By Ralph Palmer Agnew

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**Extra resources for Calculus: Analytic Geometry and Calculus, with Vectors**

**Example text**

We can suppose that a body is, at time t = 0, falling or just being dropped so that it travels past the markings on our meter sticks with increasing rapidity as time passes. On the other hand, we can suppose that the body is rising at time t = 0 so that it rises for a while before it begins its descent. 5 when a timing device shows a half-second after our time origin or zero-hour. Anyone who tosses a body upward and observes the ensuing motion must realize that it is not an easy matter to use an ordinary clock to obtain accurate data giving the coordinate s of the body at various times t.

Analytic geometry in two dimensions 34 Finally, tell why (5) must hold. Remark: In order to appreciate the significance of this work, we must do a little thinking about "elementary" mathematics. It is sometimes said that a straight line is the shortest distance between two points. If this silly collection of words means anything it means that the length (a number) of the line segment (a point set) joining two points P1 and P2 is less than the length (a number) of each other path (a point set) joining P1 and P2.

Prove that if the diagonals are perpendicular, then the rectangle is a square. 11 Sketch a figure showing the triangle having vertices at the points Pi(xl,y,), P2(x2,y2), and Pi(xa,Y3) For each k = 1, 2, 3, mark the mid-point Qk of the side opposite Pk and find the coordinates of Qk. Supposing that the line QsQ1 is not vertical, calculate its slope and show that it is parallel to the line P1P2. 12 Prove analytically (by calculating slopes) that the mid-points of the sides of a convex quadrilateral are vertices of a parallelogram.

### Calculus: Analytic Geometry and Calculus, with Vectors by Ralph Palmer Agnew

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