Calculus Review
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Review of CalculusMATH 375 Numerical Analysis
J. Robert Buchanan
Department of Mathematics
Fall 2010
J. Robert Buchanan Review of Calculus
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Motivation
The subject of this course is numerical analysis.
Numerical analysis includes, but is not limited to, a study ofnumerical methods.In addition to numerical methods we will use the tools ofcalculus (real analysis) to understand the limitations anderrors present in given numerical methods.We begin with a review of single variable calculus,particularly Taylor’s Theorem.
J. Robert Buchanan Review of Calculus
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Motivation
The subject of this course is numerical analysis.Numerical analysis includes, but is not limited to, a study ofnumerical methods.
In addition to numerical methods we will use the tools ofcalculus (real analysis) to understand the limitations anderrors present in given numerical methods.We begin with a review of single variable calculus,particularly Taylor’s Theorem.
J. Robert Buchanan Review of Calculus
![Page 4: Calculus Review](https://reader034.fdocuments.us/reader034/viewer/2022042822/5695cf821a28ab9b028e6533/html5/thumbnails/4.jpg)
Motivation
The subject of this course is numerical analysis.Numerical analysis includes, but is not limited to, a study ofnumerical methods.In addition to numerical methods we will use the tools ofcalculus (real analysis) to understand the limitations anderrors present in given numerical methods.
We begin with a review of single variable calculus,particularly Taylor’s Theorem.
J. Robert Buchanan Review of Calculus
![Page 5: Calculus Review](https://reader034.fdocuments.us/reader034/viewer/2022042822/5695cf821a28ab9b028e6533/html5/thumbnails/5.jpg)
Motivation
The subject of this course is numerical analysis.Numerical analysis includes, but is not limited to, a study ofnumerical methods.In addition to numerical methods we will use the tools ofcalculus (real analysis) to understand the limitations anderrors present in given numerical methods.We begin with a review of single variable calculus,particularly Taylor’s Theorem.
J. Robert Buchanan Review of Calculus
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Limits
DefinitionA function f defined on a set S of real numbers has a limit L atx0, denoted
limx→x0
f (x) = L,
if for every ε > 0, there exists a real number δ > 0 such that
|f (x)− L| < ε, whenever x ∈ S and 0 < |x − x0| < δ.
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Graphical Interpretation
x0-∆ x0 x0+∆x
L-Ε
LL+Ε
fHxL
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Limit of a Sequence
DefinitionLet {xn}∞n=1 be an infinite sequence of real numbers. Thesequence has the limit L (or is said to converge to L) if, forevery ε > 0 there exists a positive integer N(ε) such thatwhenever n > N(ε) then |xn − L| < ε.
The notations
limn→∞
xn = L, or
xn → L as n→∞
denote the convergence of {xn}∞n=1 to L.
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Limit of a Sequence
DefinitionLet {xn}∞n=1 be an infinite sequence of real numbers. Thesequence has the limit L (or is said to converge to L) if, forevery ε > 0 there exists a positive integer N(ε) such thatwhenever n > N(ε) then |xn − L| < ε.
The notations
limn→∞
xn = L, or
xn → L as n→∞
denote the convergence of {xn}∞n=1 to L.
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Graphical Interpretation
NHΕL
LL-Ε
L+Ε
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Continuity
DefinitionLet function f be defined on a set S of real numbers andsuppose x0 ∈ S. We say that f is continuous at x0 if
limx→x0
f (x) = f (x0).
The function f is continuous on the set S if it is continuous ateach x ∈ S.
DefinitionThe set of functions continuous on set S will be denoted C(S).
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Continuity
DefinitionLet function f be defined on a set S of real numbers andsuppose x0 ∈ S. We say that f is continuous at x0 if
limx→x0
f (x) = f (x0).
The function f is continuous on the set S if it is continuous ateach x ∈ S.
DefinitionThe set of functions continuous on set S will be denoted C(S).
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Continuity and Sequences
TheoremLet f be a function defined on a set S and let x0 ∈ S, then thefollowing statements are equivalent.
1 Function f is continuous at x0.2 If {xn}∞n=1 is any sequence in S converging to x0, then
limn→∞
f (xn) = f (x0).
