Lecture 14: More on Functions

Problem of the day

This is due on Friday, Sept. 25, test day. The temperature at a point (x,y) in the plane is given by P(x,y)= x²+y²-4x+2y.
Find the coldest point in the plane and the temperature at that point.

Assignment

Prepare for the first test.

Brief review of the lecture

We're going to have a problem session assuming we finish discussing section 2.2.

Left from last time: Analytic Geometry

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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