Math 163: Lecture notes

Transcription

1 Math 63: Lecture notes Professor Monika Nitsche March 2, 2

2 Special functions that are inverses of known functions. Inverse functions (Day ) Go over: early exam, hw, quizzes, grading scheme, attendance policy, your resources (I hope to be your main resource for this class), office hours Today s material should mostly be review for you, so I will phrase it in terms of questions for you to answer. Given a function f(x), how is the inverse f of f defined? Answers: (f f )(x) = x, f(f (x)) = x, f(x) = y f(y) = x What are the implications for the graph of f? Answer: if (a,b) on graph of f, then (b, a) on graph of inverse, symmetry about y = x. Show it. When is f invertible? Answers: horizontal line test, -. Example: f(x) = x 2 not invertible, but if restricted to x then yes. If f continuous then to be invertible must either be always increasing or always decreasing. What happens if f (x) = at a point? Example f(x) = x 3. Note: if f is differentiable in D, then it is invertible if f or f, with isolated zeros only! How do we find formula for f (x)? Answer: use definition. Solve for x =.... The dots are f (y). Can switch y for x if you wish. Examples: f(x) = x 2 +, x, f(x) = x 2 +, x, f(x) = x 2 How to find (f ) (x)? Use definition. Implicit differentiation. Chain rule. Example: f(x) = x 2, (f ) (x) = = 2 x What it means graphically Example: if f(4) = 5, f (4) = 2/3, find equation for tangent line to f at x = 5. 2

3 .2 The exponential function f(x) = a x, a > (Day 2) What is a x, x R? Is a < allowed? Go through a p, a p/q, continuous extension, why a cant be negative Graphs Graphs of 2 x, 4 x, x, (/2) x, (/4) x Notes:. a x is always positve, for any x (a > ) 2. a = for any a. But different slopes at x = Algebra a x = a x, ax b x = (ab) x, (a x ) y = a xy, a x a y = a x+y Limits lim x 2x, lim x 2x, lim (/2) x, x e x e x lim x e x + e, x lim x e 2x cos x Derivative of f(x) = a x Use the definition of the derivative to show that f (x) = Ca x a h, C = lim h h Plot g(h) = (a h )/h for a couple of values of a, in Matlab, and get a feeling for what the limit as h is. Note: C is a number. So derivative of f is proportional to itself. If C positive, f always positive, f always increasing! So f always increasing! similarly if C is negative The number e Definition of the number e: it is the one value of a for which f () = d dx [ex ] = e x Graph of e x. Tangent line at origin Examples of derivatives Examples of integrals Question: d dx ax = Ca x. What is C??? 3

4 2 Series k=k a k 2. The Value of a Series Simple example. Given an infinite series, for example 2 = k , k= what is the most natural thing to do to try to find its value? Answer: Compute the partial sums, s = 2 s 2 = = 3 4 s 3 = = 7 8 s 4 = = It is evident by visualizing these partial sums on a line (below, in black) S 4 S 3 S 2 S /2 3/4 7/8 /4 /8 /6 that the partial sums s N (red) approach as N : since s N+ consists of adding half of the distance between and s N (light blue) to s N, we see that as N increases, s N gets closer and closer to, and is never bigger than. Alternatively, we can recognize the pattern that s N = 2N 2 N (which we will prove in a bit) and deduce that lim N s N =. Definition. The value of a series is defined as the limit of the partial sums k= a k = lim N s N where s N = N a k = a + a 2 + a a N If this limit exists, we say the series converges, otherwise it diverges. k= 4

5 summands a n Examples.To illustrate, below we plot the sequence of summands a n and the corresponding partial sums for several examples. It looks like the first and last series converge, but it is not so clear from the other two. We need tests to determine whethe a series converges. In that case we know we can approximate it by partial sums. Series : n= (a geometric series) 2 n partial sums S N summands a n n n Series 2: n= (a p-series) n 2 Series 3: n= n partial sums S N N N (the Harmonic Series) 4 summands a n partial sums S N n N summands a n Series 4: ( ) n n= n (the Alternating Harmonic Series) partial sums S N n N

