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Unformatted text preview: 11/11/2001 SUN 14:13 FAX 6434330 MOFFITT LIBRARY 001
George M. Bergman F0,” 1997, Math HlA 18 December, 1997 61 Evans Hall Final Exam 12:30—3z30 PM 1. (30 points, 5 points apiece) Compute each of the following. (a) d/dx (ln(ln x)).
(b) limx_m/4 ((tan x) —1)/((coszx) — 1/1).
(c) limx_>0— (tanx)/((coszx) —l). (d) An expression for the “telescoping sum” Elli"? f(i+1) —f(1') in which cancelling summands have been dropped. (6) limn_)+m (2;;1 10’/”)/n. (t) j): cos (x2/3) dx. 2. (18 points) The Riemann integral of a function f was deﬁned by the formula I:f(x)dx = ilm1|P[|__>0 221:1 f(xi*)Axl-. (a) (9 points) In the above formula, P denotes a partition of [a, b], denotes its norm, and the Axi denote certain related numbers. Deﬁne each of these: A partition P of [42.19] means
Given such a partition P, we deﬁne Axi = For such apartition P, its norm [|P|| means (b) (9 points) Now complete the sentence below to give a precise (“s-6”) deﬁnition of
the limit expression in the above deﬁnition of the Riemann integral. (You do not, of
course, have to explain Z-notation. On the other hand, your deﬁnition should make clear
what the symbol x5“ refers to.) If f is a function on an interval [a, b], and L is a real number, we write . n I f(xfk)Axi = L 1f 11/11/2001 SUN 14:13 FAX 6434330 MOFFITT LIBRARY 002 3. (15 points) (a) (5 points) Suppose f is a continuous function on (0,+oo), and a
and b are positive real numbers such that f(ax) bf(x) for all x > 0. Show that if
F is any antiderivative of f, there will exist a constant k such that F(ax) = abF(x) + k for all x>0. II (b) (5 points) Letting f(x) = l/xz, and a be an arbitrary positive real number, give a
value of b that makes the equation ﬂax) : bf(x) hold. Give the general antidcrivativc F of this function f, and for each such antiderivative ﬁnd a constant k satisfying the equation given in (a).
(c) (5 points) Do the same for f(x) = l/x. 4.. (20 points, 5 points apiece) Give an example of each of the following. You do not
have to prove that your examples have the properties asked for. When asked to give a
function with a given domain [(1.19], you may give a function with larger natural domain, understanding it to be restricted to the domain asked for. (a) A function f on [0,1] having no maximum value. (b) A continuous function f on [1,101 which has a local maximum that is not an absolute maximum. (c) A continuous function f on [0,5] which is concave upward on [0, 3] and concave downward on [3, 5]. (d) A one—to-one function f with domain [0,1] and range [3,5], and the function g inverse to f. b 5. (7 points) Given that d/dx tanx = 3602):, get a formula for _[ seczx dx, and state
(1 conditions on a and b under which your formula is valid. (For full credit, you should give the most general such conditions.) 6. (10 points) In the last reading, we deﬁned the exponential function exp as the inverse
of the natural logarithm function In. Using this deﬁnition, prove the identity exp(x +y) =
exp(x) exp(y). The proof uses one or more properties of the natural logarithm function;
state explicitly the properties you use. (You are not asked to prove those properties of the
logarithm, or to say anything about how the logarithm function is deﬁned.) ...
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