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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.