1 |
- Give an example of (a) a finite abelian group; (b) an infinite non-abelian group;
- Ex. 1 (except Thms I.6, I.11) on p. 19;
- Ex. 1 on p. 21;
- Ex. 12 on p. 36-37, ex. 2 on p. 40, ex. 12 on p. 41.
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2 |
- Read subsections I.4.5 and I.3.11 in the book;
- Ex. 1, 2, 3, 4, 5, 7, 8 on p. 28.
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3 |
- Ex. 1abchij on p. 43;
- Ex. 2def, 4ab on p. 63-64;
- Ex. 1bc, 5, 12 on p. 70-71.
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4 |
- Prove that if f(x) is bounded on [a, b] and monotonic on (a, b), then it is integrable on [a, b];
- Give an example of a function on [a, b] which is monotonic on (a, b) and not integrable on [a, b];
- Ex. 13, p. 45-46.
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5 |
- Ex. 9, 21, 25 on p. 83-84;
- Ex. 17b, p. 94.
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6 |
- Review integration: ex. 19, 25, 27 on p. 105, ex. 8, 15 on p. 124, ex. 15 on p. 114;
- Prove using the \epsilon-\delta language: f(x)=\sqrt(x) is continuous for all x>0;
- Ex. 28, p. 139.
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7 |
- Read section 3.4 in the book;
- Prove that if f(x)= x^2 sin(1/x) for x not equal to 0, and f(0)=0, then f(x) is continuous at x=0;
- Ex. 1, 3, 11, 14, 16, 19 on p. 142;
- Ex. 1, 5, 6 on p. 145.
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8 |
- Let f(x)=x^4 +2 x^2 +1 for x in [0, 2]. Show that f(x) is strictly increasing and find the domain of the inverse function g(y). Find an expression for g(y);
- Show by example that Intermediate Value Theorem can fail if f(x) is continuous only on (a, b) and bounded on [a, b];
- Ex. 7, p. 155.
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9 |
- Ex. 12, 14, 15, 24, 35, 38 on p. 167-168;
- Find f'(x) by definition (if it exists):
a) f(x) = x sin(1/x) for nonzero x, and f(0)=0; b) f(x) = x^2 sin(1/x) for nonzero x, and f(0)=0. Are the derivatives continuous at x =0? - Ex 4, 6, 7, 10, 14, 15 on p. 179-180.
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10 |
- =Find the derivative dy/dx by implicit differentiation:
(a) a cos^2(x+y) = b, where a, b are nonzero numbers; (b) x^3 + y x^2 + y^2 = 0; - Read sections 4.17, 4.18 (in particular, thm. 4.10 on convexity);
- Ex. 9, 14 on p. 191; Ex. 5 on p. 186.
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11 |
- Ex. 3, 4, 6, 8, 12, 22 on p. 208;
- Obtain an estimate for \pi: 3 < \pi < 2 \sqrt(3) by using estimates for the integral of cos(x) from 0 to (\pi)/6;
- Ex. 2 to 17 on p. 216.
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12 |
- Ex. 2-9 on p. 220.
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13 |
- Show that (1 + 1/n)^n < e < (1 + 1/n)^{n+1} (it may be helpful to consider the integral of e^x from 0 to 1/n);
- Ex. 10-20 on p. 236;
- Ex. 9-16, 21-29 on p. 248-249;
- Ex. 30-37 on p. 258.
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14 |
- Ex. 5, 11, 12, 19, 20, 21, 22 on p. 267;
- Ex. 33-38 on p. 267.
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15 |
- Ex. 6-14, p. 291 (use Taylor's formula);
- Ex. 3, 4, 5, 7, 8 p. 278;
- Ex. 4 on p. 285;
- Read Section 7.3.
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16 |
- Ex. 4-14, p. 295;
- Ex. 1-7, 11, 12, p. 303.
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17 |
- Ex. 15-29, p. 291;
- Ex. 14-25, p. 303;
- Ex. 1-8, 24, 27, 30, p. 382;
- Ex. 2, 4, 11, 12, 15, 17, 18, p. 391.
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18 |
- Review for the quiz.
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19 |
- Ex. 1, 2, 4-11, p. 402;
- Ex. 1-14, p. 398;
- Ex. 1-9, p. 409.
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20 |
- Ex. 13-21, 27-32 p. 409;
- Ex. 49-51, p. 411.
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21 |
- Ex. 1-10, p. 420;
- Ex. 1-10, p. 430.
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22 |
- Ex. 1-4, 6-10, p. 438;
- Expand the function f(x)= x^3 -2x^2 -5x -2 in a series of powers of (x+4).
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23 |
- Ex. 11, 13-17, p. 439;
- Expand the integral from 0 to x of sin(t)/t and find the radius of convergence;
- Expand 1/(4-x^4) and find the radius of convergence.
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