Solving Cubic Polynomials
So how do we find these rational solutions when they occur? If x = p q is a rational solution to the polynomial equation f(x) = 0 then
Download File PDF Real Imaginary Solutions Polynomials
il y a 4 jours Finding Real and Imaginary Roots of a Polynomial Equation . ... Polynomial Function Using The Rational Zero Theorem Determine.
7.2 Finding Complex Solutions of Polynomial Equations
19 mars 2014 What are the real zeros of p (x)? Solve x 2 + 4x + 16 = 0 using the quadratic formula. x = -b ±? ???.
Practice Masters Levels A B
https://www.monroe.kyschools.us/userfiles/1010/file/5-4%20practice%20levela.pdf
Online Library Real Imaginary Solutions Polynomials
9 mai 2022 ties Partial Fractions Polynomials Rational Expressions Sequences ... Find the real or imaginary solutions of the equation by factoring.
Answers and Solutions
Alternatively we need to find numbers a
Elementary Number Theory and Algebra
Find all rational solutions ? ? Q of the equation f(x) = x3 + x2 ? 5x +3=0. As both terms n and b are divisible by a the equality (3.4.7.2) is ...
rational normal forms and stability of small solutions to nonlinear
7.2. Elimination of the quintic term by the cubic Key words and phrases. ... for some solutions of a nonlinear wave equation in 1d with periodic ...
Lesson 7-2 worksheet answers.pdf
Answers. Finding Complex Solutions of Polynomial Equations. Practice and Problem Solving: A/B. Write the simplest polynomial function with the given roots.
Chapter 5. Integration §1. The Riemann Integral Let a and b be two
In this section we establish some basic properties of the Riemann integral. Theorem 2.1. Let f and g be integrable functions from a bounded closed interval [a
Chapter 5. Integration
§1. The Riemann Integral
Letaandbbe two real numbers witha < b. Then [a;b] is a closed and bounded interval in IR. By apartitionPof [a;b] we mean a finite ordered set{t0;t1;:::;tn}such that a=t0< t1<···< tn=b: ThenormofPis defined by∥P∥:= max{ti-ti-1:i= 1;2;:::;n}. Supposefis a bounded real-valued function on [a;b]. Given a partition{t0;t1;:::;tn} of [a;b], for eachi= 1;2;:::;n, let m Theupper sumU(f;P) and thelower sumL(f;P) for the functionfand the partitionPare defined by
U(f;P) :=n∑
i=1M i(ti-ti-1) andL(f;P) :=n∑ i=1m i(ti-ti-1):Theupper integralU(f) offover [a;b] is defined by
U(f) := inf{U(f;P) :Pis a partition of [a;b]}
and thelower integralL(f) offover [a;b] is defined byL(f) := sup{L(f;P) :Pis a partition of [a;b]}:
A bounded functionfon [a;b] is said to be (Riemann)integrableifL(f) =U(f). In this case, we write∫b a f(x)dx=L(f) =U(f):By convention we define
a b f(x)dx:=-∫ b a f(x)dxand∫ a a f(x)dx:= 0: A constant function on [a;b] is integrable. Indeed, iff(x) =cfor allx∈[a;b], then L(f;P) =c(b-a) andU(f;P) =c(b-a) for any partitionPof [a;b]. It follows that b a cdx=c(b-a): 1 Suppose thatP={t0;t1;:::;tn}is a partition of [a;b], and thatP1is a partition obtained fromPby adding one more pointt∗∈(ti-1;ti) for somei. The lower sums forPandP1 m L(f;P1)-L(f;P) =m′(t∗-ti-1) +m′′(ti-t∗)-mi(ti-ti-1): m mConsequently,
Now suppose thatPNis a mesh obtained fromPby addingNpoints. An induction argument shows thatSimilarly we have
By the definition ofL(f) andU(f), for eachn∈IN there exist partitionsPandQof [a;b] such that Consider the partitionP∪Qof [a;b]. SinceP⊆P∪QandQ⊆P∪Q, by (1) and (2) we get We are in a position to establish the following criterion for a bounded function to be integrable. 2 Theorem 1.1.A bounded functionfon[a;b]is integrable if and only if for each" >0 there exists a partitionPof[a;b]such thatU(f;P)-L(f;P)< ":
Proof.Suppose thatfis integrable on [a;b]. For" >0, there exist partitionsP1andP2 such thatL(f;P1)> L(f)-"
2 andU(f;P2)< U(f) +" 2ForP:=P1∪P2we have
L(f)-"
2 2SinceL(f) =U(f), it follows thatU(f;P)-L(f;P)< ".
