By Jerrold Marsden, Alan Weinstein
Undergraduate Texts in arithmetic
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Additional resources for Calculus I (Undergraduate Texts in Mathematics)
Thus the symbol 6 represents a real number only when r > 0, in which case we always take fi to mean the non-negative number whose square is r. Example 6 Solution Solve for x: (a) 4x2 = 2x + 5 and (b) 2x2 + 4x - 6 = 0. (a) Subtracting 2x + 5 from both sides of the equation gives 4x2- 2x - 5 = 0, which is in the form ax2 + bx + c = 0 with a = 4, b = -2, and c = - 5. The quadratic formula gives the two roots (b) An alternative to the quadratic formula is to factor. From Example 4, 2x2 + 4x - 6 = 2(x + 3)(x - 1).
6. 6. The slope of this line is (y2 -y , ) / ( x 2 - XI). , If (x ,y ,) and (x, , y,) lie on the line I, the slope of I is m=- An important feature of the slope m is that it does not depend upon which two points we pick, so long as they lie on the line I. To verify this, we observe (see Fig. 7)that the right triangles PIP$ and P;P;R' are similar, since corresponding angles are equal, so P,R/ P,R = P;R'/ PiR'. In other words, the slope calculated using PI and P2 is the same as the slope calculated using Pi and Pi.
X 2 - 5 x 7 = 0 36. 1 x 2 - 1 . 7 = 0 Solve for x in Exercises 37-42. 37. x 2 + 4 = 3x2 - x 38. 4 x = 3x2 7 39. 2 x x 2 = 9 + x 2 40. (5 - x)(2 - X ) = 1 41. 2 x 2 - 2 S ' j x + i = O 42. x 2 9 x = 0 25. 26. 27. 28. + + + + + + + 43. Put the following list of numbers in ascending order. ) 44. Put the following numbers in ascending order. (Do not use a calculator): Copyright 1985 Springer-Verlag. All rights reserved. 2 Intervals and Absolute Values Simplify the inequalities in Exercises 45-50. 45.