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Differentiability
DefinitionLet f be a function defined in an open interval containing x0.Then f is differentiable at x0 if
limx→x0
f (x)− f (x0)
x − x0
exists. Provided the limit exists we denote it f ′(x0) and call it thederivative of f at x0. If function f has a derivative at each x in aset S, we say f is differentiable on S.
TheoremIf function f is differentiable at x0, then f is continuous at x0.
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Differentiability
DefinitionLet f be a function defined in an open interval containing x0.Then f is differentiable at x0 if
limx→x0
f (x)− f (x0)
x − x0
exists. Provided the limit exists we denote it f ′(x0) and call it thederivative of f at x0. If function f has a derivative at each x in aset S, we say f is differentiable on S.
TheoremIf function f is differentiable at x0, then f is continuous at x0.
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Graphical Interpretation
Hx0,fHx0LLslope: f'Hx0L
x0
x
fHx0L
fHxL
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Sets of Differentiable Functions
We will frequently use the following notation.Cn(S): the set of all functions that have n continuous
derivatives on set S.C∞(S): the set of all functions that have derivatives of all
orders on set S.
Example1 If p(x) is a polynomial, then p ∈ C∞(R).2 If f (x) = ln x , then f ∈ C∞(0,∞).
3 If f (x) =
∫ x
0|t |dt , then f ∈ C1(R).
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Sets of Differentiable Functions
We will frequently use the following notation.Cn(S): the set of all functions that have n continuous
derivatives on set S.C∞(S): the set of all functions that have derivatives of all
orders on set S.
Example1 If p(x) is a polynomial, then p ∈ C∞(R).2 If f (x) = ln x , then f ∈ C∞(0,∞).
3 If f (x) =
∫ x
0|t |dt , then f ∈ C1(R).
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Rolle’s Theorem
TheoremSuppose f ∈ C[a,b] andf is differentiable on(a,b). If f (a) = f (b), thenthere exists c ∈ (a,b)such that f ′(c) = 0.
f'HcL=0
a bcx
fHaL=fHbL
y
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Mean Value Theorem
TheoremSuppose f ∈ C[a,b] andf is differentiable on(a,b). There existsc ∈ (a,b) such that
f ′(c) =f (b)− f (a)
b − a.
slope: f'HcL
slope:f HbL - f HaL
b - a
a bcx
fHaL
fHbL
y
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Extreme Value Theorem
TheoremIf f ∈ C[a,b] then there exist c1 ∈ [a,b] and c2 ∈ [a,b] such thatf (c1) ≤ f (x) ≤ f (c2) for all x ∈ [a,b]. If additionally f isdifferentiable on (a,b), then the numbers c1 and c2 occur eitherat the endpoints of [a,b] or where f ′(c) = 0.
In the definition above f (c1) and f (c2) are referred to as theabsolute minimum and absolute maximum of f on [a,b]respectively.
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Extreme Value Theorem
TheoremIf f ∈ C[a,b] then there exist c1 ∈ [a,b] and c2 ∈ [a,b] such thatf (c1) ≤ f (x) ≤ f (c2) for all x ∈ [a,b]. If additionally f isdifferentiable on (a,b), then the numbers c1 and c2 occur eitherat the endpoints of [a,b] or where f ′(c) = 0.
In the definition above f (c1) and f (c2) are referred to as theabsolute minimum and absolute maximum of f on [a,b]respectively.
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Intermediate Value Theorem
TheoremIf f ∈ C[a,b] and K isany number betweenf (a) and f (b), then thereexists a numberc ∈ (a,b) for whichf (c) = K .
Ha,fHaLL
Hb,fHbLL
a bcx
fHaL
fHbL
K
y
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Example
Example
Show that p(x) = x5 − 3x3 + 2x − 1 has a root in the interval[0,2].
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Generalized Rolle’s Theorem
On occasion we will need a stronger version of Rolle’sTheorem.
TheoremSuppose f ∈ C[a,b] is n times differentiable on (a,b). If f hasn + 1 distinct roots
a ≤ x0 < x1 < · · · < xn ≤ b,
then there exists c ∈ (x0, xn) ⊂ (a,b) such that f (n)(c) = 0.
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Riemann Integrals (1 of 2)
DefinitionLet P = {x0, x1, . . . , xn} be a set of numbers in [a,b] such that
a = x0 ≤ x1 ≤ · · · ≤ xn = b.
