Math 19: Calculus Winter 2008 Instructor: Jennifer KlokeLecture Notes (Intermediate Value Theoremand Tangent Lines)Wednesday, January 23Announcements1. Quiz 2 is on Monday 01/28/082. Homework 2 is due Wednesday 01/30/083. Midterm 1 is on Thursday 01/31/08. Any make-up exams must be taken before thescheduled exam. Please let me know by Sunday evening if you need to take your examat a different time.4. Review Quiz scores will not be included in the end-of-quarter analysis for curving ofgrades.RecapDefinition: If a function satisfies limx→af(x) = f(a), we say that f(x) is continuous atx = a. If f(x) is continuous at every point in its domain, we say that f (x) is a continuousfunction.Also, if f(x) is a combination of continuous functions, then f(x) is continuous on itsdomain.Wrapping up ContinuityThere are a few things that I did not get to cover last Friday.First is the types of discontinuities that are possible. Please read the bottom half of page118 from your textbook for the definitions of the different types of discontinuities.We also have one more definition that is just and extension on our previous definition ofcontinuity.1Definition: A function f is continuous on an interval if it is continuous at everynumber in the interval.Example: Is the function f (x) = 1/x continuous on the interval [1,2]? On [-1,1]?Intermediate Value TheoremIn addition to making the evaluation of limits easier, there is another very useful propertyof continuous functions.Intermediate Value Theorem Suppose f(x) is a continuous function on the interval[a,b]. Suppose N is between f(a) and f(b) (e.g. f(a) ≤ N ≤ f(b).) Then there is a numberc in the interval (a, b) so that f(c) = N.Examples: Show that the function f(x) = x2008+ 2008x − 1 has a root between 0 and1.Solution: We are asked to say that there is some number c between 0 and 1 (notice thatwe don’t need to say what c is, just that it exists), that has f (c) = c2008+ 2008c − 1 = 0.Since the IVT is the only theorem we have so far that just tells us that some number existsthat satisfies a given property, let’s see if we can apply it.First we must check the hypotheses of the theorem:-Is this function continuous on the interval [0,1]?- Is 0 between f(0) and f (1)?Back to VelocityRecall from two weeks ago the problem that led us to learn about limits: finding the instan-taneous velocity if we know the distance over time function. Let’s return to that examplenow that we know a lot more about limits.We concluded that if d(t) was a function of the distance traveled over time, the instan-taneous velocity at time t = t0could be found by approximating the slope of the secant linebetween t = t0and t = x when x is close to t0.In fact we said that the instantaneous velocity was the limit of these slopes. Using ourlimit notation, we can now say, the instantaneous velocity at time t = t0islimx→t0d(t0) − d(x)t0− x.Example: Suppose a ball is dropped from the top of the Hoover tower. From physicswe know that the distance traveled by a falling object at time t is given by the function2d(t) = 4.9t2(where the distance is measured in meters and time in seconds). What is thevelocity of the ball after 3 seconds?Tangent LineAbove we concluded that the velocity could be found by finding the limit of the slope of thesecant lines. The line that is the limit of these secant lines is called the tangent line andit is a line that comes up in real life all the time (that is the part of life that has anythingto do with functions).Definition: The tangent line to the curve y = f(x) at the point P = (a, f(a)) is theline through P with slopem = limx→af(x) − f(a)x − aprovided that this limit exists.Example: Find the equation for the tangent line to the function f(x) = x2at the point(-1,1).What to Know/Memorize1. The definition of a function being continuous on an interval [a, b].2. The Intermediate Value Theorem - know its hypotheses and conclusion. Note thatthis is the only theorem we have up to now that tells you that a number exists with acertain property but do es not tell you what that number is.3. How to find the velocity in terms of limits.4. The definition of the tangent