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* indicates the problem will be collected |
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| Friday April 19 |
SOS Ch
3 pp 49,51, 53 Ch 11 pp279-282 Uniform continuity Beginning series Tests for convergence |
0. SOS
3.10a, 3.13, 3.21, 3.29, 3.30; 11.2-11.3, 1*. Let `f(x) = 5x -2` for all ` x in R`. i)Use the `epsilon-delta` definition of continuity to prove `f` is continuous on `R`. ii)Use the `epsilon-delta` definition of uniform continuity to prove `f` is uniformly continuous on `R`. 2*. Suppose `f: R ->R` and `g:R->R` are continuous functions. Let `h(x) = f(g(x))` for all `x in R`. i) Use the `epsilon-delta` definition of continuity to prove `h` is continuous on `R`. ii) Use the topological characterization of continuity to prove `h` is continuous. 3.* Suppose `A sub R`. Define `I(A) = cup{O sub R: O` is open and `O subA`}. a.* Find (and justify your result) `I([4,5])`. b.* Prove: i) `I(A) sub A`. ii)If `O` is an open sub set of `A` then `O sub I(A)`. iii)If A is an open set then `I(A)=A`. c.* Prove: Let `C` denote the Cantor set. Prove `I(C)=O/`. |
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| Friday April 12 |
SOS Ch 12 p324, 326 SOS Chapter 11 p279-283 Infinite series and function. Connectedness Continuous functions and connected sets The Intermediate Value Theorem (revisited) Compact sets(topological definition) Uniform continuity The real line is uncountable |
The Cantor Set `= C`.
Review the definition of the Cantor Set .See Integration
Notes from Week 10 1*. Suppose `a_n` is a sequence of numbers in the Cantor Set. Prove there is an element `b` in the Cantor set where `b ne a_n` for any `n in N`. [This the Cantor Set is "uncountably infinite".] 2*. Suppose `f : [0,1] -> R` with `f(x) = 1` for ` x in C` and `f(x) = 0` for `x in [0,1]-C`. a) Prove `f` is continuous for `[0,1]-C`. b) Prove `f` is Darboux integrable and `\int_0^1 f = 0.` 3*. Prove that `[0,1] - C ` is an open set. 4*. Suppose `a,b in R, a ne b`. Prove there exist open sets, `U` and `W` with ` a in U, b in W` and `U cap W = O/`. |
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| Friday, April 5 |
Chapter 4: Continuous Functions on RLimits of functionsLimits of functions in terms of limits of sequences Continuous functions Open and closed sets in metric spaces Continuity and open sets Properties of open and closed sets Constructing and recognizing open and closed sets |
0. SOS 5.9, 5.13, 5.28 1*. SOS 5.44 2*. SOS 5.46 3*. SOS 5.80 4. Suppose `f` is a function on `[a,b]` where for all `x,y in [a,b]`, if `x<y` then `f(x) \le f(y)`. a*. If `P = {t_0, t_1, ..t_n}` is any partition of `[a,b]`, what is `L(f,P)` and `U(f,P)`? b*. Suppose `t_k-t_{k-1} = delta` for each `k`. Prove `U(f,P)-L(f,P) = delta [f(b)-f(a)]`. c*. Prove `f` in Darboux integrable. |
Spivak Ch 13 Integrals Ch 14 FTofC Ch 5 Limits |
| Thursday March 28 |
SOS Ch 5 Interactive Real Analysis 7.1 The Riemann Integral Analysis Web Note: Chapter 8: Integration |
0. SOS 4.25, 4.26, 5.2, 5.3, 5.5, 5.6 Definition: Suppose `f : R -> R`. We say `lim_{x -> oo}f(x) = L` if for any unbounded sequence `{a_n}` with `a_{n+1} > a_n ` for all `n`, `lim_{n -> oo}f(a_n) = L`. 1 a.* Prove `lim_{x -> oo}1/x = 0`. b.* Prove if `lim_{x -> oo}f(x) = L` and `lim_{x -> oo}g(x) = M` then `lim_{x -> oo}(alpha f(x) + g(x)) = alpha L + M` where `alpha in R`. 2*. SOS. 5.34 3*. SOS. 5.42 4*. Suppose `f : [0,4] -> R` and `f(x) = 5` for `x in [0,2)` and `f(x) = 7` for ` x in [2,4]. Use the definition of the definite integral [ Riemann or Darboux- your choice] to show that `\int_0^4 f = 24.`
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| Friday March 8 |
SOS Ch.4 Differentiable Functions 6.5 |
0. SOS: 4.1, 4.4, 4.5, 4.12, 4.13, 4.15, 4.22,
