By F. H. C. Marriott (Auth.)
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Additional resources for Basic Mathematics for the Biological and Social Sciences
X —2 >· 6. Sketch the curves, in polar coordinates: (i) r (cos Θ + 3 sin Θ) = 5. (iv) r = sin 0 tan 0. (v) r = 1 + cos 0. (ii) r = cos Θ + 2 sin 0. (iii) r0 - a2. 7. Show that the equation r = ek/(l — e cos Θ) represents a conic section, and that it is an ellipse, parabola, or hyperbola according as e < , = , > 1. [The origin is at the focus, and e is called the eccentricity; this is the standard equation for conies in polar coordinates. ] In examples 8-10, describe the surfaces and, where two or more surfaces are given together, the curves in which they intersect.
If k is irrational, the number of petals is infinite. The toy called the spirograph is based on this principle. 8. Three-dimensional problems Cartesian coordinates are easily generalized to three dimensions by including a third coordinate (z) measured along an axis perpendicular to the other twof (Fig. 17). In three dimensions, an equation in x, y and z generally defines a surface \ a line or curve in space is defined by two simultaneous equations representing the intersection of two surfaces. t In some applications it is necessary to distinguish between "left-handed" and "right-handed'' sets of axes, but the point does not arise in this book.
Iv) y = x + | x |. 2 (ii) v = 5 + Ax — x . ,λ (v) ^ = (iii) j>2 — x3. x —2 >· 6. Sketch the curves, in polar coordinates: (i) r (cos Θ + 3 sin Θ) = 5. (iv) r = sin 0 tan 0. (v) r = 1 + cos 0. (ii) r = cos Θ + 2 sin 0. (iii) r0 - a2. 7. Show that the equation r = ek/(l — e cos Θ) represents a conic section, and that it is an ellipse, parabola, or hyperbola according as e < , = , > 1. [The origin is at the focus, and e is called the eccentricity; this is the standard equation for conies in polar coordinates.
Basic Mathematics for the Biological and Social Sciences by F. H. C. Marriott (Auth.)