Conversely, suppose that for each" >0 there exists a partitionPof [a;b] such that U(f;P)-L(f;P)< ". ThenU(f;P)< L(f;P) +". It follows that that is,fis integrable. Letfbe a bounded real-valued function on [a;b] and letP={t0;t1;:::;tn}be a partition of [a;b]. For eachi= 1;2;:::;n, choosei∈[xi-1;xi]. The sum n i=1f(i)(ti-ti-1) is called aRiemann sumoffwith respect to the partitionPand points{1;:::;n}. Theorem 1.2.Letfbe a bounded real-valued function on[a;b]. Thenfis integrable on [a;b]if and only if there exists a real numberIwith the following property: For any" >0 there exists some >0such that n wheneverP={t0;t1;:::;tn}is a partition of[a;b]with∥P∥< andi∈[ti-1;ti]for i= 1;2;:::;n. If this is the case, then b a f(x)dx=I: 3 Proof.Let"be an arbitrary positive number. Suppose that (3) is true for some partition P={t0;t1;:::;tn}of [a;b] and pointsi∈[ti-1;ti],i= 1;2;:::;n. ThenL(f;P) = inf{
n∑ i=1f(i)(ti-ti-1) :i∈[xi-1;xi] fori= 1;2;:::;n} ≥I-" andU(f;P) = sup{
n∑ i=1f(i)(ti-ti-1) :i∈[xi-1;xi] fori= 1;2;:::;n} on [a;b]. Moreover,L(f) =U(f) =I. Conversely, suppose thatfis integrable on [a;b]. LetM:= sup{|f(x)|:x∈[a;b]} andI:=L(f) =U(f). Given an arbitrary" >0, there exists a partitionQof [a;b] such thatL(f;Q)> I-"=2 andU(f;Q)< I+"=2. Suppose thatQhasNpoints. Let P={t0;t1;:::;tn}be a partition of [a;b] with∥P∥< . Consider the partitionP∪Qof [a;b]. By (1) and (2) we have :="=(4MN). Since∥P∥< , we deduce from the foregoing inequalities that Thus, withi∈[ti-1;ti] fori= 1;2;:::;nwe obtainThis completes the proof.
Theorem 1.3.Letfbe a bounded function from a bounded closed interval[a;b]toIR. If the set of discontinuities offis finite, thenfis integrable on[a;b]. Proof.LetDbe the set of discontinuities off. By our assumption,Dis finite. So the setD∪ {a;b}can be expressed as{d0;d1;:::;dN}witha=d0< d1<···< dN=b. Let M:= sup{|f(x)|:x∈[a;b]}. For an arbitrary positive number", we choose >0 such 4 that < "=(8MN) and <(dj-dj-1)=3 for allj= 1;:::;N. Forj= 0;1;:::;N, let x j:=dj-andyj:=dj+. Then we have a=d0< y0< x1< d1< y1<···< xN< dN=b: LetEbe the union of the intervals [d0;y0], [x1;d1];[d1;y1];:::;[xN-1;dN-1];[dN-1;yN-1], and [xN;dN]. There are 2Nintervals in total. Forj= 1;:::;N, letFj:= [yj-1;xj]. Further, letF:=∪Nj=1Fj. The functionfis continuous onF, which is a finite union of bounded closed intervals. Hencefis uniformly continuous onF. There exists some >0 such that|f(x)-f(y)|< "=(2(b-a)) wheneverx;y∈Fsatisfying|x-y|< . For each j∈ {1;:::;N}, letPjbe a partition ofFjsuch that∥Pj∥< . LetP:={a;b} ∪D∪(∪Nj=1Pj):
The setPcan be arranged as{t0;t1;:::;tn}witha=t0< t1<···< tn=b. ConsiderU(f;P)-L(f;P) =n∑
i=1(Mi-mi)(ti-ti-1); [ti-1;ti] is either contained inEor inF, but not in both. Hence n i=1(Mi-mi)(ti-ti-1) =∑ [ti1;ti]⊆E(Mi-mi)(ti-ti-1) +∑ [ti1;ti]⊆F(Mi-mi)(ti-ti-1): There are 2Nintervals [ti-1;ti] contained inE. Each interval has length < "=(8MN). 2 If [ti-1;ti]⊆F, thenti-ti-1< ; henceMi-mi< "=(2(b-a)). Therefore,2(b-a)∑
[ti1;ti]⊆F(ti-ti-1)<"2(b-a)(b-a) ="
2 From the above estimates we conclude thatU(f;P)-L(f;P)< ". By Theorem 1.1, the functionfis integrable on [a;b]. Example 1.Let [a;b] be a closed interval witha < b, and letfbe the function on [a;b] given byf(x) =x. By Theorem 1.3,fis integrable on [a;b]. LetP={t0;t1;:::;tn}be a partition of [a;b] and choosei:= (ti-1+ti)=2∈[ti-1;ti] fori= 1;2;:::;n. Then n i=1f(i)(ti-ti-1) =1 2 n i=1(ti+ti-1)(ti-ti-1) =1 2 n i=1(t2i-t2i-1) =1 2 (t2n-t20) =1 2 (b2-a2): 5By Theorem 1.2 we have∫b