4.23, 4.25 1.* 4.68 2.* 4.72 [Use the MVT.] 3.* Suppose f is a continuous function on an interval `(a,b)` and `c in (a,b)` with `f(c) = L >0`. Prove: There is a natural number M > 0 so that for all `x in (c-1/M, c+1/M)`, `f(x) > 0`. 4. a. Prove: If `f(x) = 1/x` for `x \ne 0` , then `f '(x) = - 1/{x^2}` for `x \ne\ 0`. [Not to be submitted.] b.* Use the chain rule, the product rule, and part a to prove the quotient rule. 5. Let `D =[-2,-1] uuu (2,3] `and `f(x) = x^2` for ` x in D`. Let `E = [1,9]` and `g(x) = -sqrt(x)` for `x in [1,4]` and `g(x) = sqrt(x)` for `x in (4,9]`. a.* Prove: `g (f (x)) = x ` for all `x in D` and `f (g (y)) = y` for all `y in E`, so `g = f^{-1}`. b.* Explain why `f ` is continuous on `D` , but `g` is not continuous on `E`. [Give details to show why `g` is not continuous at 4.] 6. Suppose `f ` is continuous on an interval `[a,b]` and for all `x, y in [a,b]` if `x <y` then `f(x)<f(y)`. a.* Prove: `f([a,b]) = {y in R: y = f(x)` for some `x in [a,b]}= [f(a), f(b)]`. [This is an equality of sets.] b.* Prove: There is a unique function `g: [f(a),f(b)] -> [a,b]` so that `g (f (x)) = x ` for all `x in [a,b]` and `f (g (y)) = y ` for all `y in [f(a),f(b)]`. c.* Prove: The function `g ` described in part b is continuous on [f(a), f(b)]. |
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| Thursday Feb 28 |
SOS: Ch.3. Continuity of Real functions The boundedness theorems Images of intervals |
0. SOS: 2.16, 2.18; 3.3, 3.6, 3.16 1. * Prove: If `{a_n}` is a sequence with `a_n > 0 ` for all `n` and `lim_{n -> oo}a_n = L`, then `L >= 0`. 2.* Prove that if a subsequence of a Cauchy sequence converges, then the original Cauchy sequence converges. 3.* Suppose the sequence `{a_n} ` is bounded and monotonic and `{b_n}` is a sequence with `|a_n - b_n| = (b_0-a_0)/ {2^{n}}`. Show that `{b_n}` converges, and `lim_{n -> oo}a_n = lim_{n -> oo}b_n `. 4.a*. Prove that ` 1/{n+1} < ln(n+1)- ln(n) < 1/ n `. b. * Let `a_n = 1 + 1/2 + 1/ 3 + ... +1/n - ln(n)`. Show that `a_n >= 0` and `a_n > a_{n+1}` for all `n`. [thus there is a limit for the `a_n` which is called Euler's constant and denoted `gamma`.] 5.a* Give a definition of piecewise continuous based on the definition of continuity for a set D using sequences. b*Use your definition to prove that the function `h` where `h(x) = 1 ` for `x >= 0` and `h(x) = 0` for `x<0 ` is piecewise continuous. |
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| Thursday Feb 21 |
SOS: Ch.2 p25-27. Cauchy sequences Limits of functions The epsilon-delta definition Continuity of Real functions Some horrible functions |
0. 2.13, 2.14, 2.16, 2.18, 2.22, 2.24, 2.35
1. Suppose f and g be real valued functions which are continuous on `(-oo,oo)` and `f (0) = g(0)`. a.* Prove that the function defined by h(x) = f(x) for x `<=` 0 and h(x) = g(x) for x > 0 is continuous everywhere. b.* Prove that the absolute value function | x | is continuous everywhere. c.* If f is a continuous function on R, prove that the function | f(x) | is also continuous on R. d.* If f and g are continuous functions prove that the function M(x) = max{f (x), g (x)} is also continuous. [Hint: Prove that max(a, b) =((a + b) + |a - b|)/2 for any real numbers a and b.] e. Prove that the function m(x) = min{f (x), g (x)} is continuous if f and g are. 2.a.* Use the geometry of the sine function on the unit circle to prove that `| sin x - sin y | <= | x - y |` b. * Prove that the function f(x) = sin x is continuous everywhere. c. Prove that the function cos x is continuous everywhere. 3.* Suppose f :[0,1] `->` [0,1] is a continuous function. Prove there is some `a in [0,1]` where `f(a) = a`. |
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Thursday Feb 14 |
SOS: p6 (Bounds) Ch.2 p25-26. Axioms for the Real numbers Sequences of Numbers The intermediate value theorem |
0. SOS: 1.21(a,b); 1.64(a,b); 1.65(a,b);
