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I. Definitions: 1. Let f : D - JR and let c be an accumulation point of D. lim f ( x) = L if to each number x---+c E > 0 ... 2. Let f : D - JR and let c E D. f is continuous at c if to each number E > 0 ... 3. Let f : D - R f is uniformly continuous if to each number E > 0 ... 4. Let f : D - JR and let c ED. f is differentiable at c if ... 5. If f is n-times differentiable on I and a EI, then the Taylor polynomial for f of degree n. in powers of x - a is • • • 6. Let I= [a, b] and let P = {xo, xi, x2, ... , xn} be a partition of [a, b]. The upper sum of f with respect to P is · · • 1 II. Theorems: 1. Let f : D--+ JR be continuous. If D is compact, then f(D) is 2. Let f: [a.b]--+ JR be continuous. If f(a) < k < f(b), then ... 3. Let f : [a.b]--+ R If f is continuous on [a, b] and differentiable on (a, b), then there is at least one number c E ( a, b) such that ... 4. Let I and J be intervals. Let f : I--+ JR and g : J--+ JR, where f(I) � J. Let c E J. If f is differentiable at c, and g is differentiable at f ( c), then · · · 5. (Taylor's Theorem) Let f be (n + 1)-times differentiable on an open interval I and let a E J. Then for each x E J, x-::/- a, there is a number c ... 6. Let f be integrable on [a, b]. Define F: [a, b]--+ JR by F(x) = 1 x f(t) dt. Then F is: 2 III. Chapter 5. Limits and Continuity 1. TRUE-FALSE. If your answer is "True," justify by quoting a definition or theorem, or by giving an explanation. If your answer is "False," give a counter-example, which could be a figure. (a) Let f : [a, b] --t JR and let c E (a, b). If lim f(x) = L and L > 0, then there is a x-+c deleted neighborhood N*(c) such that f(x) > 0 for all x E N*(c). (b) Let f: D - JR be continuous. If D is bounded, then f(D) is bounded. (c) Let f: D --t JR be continuous. If D is open, then f(D) is open. ( d) Let f, g : D - JR . If f and f + g are continuous on D, then g is continuous on D. 3 2. Given that f : [1, 2] --+ R Which of the following statements are always true? Which are sometimes true? Which are never true? Give a reason for your answer. (a) If f is continuous and f(l) = e and /(2) = 1r, then there must be at least one point c E (1, 2) such that f(c) = 3. (b) If f(l) < 0 and /(2) > 0, then there must be at least one point c E (1, 2) such that f(c) = 0. (c) If f is continuous on [1, 2] and there is a point c in (1, 2) such that f(c) = 0, then f(l) and /(2) have opposite sign. (d) If there is a point c in (1, 2) such that f(c) > 0, then f(x) must be positive on some sub-interval of (1, 2). 4 3. Use the definition of limit to prove that lim (5x + 2) = 12. x-+2 x 2 - X - 6 4. Let f(x) = x-3 on (1,5), x/3. (a) Show that f can be extended to a function F which is continuous on [l, 5]. What is F(l), F(3), F(5)? (b) Is the extended function F uniformly continuous on (1, 5)? If so, why? 5 IV. Chapter 6. Differentiation 1. TRUE-FALSE: If your answer is "True," justify by quoting a definition or theorem, or by giving a proof. If your answer is "False," give a counter-example, which could be a figure. (a) If f : I-+ JR is continuous at x = c, then f is differentiable at x = c. (b) If f : I -+ JR is differentiable at x = c, then f is continuous at x = c. ( c) There does not exist a differentiable function f : JR -+ JR such that f(-1) = 1, J(3) = 9 and J'(x) � 1 for all x E [-1, 3]. ( d) If f is differentiable and strictly decreasing on the open interval I then f' ( x) < 0 on I. 6 2. Set f(x) -{ x 2 - 2x, 3, x 2- 2x+4, X � -l -1 < X < 1 X 2: 1 (a) Is f discontinuous at any point x? If so, where? (b) Does f fail to be differentiable at any point x? If so, where? 3. Evaluate the following limits. ( ) 1. x - sin x a im 3 . x-+O 2x (b) lim (1 + 3x) l/x_ X->00 7 4. Suppose that f is a function with the property that I j(n) (x) I � 3 for all n and for all x on [-1, 1]. (a) Estimate the error if the Taylor polynomial p4(x) (in powers of x) is used to approximate f(l/2). (b) Find the least positive integer n such that Pn(x) (in powers of x) will approximate f on [-1, 1] to within 0.001. 8 V. Chapter 7. Integration 1. Given that fo 1 f(x) dx = 3, 1 2 f(x) dx = -l, and fo 3 f(x) dx = 5, find 1 2 f(x) dx. 2. Suppose that f is continuous on [a, b]. Prove that there exists a point c E (a, b) such that 1 b f(t) dt = f(c)(b- a). by setting F(x) = 1 x f(t) dt and showing that F satisfy the hypotheses of the mean-value theorem for derivatives? 9 3. Suppose that f(x) = x rational { 4, 2, x irrational Let P be the partition {1,2,�,3,4}. (a) What is the lower sum .C(f, P)? (b) What is the upper sum U(f, P)? on [1,4]. (c) What is .C(f, Q) for any partition Q of [1, 4]? (d) What is U(f, Q) for any partition Q of [1, 4]? (e) Is f integrable on [1,4]? If not, why not? 10 4. Let f be continuous on [a, b] and suppose that f(x) � 0 for all x E [a, b] (a) Suppose that there is a point c E [a, b] such that f(c) > 0. Explain why l b f(x) dx > 0. (b) Give an example, which could be a figure, to show that this result might not hold if the continuity requirement is dropped from the hypothesis. 5. Suppose that f is a continuous function and Calculate F"(2). 11

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