a xdx=1 2 (b2-a2): More generally, for a positive integerk, letfkbe the function given byfk(x) =xkfor x∈[a;b]. Choose i:=(tki-1+tk-1 i-1ti+···+tki k+ 1) 1=k ; i= 1;2;:::;n: n i=1f k(i)(ti-ti-1) =1 k+ 1n i=1(tk+1 i-tk+1 i-1) =1 k+ 1(tk+1n-tk+10) =1 k+ 1(bk+1-ak+1):By Theorem 1.2 we conclude that
b a xkdx=1 k+ 1(bk+1-ak+1): andg(0) := 0. The only discontinuity point ofgis 0. By Theorem 1.3,gis integrable on [0;1]. Note thatgis not uniformly continuous on (0;1). Indeed, letxn:= 1=(2n) and y n:= 1=(2n+=2) forn∈IN. Then limn→∞(xn-yn) = 0. But |f(xn)-f(yn)|=|cos(2n)-cos(2n+=2)|= 1∀n∈IN: Hencegis not uniformly continuous on (0;1). On the other hand, the functionugiven continuous on (0;1]. Theorem 1.3 is not applicable tou, becauseuis unbounded. Example 3.Lethbe the function on [0;1] defined byh(x) := 1 ifxis a rational number in [0;1] andh(x) := 0 ifxis an irrational number in [0;1]. LetP={t0;t1;:::;tn}be a partition of [0;1]. Fori= 1;:::;nwe have m i:= inf{h(x) :x∈[ti-1;ti]}= 0 andMi:= sup{h(x) :x∈[ti-1;ti]}= 1: HenceL(h;P) = 0 andU(h;P) = 1 for every partitionPof [0;1]. Consequently,L(h) = 0 andU(h) = 1. This shows thathis not Riemann integrable on [0;1]. 6§2. Properties of the Riemann Integral
In this section we establish some basic properties of the Riemann integral. Theorem 2.1.Letfandgbe integrable functions from a bounded closed interval[a;b] toIR. Then (1)For any real numberc,cfis integrable on[a;b]and∫b a(cf)(x)dx=c∫b af(x)dx; (2)f+gis integrable on[a;b]and∫b a(f+g)(x)dx=∫b af(x)dx+∫b ag(x)dx. Proof.Suppose thatfandgare integrable functions on [a;b]. WriteI(f) :=∫b af(x)dx andI(g) :=∫b ag(x)dx. Let"be an arbitrary positive number. By Theorem 1.2, there exists some >0 such that n wheneverP={t0;t1;:::;tn}is a partition of [a;b] with∥P∥< andi∈[ti-1;ti] for i= 1;2;:::;n. It follows that n i=1(cf)(i)(ti-ti-1)-cI(f)=|c|nHencecfis integrable on [a;b] and∫b
a(cf)(x)dx=c∫b af(x)dx. Moreover, n i=1(f+g)(i)(ti-ti-1)-[I(f) +I(g)] n i=1f(i)(ti-ti-1)-I(f)+nThereforef+gis integrable on [a;b] and∫b
a(f+g)(x)dx=∫b af(x)dx+∫b ag(x)dx. Theorem 2.2.Letfandgbe integrable functions on[a;b]. Thenfgis an integrable function on[a;b]. Proof.Let us first show thatf2is integrable on [a;b]. Sincefis bounded, there exists for any partitionPof [a;b]. Let" >0. Sincefis integrable on [a;b], by Theorem 1.1 7 there exists a partitionPof [a;b] such thatU(f;P)-L(f;P)< "=(2M). Consequently, U(f2;P)-L(f2;P)< ". By Theorem 1.1 again we conclude thatf2is integrable on [a;b]. Note thatfg= [(f+g)2-(f-g)2]=4. By Theorem 2.1,f+gandf-gare integrable on [a;b]. By what has been proved, both (f+g)2and (f-g)2are integrable on [a;b]. Using Theorem 2.1 again, we conclude thatfgis integrable on [a;b]. functionfis integrable on[a;b], thenf|[c;d]is integrable on[c;d]. Proof.Suppose thatfis integrable on [a;b]. Let"be an arbitrary positive number. By Theorem 1.1, there exists a partitionPof [a;b] such thatU(f;P)-L(f;P)< ". It follows thatU(f;P∪ {c;d})-L(f;P∪ {c;d})< ". LetQ:= (P∪ {c;d})∩[c;d]. ThenQis a partition of [c;d]. We haveHencef|[c;d]is integrable on [c;d].
