Homework for M317

 

HW1 Let a, b, c (Mathtypeís symbol for the set of real numbers.) and define a-b=a+(-b).

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

(b)Prove that Ė(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

 

HW3Use 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, FitzpatrickDue 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

 

HW8Prove that is irrational.Due 1/29

 

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

 

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

 

HW11Use the definition to prove that .

 

HW12Use the definition to prove that .

 

HW13Suppose .Prove that there exists an integer N and a real K such that for , .

 

HW14Suppose .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 .

 

HW24If andis 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 isnít written as a two piece functions because I couldnít 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 3/23/07

 

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

 

HW56For 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 .

 

HW65Use the definition to prove that .

 

HW66Use the definition to prove that .

 

HW67Use Prop 4.9 () to prove that is continuous at x=1.

 

HW68Use 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.

 

HW69Use the definition of the derivative to compute f Ď(-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 wonít let me do it the way that we all know it should be done.)

 

HW70Decide where the function is differentiable.Prove all of your claims.

 

HW71Use 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 .

 

 

PsI donít know why the MathType made all of the mathematics so ugly all of the sudden.

 

 

HW73Prove that .

 

HW74Decide whether or not the limit exists for .If not, show why.If yes, prove it.