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Published byMyron Jones Modified over 8 years ago
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1.2 Slope of a Curve at a Point
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We want to extend the concept of slope from straight lines to more general curves. To do this, we must introduce the notion of a tangent line to a curve at a point.
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The tangent line to a circle at a point P is the straight line that touches the circle just at P.
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If we enlarge the region near P, we will notice that the magnified section of the circle looks almost straight. Further magnifications would make the section of the circle near P more closely resemble the tangent line at P.
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We notice that the tangent line at P reflects the steepness of the circle at P. It is reasonable to define the slope of the circle at P to be the slope of the tangent line at P. Similar reasoning leads to a suitable definition of slope for an arbitrary curve at a point P.
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If we magnify the region close to a point P for an arbitrary curve, we will notice that like the circle, as we increase magnification, the region of the curve near P more closely resembles a certain straight line.
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This straight line is called the tangent line to the curve at P. This line best approximates the curve near P. We can then define the slope of a curve at a point P to be the slope of the tangent line to the curve at P.
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We can usually use calculus to compute slopes by using formulas. For now, suppose we have the graph of y = x 2. The slope at any point on this graph is given by [slope of the graph y = x 2 at point (x,y)] = 2x So, the slope at (3,9) is 2(3) = 6, and the slope at (0,0) is 2(0) = 0.
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EXERCISES
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