By Hans Sagan
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Additional resources for Integral and Differential Calculus: An Intuitive Approach
Example text
18. f(x) = fix. f(x + h) and 1(x + h) -fl (x) x) latter. h Simplify the 4. STRAIGHT LINE We have seen in the preceding section that the function y = 2x represents a straight line in a right coordinate system. In this section we will establish the mathematical representation of a straight line in general. First, however, let us consider an example. We know from plane geometry that a straight line is uniquely determined by two points. Let us consider the two points P1(l, 1) and P2(3, 5) (see Fig.
First, we factor out the coefficient of x2: a(x2+bx+e)=0. 26) in the form \x+ 2ab/2 C. 9. THE QUADRATIC EQUATION 43 x Fig. 29) is called the quadratic formula. , the parabola y = axe + bx + c intersects the x-axis in two distinct points (see Fig. 28). If the quadratic equation arises out of a problem concerning the intersection of a parabola and a line, as we discussed it at the end of the preceding section, then there are two distinct intersection points of parabola and Fig. 29 line in this case (see Fig.
Introduce a new coordinate system x,9 such that the equation of this parabola in the new coordinate system becomes 9 = 2. 61. 60 for y = -x2 + 4x + 1 (9 = -0). 62. Given y = x,2 + 2x - 2. Stretch the y-axis in the ratio 1:4 and write the equation of the stretched parabola. Sketch the parabola before and after the stretching process. 63. , to the form y = x2 in some 5,y-coordinate system. 64. G:ven y = x2 - 4x - 1. (a) Carry out the translation x = x + 1, y = 9 - 2 and then stretch the 9-axis in the ratio 1 :2.