By Serge Lang

This 5th variation of Lang's publication covers all of the issues routinely taught within the first-year calculus series. Divided into 5 components, every one part of a primary path IN CALCULUS comprises examples and purposes on the subject of the subject coated. furthermore, the rear of the publication comprises specific suggestions to a number of the routines, letting them be used as worked-out examples -- one of many major advancements over past versions.

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**Extra resources for A first course in calculus, Fifth Edition**

**Example text**

CHAP. 1. 18 FUNCTIONS Y I -3 -2 1l I -1 1 -x 2 Fig. , it has a y-value for every value of x. Here is another example of a function: 0 for all x which are less than -3 1 for x = -3 y= 1 for all x that are greater than -3 but less than 0 Oforx=0 2x for all x which are positive. In this case, we can even supply a geometric representation of this function (graph) as indicated in Fig. 19. (The student may investigate the possibility of representing by a graph the function which we mentioned before this one..

36). 10. TANGENT TO A PARABOLA 49 y Fig. 36 or, after some algebraic manipulations, A2 - (2b + 4axo)A + b2 + 4a2xo2 + 4abxo = 0. 38) A = 2axo + b. Thus we can say: The tangent line to the parabola y = axe + bx + c at the point with the x-coordinate x0 has the slope 2axo + b. ) As an example, let us consider y=2x2-4x+3 CHAP. I. 50 FUNCTIONS and the point on the parabola with the x-coordinate x0 = 2. 38), A=8-4=4. The y-value corresponding to x0 = 2 is found to be. yo = 3. Hence we find the equation of the tangent line from the point-slope formula to be y - 3 = 4(x - 2), and from this y = 4x - 5.

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. 29). 26) are complex numbers. ) In this case, the parabola y = axe + bx + c does not have any real intersection points with the x-axis (see Fig.