18.01 Calculus Jason Starr Fall 2005 Lecture 2. September 9, 2005 Homework. Problem Set 1 Part I: (f)–(h); Part II: Problems 3. Practice Problems. Course Reader: 1C2, 1C3, 1C4, 1D3, 1D5. 1. Tangent lines to graphs. For y = f (x), the equation of the secant line through (x0 , f (x0 )) and (x0 + Δx, f (x0 + Δx)) is, y= f (x0 + Δx) − f (x0 ) (x − x0 ) + f (x0 ). Δx In the limit, the equation of the tangent line through (x0 , f (x0 )) is, y = f � (x0 )(x − x0 ) + y0 . Example. For the parabola y = x2 , the derivative is, y � (x0 ) = 2x0 . The equation of the tangent line is, y = 2x0 (x − x0 ) = 2x0 x − x20 . For instance, the equation of the tangent line through (2, 4) is, y = 4x − 4. Given a point (x, y), what are all points (x0 , x20 ) on the parabola whose tangent line contains (x, y)? To solve, consider x and y as constants and solve for x0 . For instance, if (x, y) = (1, −3), this gives, (−3) = 2x0 (1) − x20 , or, x20 − 2x0 − 3 = 0. 18.01 Calculus Jason Starr Fall 2005 Factoring (x0 − 3)(x0 + 1), the solutions are x0 equals −1 and x0 equals 3. The corresponding tangent lines are, y = −2x − 1, and y = 6x − 9. For general (x, y), the solutions are, x0 = x ± � x2 − y. 2. Limits. Precise deﬁnition is on p. 791 of Appendix A.2. Intuitive deﬁnition: limx→x0 f (x) equals L if and only if all values of f (x) can be made arbitrarily close to L by choosing x suﬃciently close to x0 . One interpretation is the “microscope/laser illuminator” analogy: An observer focuses a microscopes ﬁeldofview on a thin strip parallel to the xaxis centered on y = L. The goal of the illuminator is to focus a laserbeam centered on x0 parallel to the yaxis (but with the line x = x0 deleted) so that only the portion of the graph in the ﬁeldofview is illuminated. If for every magniﬁcation of the microscope, the illuminator can succeed, then the limit is deﬁned and equals L. There is a beautiful Java applet on the webpage of Daniel J. Heath of Paciﬁc Lutheran University, http://www.plu.edu/~heathdj/java/calc1/Epsilon.html If you use this, try a = −1. For lefthand limits, use a laser that illuminates only to the left of x0 . For righthand limits, use a laser that illuminates only to the right of x0 . 3. Continuity. A function f (x) is continuous at x0 if f (x0 ) is deﬁned, limx→x0 f (x) is deﬁned, and limx→x0 f (x) equals f (x0 ). Also, f (x) is continuous on an interval if it is contin uous at every point of the interval. The types of discontinuity are: removable discontinuity, jump discontinuity, inﬁnite discontinuity and essential discontinuity.