Homework for M317

HW1 Let a, b, c _{} _{} (Mathtype’s symbol for the set of real
numbers.) and define a-b=a+(-b).

(a) Prove that a(b-c)=ab-ac.

(b) Prove that –(b-c)=-b+c. Due 1/19

HW2
(a) Prove that _{}is not
rational, i.e. prove that there is no rational number r such that r^{ 2}
= 7.

(b) Prove that _{}is not
rational. Due 1/22

HW3
Use math induction to prove that _{}. Due 1/23

HW4 Use math induction to prove that _{} for all negative integers n.

[Make clear what calculus results you assume to be true as a part of your proof.] Due 1/24

HW5 #2, page 10, Fitzpatrick Due 1/26: Note that 5(c) has been corrected. Thank you Amy Clark.

HW6 #11, page 11, Fitzpatrick. Due 1/26

HW7 #13, page 11 Fitzpatrick. Due 1/29

HW8
Prove that _{} is irrational.
Due 1/29

HW9 #1, page 16 Fitzpatrick. Due 1/30

HW10 #4, page 16, Fitzpatrick Due 1/20

HW11 Use the definition to prove that _{}.

HW12 Use the definition to prove that _{}.

HW13 Suppose _{}. Prove that there exists an integer N and a
real K such that for _{}, _{}.

HW14 Suppose _{}. Prove that there exists an integer N and a
real K such that for _{}, _{}.

HW15 #1, page 32, Fitzpatrick Due 2/6

HW16 #11, page 34 Fitzpatrick Due 2/7

HW17 #1, page 37, Fitzpatrick Due 2/9

HW18 #2, page 37, Fitzpatrick Due 2/9

HW19 #3, page 37, Fitzpatrick Due 2/9

HW20 #1, page 42, Fitzpatrick Due 2/12

HW21 #2, page 43, Fitzpatrick Due 2/12

HW22 #3, page 43, Fitzpatrick Due 2/12

HW23
(a) Use the _{}-_{}definition
to prove that _{}.

(b) Use the _{}-_{}definition
to prove that _{}.

HW24 If _{}and _{}is some
sequence such that _{}, then _{}. If true, prove it. If false, explain why and give a counter
example.

HW25 #1, p. 57, Fitzpatrick

HW26 #3, p. 57, Fitzpatrick

HW27 #9, p. 58, Fitzpatrick (you may ignore the hint).

HW28 Define _{} for _{}and *f(0)=0*.
(The function isn’t written as a two piece functions because I couldn’t
get my typing software to do it.)

(a) Graph the function (a reasonably rough graph is acceptable).

(b) Discuss the
continuity of* f*---assuming that *sin(x)* is continuous.

(c) Specifically,
prove that *f* is continuous at *x=0*.

HW29. #13, p.58, Fitzpatrick (again, you may ignore the hint.)

HW30 #1, page 61, Fitzpatrick.

HW31 #1, page 65, Fitzpatrick Due 3/2

HW32 #2, page 65, Fitzpatrick Due 3/2

HW33 #5, page 65, Fitzpatrick Due 3/2

HW34 #1, page 69 Fitzpatrick

HW35 #3, page 69 Fitzpatrick

HW36 #6, page 69 Fitzpatrick

HW37 #1, page 80, Fitzpatrick Due 3/5

HW38 #4, page 80, Fitzpatrick Due 3/9

HW39 #13, page 80, Fitzpatrick Due 3/9

HW40 #14, page 80, Fitzpatrick Due 3/9

HW41 #15, page 80, Fitzpatrick Due 3/9

Description of why rotating the curve about y=x gives the plot of the inverse function.

HW42 #1, page 93, Fitzpatrick, Due 3/20

HW43 #4, page 94, Fitzpatrick. Due 3/21

HW44 #11, page 95, Fitzpatrick. Due

HW45 #12, page 95, Fitzpatrick, Due 3/23

HW46 #9, page 95, Fitzpatrick, Due 3/26

HW47 #10, page 95, Fitzpatrick, Due 3/26

HW48 #4, page 101, Fitzpatrick, Due 3/28

HW49 #19, page 110 Fitzpatrick, Due 4/2

HW50 #1, page 108 Fitzpatrick

HW51 #4, page 108, Fitzpatrick

HW52 #15, page 109, Fitzpatrick

HW53 #2, page 141, Fitzpatrick

HW54 #5, page 141, Fitzpatrick

HW55 #7, page 142, Fitzpatrick

HW56
For _{} on [0,1] compute _{} and _{}.

HW57 #3, page 149, Fitzpatrick

HW58 #7, page 149, Fitzpatrick

HW59 #10, page 150, Fitzpatrick

HW60 _{}, bounded and _{}. If f is integrable
on [a,c] and [c,b], then f is integrable on [a,b] and _{}.

HW61 #1, page 159 Fitzpatrick

HW62 (a) _{}, f integrable on [a,b] and _{} for all _{}. Then _{}.

(b) _{}, f continuous on [a,b] and _{}. Then there exists _{} such that _{}.

HW65
Use the definition to prove that _{}.

HW66
Use the definition to prove that _{}.

HW67
Use Prop 4.9 (_{}) to prove that _{} is continuous at x=1.

HW68
Use Prop 4.15 (sequential equivalent def of cont---books def) to prove
that the function f defined by _{}when x is rational and _{} when x is irrational
is not continuous at x=1. What about
x=0? Prove it.

HW69 Use the definition of the derivative to
compute f ‘(-1) where f is defined to be _{} when x<0 and _{} when x>0. (Please excuse the ugly way of writing the
functions in this problem and HW68. As
far as I can tell, MathType won’t let me do it the way that we all know it
should be done.)

HW70 Decide where the function _{} is
differentiable. Prove all of your
claims.

HW71
Use the definition or the Archimedes-Riemann Theorem to prove that the
following integral exists and to compute its value: _{}.

HW72.
Define _{} for x rational and _{} for x irrational. Define _{} when _{} and _{} when x>0. If they exist, compute _{}

and _{}.

Ps I don’t know why the MathType made all of the mathematics so ugly all of the sudden.

HW73 Prove that .

HW74 Decide whether or not the limit exists for . If not, show why. If yes, prove it.