Theorem 2.4.Letfbe a bounded real-valued function on[a;b]. Ifa < c < b, and iffis integrable on[a;c]and[c;b], thenfis integrable on[a;b]and b a f(x)dx=∫ c a f(x)dx+∫ b c f(x)dx: Proof.Suppose thatfis integrable on [a;c] and [c;b]. We writeI1:=∫c af(x)dxand I2:=∫b
cf(x)dx. Let" >0. By Theorem 1.1, there exist a partitionP1={s0;s1;:::;sm} of [a;c] and a partitionP2={t0;t1;:::;tn}of [c;b] such thatU(f;P1)-L(f;P1)<"
2 andU(f;P2)-L(f;P2)<" 2 LetP:=P1∪P2={s0;:::;sm-1;t0;:::;tn}. ThenPis a partition of [a;b]. We have L(f)≥L(f;P) =L(f;P1) +L(f;P2)> U(f;P1) +U(f;P2)-"≥I1+I2-" andIt follows that
c a f(x)dx+∫ b c c a f(x)dx+∫ b c f(x)dx+": 8 Since the above inequalities are valid for all" >0, we conclude thatfis integrable on [a;b] and∫b af(x)dx=∫c af(x)dx+∫b cf(x)dx. Leta;b;cbe real numbers in any order, and letJbe a bounded closed interval con- taininga,b, andc. Iffis integrable onJ, then by Theorems 2.3 and 2.4 we have∫b a f(x)dx=∫ c a f(x)dx+∫ b c f(x)dx: then∫b ag(x)dx. Proof.By Theorem 2.1,h:=g-fis integrable on [a;b]. Sinceh(x)≥0 for allx∈[a;b], it is clear thatL(h;P)≥0 for any partitionPof [a;b]. Hence,∫b ah(x)dx=L(h)≥0.Applying Theorem 2.1 again, we see that∫b
a g(x)dx-∫ b a f(x)dx=∫ b a h(x)dx≥0: Theorem 2.6.Iffis an integrable function on[a;b], then|f|is integrable on[a;b]and∫ b a b a |f(x)|dx: Proof.LetP={t0;t1;:::;tn}be a partition of [a;b]. For eachi∈ {1;:::;n}, letMiand m idenote the supremum and infimun respectively offon [ti-1;ti], and letM∗iandm∗i denote the supremum and infimun respectively of|f|on [ti-1;ti]. Then M i-mi= sup{f(x)-f(y) :x;y∈[ti-1;ti]} and M ∗i-m∗i= sup{|f(x)| - |f(y)|:x;y∈[ti-1;ti]}:By the triangle inequality,
and n∑ i=1(Mi-mi)(ti-ti-1): number. By our assumption,fis integrable on [a;b]. By Theorem 1.1, there exists a partitionPsuch thatU(f;P)-L(f;P)< ". HenceU(|f|;P)-L(|f|;P)< ". By using Theorem 1.1 again we conclude that|f|is integrable on [a;b]. Furthermore, since a b a |f(x)|dxand-∫ b a b a |f(x)|dx:Therefore
∫b a|f(x)|dx. 9§3. Fundamental Theorem of Calculus
In this section we give two versions of the Fundamental Theorem of Calculus and their applications. Letfbe a real-valued function on an intervalI. A functionFonIis called an antiderivativeoffonIifF′(x) =f(x) for allx∈I. IfFis an antiderivative off, then so isF+Cfor any constantC. Conversely, ifFandGare antiderivatives offonI, then G ′(x)-F′(x) = 0 for allx∈I. Thus, there exists a constantCsuch thatG(x)-F(x) =C for allx∈I. Consequently,G=F+C. The following is the first version of the Fundamental Theorem of Calculus. Theorem 3.1.Letfbe an integrable function on[a;b]. IfFis a continuous function on [a;b]and ifFis an antiderivative offon(a;b), then b a f(x)dx=F(x)b a :=F(b)-F(a): Proof.Let" >0. By Theorem 1.1, there exists a partitionP={t0;t1;:::;tn}of [a;b] such thatU(f;P)-L(f;P)< ". Sincet0=aandtn=bwe haveF(b)-F(a) =n∑
