## Transcribed Text

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 · · •
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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:
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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.
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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).
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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?
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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.
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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
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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.
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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?
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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?
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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).
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