Lecture 14: The Differentiation Process

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Assignment

Brief review of the lecture

We discussed the method of finding the slope of the line tangent to the graph of a function at a given point. We used the function f(x)=1/x and the point (2, 1/2). We found that the slope is -4 at that point, and recorded the infomation as follows: f'(2)=-4. Then we carried out the process again for the point (1,1) and fount that the slope is -1. We record that by saying f'(1)=-1. Finally we carried out the process at an arbitrary x, and found that the slope is -1/x^2.

Analytic Geometry

This material is left from a lecture in Math 1100.

The Fundamental Principle. There is a one-to-one correspondance between real numbers and the points on a line. This correspondance respects the relation less than in the sense that the point on the line which corresponds to zero has the property that the points on the line which correspond to positive numbers are all on one side of the point, and those which correspond to negatives are on the other side.

This principle provides the basis for the rich interplay between algebra (numbers) and geometry (points). It enable us to “see” algebraic expressions (as graphs).

The Cartesian Plane. This is a set of points each of which is described by an ordered pair of numbers. We imagine two perpendicular lines in the plane, called axes. We take the perspective that one is horizontal and the other vertical, and we call former the x-axis and the later the y-axis. The notation P=(u,v) means that P is located on the vertical line through the point u on the x-axis and on the horizontal line through v on the y-axis.

Distance. The distance beween a pair of points (x,y) and (u,v) is determined by the Pythagorean Theorem. See page 170 for the formula, which we derived in class.

Reflection across a line. Symmetry with respect to
1. the x-axis
2. the y-axis
3. the line y=x
4. the origin

Also discussed: midpoints, circles, slopes of lines.


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