2.20(a,b) ; 2.21 (a,b); 2.22 *1. Suppose A and B are nonempty sets of numbers and that for any x in A and any y in B, `x <=y`. a) Prove that the `LUB (A) <= y` for all y in B b) Prove that `LUB(A) <= GLB( B)`. *2. Suppose A and B are nonempty sets of numbers that are bounded above. Let A+B denote the set of numbers that can be expressed as a+b where a is in A and b is in B. Prove: `LUB (A + B) = LUB(A) + LUB(B)` *3 .a) Suppose `I_1 = [a_1,b_1], I_2 = [a_2,b_2], ... `is a sequence of close intervals with `a_n<b_n`, `a_n<=a_{n+1)` and ` b_n >= b_{n+1}` for all `n`. Prove: There is a number c that is in all the intervals. b) Give a counterexample to show that the statement in part a is not true if the intervals are open instead of closed. |
Spivak Ch 8 especially Proof of Theorem 7.1 SOS: 1.34 on Dedekind Cuts |
| Thursday Feb. 7 |
SOS: Chapter 3 THEOREMS ON CONTINUITY p52 |
*1.Give a statement the Extreme Value Theorem with
examples illustrating how it can be false if one of the
hypotheses is not satisfied. *2. Suppose f is a continuous function on a domain D that is the union of a finite number of intervals of the form `[a_j, b_j]` with `a_j < b_j` for `j` = 1 to n and a set of real numbers `{ c_k: k = 1 `to` m}`. Prove that there are numbers p and q ` in D`, where `f(p) <= f(x) <= f(q)` for all `x in D`. *3. Suppose f is a continuous function on a domain D that is the union of a finite number of intervals of the form `[a_j, b_j]` with `a_j < b_j` for `j `= 1 to n and set of real numbers `{ c_k: k = 1` to `m}`. Using the Intermediate Value Theorem and the Extreme Value Theorem, prove that f (D) = {y : y = f (x) for some x in D} is also the union of a finite number of intervals of the form `[r_i, s_i]` with `r_i < s_i` for `i = 1` to ` n'` and set of real numbers (possibly empty)`{ t_l: l = 1` to ` m'}`. |
Spivak: Ch 7 pp 120-122 |
| Thursday Jan 31 |
SOS: Chapter 1 Up through POINT SETS, INTERVALS (p5) Chapter 4 MEAN VALUE THEOREMS.(p77-78) [Proofs can be found in the solved problems 4.19, 4.20(Note error in F.)] |
*1.Give a statement and proof of the Mean Value
Theorem and of Rolle's Theorem. [You may use other results
for your proof. If you use Rolle's theorem to prove the
MVT- prove Rolle's Theorem without the MVT!] *2. Prove: Rolle's Theorem is equivalent to the Mean Value Theorem. *3. For any real number x let |x| be the absolute value of x, i.e., |x| = x when x>0,|x| =-x when x< 0, and |x| = 0 when x =0. Prove: For any real numbers a and b, ` |a+b| <= |a| + |b|`. *4. Prove the following: (i) `|a-b| <= |a| + |b|`. (ii) `|a| - |b| < = |a-b|`. (iii) `|( |a| - |b| )| <= |a-b|`. *5. Suppose m(x,y) = |y-x| for x and y real numbers.Prove that for any real numbers, a,b, and c: (i) m(a,b) = 0 if and only if a=b. (ii) m(a,b) = m (b,a). (iii) `m(a,b) < = m(a,c) + m(b,c)` (iv) If m(a,b) = m(a,c) + m(b,c) then either a= c, b=c, a<c< b, or b<c<a. |
Spivak: Chapter 1; pp190-192 |
| Tuesday Jan. 29 |
Chapter
1:Analysis WebNotes by John Lindsay Orr
Introduction
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1. Make a list of 5 theorems that you feel are critical
to understanding and using the material you learned in the
first year of calculus. Be sure to state clearly the theorem with all its hypotheses and conclusions. 2. For each of the 5 theorems you gave, provide at least one example of how the theorem can fail if only one of the hypotheses is not satisfied. |
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| Future possible problem(s) | Definition: Suppose
`f:D->R` . We say `f` is an open map
if for any open set `O`, there is a set `hatO`
where `f(O cap D)= hatO cap f(D). 1. For each of the following functions `f` determine if `f` is an open map. If it is - prove it, if not give an example that shows why `f` is not an open map. a. `f:R->R, f(x) = x^2` b. `f: R->R, f(x) = sqrt(|x|)` c. `f: R->R, f(x) = 0` d.`f: R->R, f(x) = 5x + 2` e. `f: [1,2]->R, f(x) = x^2.` 2. Suppose `f : R ->R` is a 1:1 function: Prove: `f` is an open map if and only if `f^{-1}` is a continuous function. |
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