Set P is called a partition of [a,b]. If ∆xi = xi − xi−1 then‖P‖ = max
i=1,...,n{∆xi} is called the norm of P.
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Riemann Integrals (2 of 2)
DefinitionLet function f be defined on [a,b] and let P be any partition of[a,b]. If zi ∈ [xi−1, xi ] for i = 1, . . . ,n then the Riemannintegral of f on [a,b] is
lim‖P‖→0
n∑i=1
f (zi)∆xi ,
provided the limit exists and is the same for every partition andchoice of zi .
If the Riemann integral exists, it will be denoted∫ b
af (x) dx .
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Riemann Integrals (2 of 2)
DefinitionLet function f be defined on [a,b] and let P be any partition of[a,b]. If zi ∈ [xi−1, xi ] for i = 1, . . . ,n then the Riemannintegral of f on [a,b] is
lim‖P‖→0
n∑i=1
f (zi)∆xi ,
provided the limit exists and is the same for every partition andchoice of zi .
If the Riemann integral exists, it will be denoted∫ b
af (x) dx .
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Riemann Integral and Continuity
TheoremIf f ∈ C[a,b] then the Riemann integral of f on [a,b] exists andcan be evaluated as∫ b
af (x) dx = lim
n→∞
b − an
n∑i=1
f(
a +b − a
ni)
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Graphical Interpretation
a=x0 xi b=xn
x
y
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Weighted Mean Value Theorem for Integrals
TheoremSuppose function f ∈ C[a,b] and suppose function g isRiemann integrable on [a,b] and that g(x) does not changesign on [a,b]. There exists c ∈ (a,b) such that∫ b
af (x) g(x) dx = f (c)
∫ b
ag(x) dx .
If g(x) = 1 on [a,b] then f (c) can be called the average valueof f on [a,b].
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Weighted Mean Value Theorem for Integrals
TheoremSuppose function f ∈ C[a,b] and suppose function g isRiemann integrable on [a,b] and that g(x) does not changesign on [a,b]. There exists c ∈ (a,b) such that∫ b
af (x) g(x) dx = f (c)
∫ b
ag(x) dx .
If g(x) = 1 on [a,b] then f (c) can be called the average valueof f on [a,b].
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Average Value: Graphical Interpretation
a c bx
fHcL
y
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Taylor’s Theorem
Theorem
Suppose function f ∈ Cn[a,b] and that f (n+1) exists on [a,b]and let x0 ∈ [a,b]. For every x ∈ [a,b], there exists a numberz(x) between x0 and x such that f (x) = Pn(x) + Rn(x), where
Pn(x) =n∑
k=0
f (k)(x0)
k !(x − x0)k
is called the nth Taylor polynomial for f about x0, and
Rn(x) =f (n+1)(z(x))
(n + 1)!(x − x0)n+1
is called the nth Taylor remainder.
In the case where x0 = 0 these are sometimes calledMaclaurin polynomials.
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Taylor’s Theorem
Theorem
Suppose function f ∈ Cn[a,b] and that f (n+1) exists on [a,b]and let x0 ∈ [a,b]. For every x ∈ [a,b], there exists a numberz(x) between x0 and x such that f (x) = Pn(x) + Rn(x), where
Pn(x) =n∑
k=0
f (k)(x0)
k !(x − x0)k
is called the nth Taylor polynomial for f about x0, and
Rn(x) =f (n+1)(z(x))
(n + 1)!(x − x0)n+1
is called the nth Taylor remainder.
In the case where x0 = 0 these are sometimes calledMaclaurin polynomials.
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Example
Example
Let f (x) = ln x and x0 = 1.1 Find the second Taylor polynomial and remainder for f
about x0. Use this polynomial to approximate ln 1.1 andestimate the error in the approximation from the remainder.
2 Find the third Taylor polynomial and remainder for f aboutx0. Use this polynomial to approximate ln 1.1 and estimatethe error in the approximation from the remainder.
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Solution: n = 2
ln x = P2(x) + R2(x)
= 0 +1/11!
(x − 1)− 1/12
2!(x − 1)2 +
2/(z(x))3
3!(x − 1)3
= (x − 1)− 12
(x − 1)2 +1
3(z(x))3 (x − 1)3
If x = 1.1 then P2(1.1) = 0.095 and
| ln 1.1− P2(1.1)| = |R2(1.1)| = 0.00031
(z(1.1))3 ≤ 0.0003
since 1 ≤ z(1.1) ≤ 1.1.
