go up at all, not matter how much we go forward. This is what happens on level ground. If we're scaling a wall, we're going up without going forward at all. We're going up from the same spot on the ground. Thus we may say that a wall has infinite slope. With any case in between, we're both going up and forward at the same time.

But these are only intuitive ideas. In order to speak of them more rigorously, we need to define slope mathematically.
See attached image file - "Slope.png"
Given a straight line, consider a section of it, as shown in the figure. Since we want to measure how much we go up as we go forward, ...

Solution Summary

This solution explains the concept and mathematical theory of the slope of a curve and shows how it can be calculated for any curve defined by a Cartesian equation, using differentiation. An explanation of the theory is followed by three examples. Links for further reference are also included.

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