6 Series and Improper Integrals.We saw in class that just as we can relate sequences {a n } to functions f(x), we can draw a parallel between Series a k and Improper Integrals k= f(x). From our experience with these types of improper integrals, we can deduce that for a series to converge, it is necessary that a k as k sufficiently fast. This is called the Divergence Test. Divergence Test. If lim a n n then the series k= a k diverges. (Note: if lim n a n =, that is not sufficient to ensure that the series converges.) 6

7 2.2 Special Series For geometric series and for telescoping series we can actually find a compact formula for the partial sums s N, and from it compute their limit in order to determine not only whether the series converges or diverges, but also its value. Geometric Series r k. Using the trick of subtracting rs N = r+r 2 +r 3 + +r N+ k= from s N = + r + r 2 + r r N, we find that ( r)s N = r N+, so that s N = N k= r k = rn+ r. This holds whether or not the series converges! However, we can also now deduce that if r <, then r k = lim s N = N r. k= This follows since in this limit r N+ vanishes, as long as r <. If r, the series diverges by the Divergence Theorem. Similarly, one can show that (do it!) k=k r k = rk r. Telescoping Series. If there is cancellation between terms in a partial sum, it may be possible to find a compact formula for the partial sums, and from it deduce the value of a series. See examples in class and in the book. p-series. Using the integral test (next) we can show that n=n n p converges if p > and diverges if p. These series will be an important series to use in comparison to others later on. 7

8 2.3 Tests for Series with Positive Summands, a n. In order to determine whether a series converges or diverges you should always first check that a n as n. Once you established that, you can use one of the tests below to check for convergence/divergence. Once you know that a series converges, you know that you can approximate it by its partial sums. Integral Test. If f is a continuous function such that f(n) = a n and f(x) decreases, then a k converges if and only if f(x) dx converges. k k This test is good for examples in which you can (fairly easily) integrate f(x). To use it you must: State f(x) Check f is decreasing Compute the improper integral as a limit, using proper notation. Direct Comparison Test. If a n b n and b n converges, so does a n. Similarly, If a n b n and a n diverges, so does b n. This test is good for examples in which the inequality is obvious. To use this test you must correctly establish the proper inequality. a n Limit Comparison Test. If lim = L, where L is finite and not zero (that is, a n n b n and b n decay to zero at the same rate), then a n and b n either both converge or both diverge. This test is good if you can tell that a sequence a n looks like another sequence b n for which you know the behaviour of its sum. To use this test you must compute the limit properly. a n+ Ratio Test. If lim = r < then the series a n converges. (In that case, the n a n tail end of the series looks like a geometric series.) This test is good for examples that include powers and factorials. To use this test you must compute the limit properly. 8

9 2.4 What if not all summands a n are positive?? Theorem: If a n converges, then a n converges. Therefore, the first thing to do is to check wether the sum of absolute values converges. For this you can apply all the tests described in the previous subsection. Absolute convergence. If a n converges, then we say that the sum a n converges absolutely. ( 2) n Example. Consider the alternating series. Since the sum of absolute values n! n= 2 n converges by the Ratio Test (show all work) we can conclude that the alternating n! n= series of interest converges absolutely. What if a n does not converge?? If a n converges, but a n does not converge, then we say that the sum a n converges conditionally. We have already seen the classical example of a series that converges conditionally. It is the Alternating Harmonic Series. By picture (see page 5 of these notes), the Alternating Harmonic Series converges. However, the sum of the absolute values, namely the Harmonic Series, does not converge. Therefore the alternating series converges conditionally. How do we check whether a series that does not converge absolutely converges? We need to make the by picture argument we used for the alternating harmonic series more precise. Answer: alternating series test. Next lecture. 9

10 2.5 Alternating Series Test.