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Solution: n = 2
ln x = P2(x) + R2(x)
= 0 +1/11!
(x − 1)− 1/12
2!(x − 1)2 +
2/(z(x))3
3!(x − 1)3
= (x − 1)− 12
(x − 1)2 +1
3(z(x))3 (x − 1)3
If x = 1.1 then P2(1.1) = 0.095 and
| ln 1.1− P2(1.1)| = |R2(1.1)| = 0.00031
(z(1.1))3 ≤ 0.0003
since 1 ≤ z(1.1) ≤ 1.1.
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Solution: n = 3
ln x = P3(x) + R3(x)
= 0 +1/11!
(x − 1)− 1/12
2!(x − 1)2 +
2/13
3!(x − 1)3
− 6/(z(x))4
4!(x − 1)4
= (x − 1)− 12
(x − 1)2 +13
(x − 1)3 − 14(z(x))4 (x − 1)4
If x = 1.1 then P3(1.1) = 0.0953 and
| ln 1.1−P3(1.1)| = |R3(1.1)| = 0.0000251
(z(1.1))4 ≤ 0.000025
since 1 ≤ z(1.1) ≤ 1.1.
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Solution: n = 3
ln x = P3(x) + R3(x)
= 0 +1/11!
(x − 1)− 1/12
2!(x − 1)2 +
2/13
3!(x − 1)3
− 6/(z(x))4
4!(x − 1)4
= (x − 1)− 12
(x − 1)2 +13
(x − 1)3 − 14(z(x))4 (x − 1)4
If x = 1.1 then P3(1.1) = 0.0953 and
| ln 1.1−P3(1.1)| = |R3(1.1)| = 0.0000251
(z(1.1))4 ≤ 0.000025
since 1 ≤ z(1.1) ≤ 1.1.
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Return to Course Objectives
Our use of Taylor’s Theorem illustrates two of the objectives ofNumerical Analysis.
1 Given a problem, find an approximation to the solution tothe problem.
2 Determine a bound for the accuracy of the approximatedsolution.
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Example
Example
Use the third Taylor polynomial and remainder for f (x) = ln xabout x0 = 1 to approximate∫ 1.1
1ln x dx
and determine a bound for the error in the approximation.
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Solution (1 of 2)
∫ 1.1
1ln x dx =
∫ 1.1
1P3(x) dx +
∫ 1.1
1R3(x) dx
=
∫ 1.1
1
[(x − 1)− 1
2(x − 1)2 +
13
(x − 1)3]
dx
−∫ 1.1
1
14(z(x))4 (x − 1)4 dx
=
[(x − 1)2
2− (x − 1)3
6+
(x − 1)4
12
]∣∣∣∣1.1
1
− 14
∫ 1.1
1
(x − 1)4
(z(x))4 dx
=(0.1)2
2− (0.1)3
6+
(0.1)4
12− 1
4
∫ 1.1
1
(x − 1)4
(z(x))4 dx
= 0.0048416− 14
∫ 1.1
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Solution (2 of 2)
∫ 1.1
1ln x dx ≈ 0.0048416
and the error is∣∣∣∣∣∫ 1.1
1(ln x − P3(x)) dx
∣∣∣∣∣ =
∣∣∣∣∣∫ 1.1
1R3(x) dx
∣∣∣∣∣=
14
∣∣∣∣∣∫ 1.1
1
(x − 1)4
(z(x))4 dx
∣∣∣∣∣≤ 1
4
∫ 1.1
1
∣∣∣∣(x − 1)4
(z(x))4
∣∣∣∣ dx
≤ 14
∫ 1.1
1(x − 1)4 dx
= 5× 10−7.
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Taylor Series
If function f ∈ C∞[a,b] and limn→∞
Rn(x) = 0 then we may call
limn→∞
Pn(x) =∞∑
k=0
f (k)(x0)
k !(x − x0)k
the Taylor series for f about x0 provided the infinite seriesconverges.
J. Robert Buchanan Review of Calculus
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Homework
Read Section 1.1.Exercises: 1, 2, 5, 9, 14, 15, 19, 27
J. Robert Buchanan Review of Calculus