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
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 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
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 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.)
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
Ps† I donít 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.