i=1[F(ti)-F(ti-1)]: By the Mean Value Theorem, for eachi∈ {1;:::;n}there existsxi∈(ti-1;ti) such that F(ti)-F(ti-1) =F′(xi)(ti-ti-1) =f(xi)(ti-ti-1):Consequently,
On the other hand,
b aThus bothF(b)-F(a) and∫b
af(x)dxlie in [L(f;P);U(f;P)] withU(f;P)-L(f;P)< ". Hence [F(b)-F(a)]-∫ b a f(x)dx< ": Since the above inequality is valid for all" >0, we obtain∫b af(x)dx=F(b)-F(a). 10Example 1.Letkbe a positive integer. Find∫b
axkdx. Solution. We know that the functiongk:x7→xk+1=(k+ 1) is an antiderivative of the functionfk:x7→xk. By the Fundamental Theorem of Calculus we obtain b a xkdx=xk+1 k+ 1 b a =bk+1-ak+1 k+ 1:Example 2.Find the integral∫2
11=xdx.
Solution. On the interval (0;∞), the functionx7→lnxis an antiderivative the function x7→1=x. By the Fundamental Theorem of Calculus we obtain 2 11 x dx= lnx21= ln2-ln1 = ln2:
This integral can be used to find the limit
lim n→∞( 1 n+ 1+1 n+ 2+···+1 2n) Indeed, letf(x) := 1=xforx= [1;2], and letti= 1 +i=nfori= 0;1;:::;n. ThenP:={t0;t1;:::;tn}is a partition of [1;2] and
1 n+ 1+1 n+ 2+···+12n=n∑
i=1f(ti)(ti-ti-1) is a Riemann sum offwith respect toPand points{t1;:::;tn}. By Theorem 1.2 we get lim n→∞( 1 n+ 1+1 n+ 2+···+1 2n) = lim n→∞n i=1f(ti)(ti-ti-1) =∫ 2 11 x dx= ln2: Example 3.A curve in plane is represented by a continuous mappingu= (u1;u2) from [a;b] to IR2. We useL(u) to denote the length of the curve. Suppose thatu′1andu′2are continuous on [a;b]. Thenuis rectifiable. Fort∈[a;b], lets(t) denote the length of the curveu|[a;t]. It was proved in Theorem 7.1 of Chapter 4 that s [u′1(t)]2+ [u′2(t)]2; t∈[a;b]: By Theorem 3.1 (the Fundamental Theorem of Calculus), we obtainL(u) =s(b)-s(a) =∫
b a s′(t)dt=∫ b [u′1(t)]2+ [u′2(t)]2dt: The following is the second version of the Fundamental Theorem of Calculus. 11 Theorem 3.2.Letfbe an integrable function on[a;b]. DefineF(x) :=∫
x a f(t)dt; x∈[a;b]: ThenFis a continuous function on[a;b]. Furthermore, iffis continuous at a point c∈[a;b], thenFis differentiable atcand F ′(c) =f(c): for allx∈[a;b]. Ifx;y∈[a;b] andx < y, thenF(y)-F(x) =∫
y a f(t)dt-∫ x a f(t)dt=∫quotesdbs_dbs21.pdfusesText_27[PDF] 7.62x39 headstamp codes
[PDF] 7/8 clarke street kennington
[PDF] 70 alcohol antibacterial hand sanitizer gel
[PDF] 70 alcohol hand sanitiser 5l litres sanitizer gel
[PDF] 70 cours de québec 33300 bordeaux
[PDF] 70 cours du quebec
[PDF] 70 denatured ethanol
[PDF] 70 ethanol alcohol
[PDF] 70 ethanol disinfectant
[PDF] 70 ethanol hand sanitizer
[PDF] 70 ethanol hazmat
[PDF] 70 ethanol merck
[PDF] 70 ethanol msds
[PDF] 70 ethanol sds