Tangent lines to Graphs

18.01 Calculus Jason Starr Fall 2005 Lecture 2. September 9, 2005Homework. Problem Set 1 Part I: (f)–(h); Part II: Problems 3.Practice Problems. Course Reader: 1C22 1C33 1C44 1D33 1D551. 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) = 2x0x − x2 0. For instance, the equation of the tangent line through (2, 4) is, y = 4x − 4. 2Given a point (x, y), what are all points (x0, x0) 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, 2(−3) = 2x0(1) − x0, or, 2 x0 − 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: limxf (x)→x0 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 fieldof­­vie on a thin strip parallel to the xaxxi centered on y = L. The goal of the illuminator is to focus a laserbeea centered on x0 parallel to the yaxxi (but with the line x = x0 deleted) so that only the portion of the graph in the fieldof­­vie 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 lefthaan limits, use a laser that illuminates only to the left of x0. For righthaan 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, limxf (x) is →x0 defined, and limxf (x) equals f (x0). Also, f (x) is continuous on an interval if it is contin↭x0 uous at every point of the interval. The types of discontinuity are: removable discontinuity, jump discontinuity, infinite discontinuity and essential discontinuity.