lesson 26: the fundamental theorem of calculus (handout)
DESCRIPTION
The fundamental theorem shows that differentiation and integration are inverse processes.TRANSCRIPT
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Sec on 5.4The Fundamental Theorem of
Calculus
V63.0121.001: Calculus IProfessor Ma hew Leingang
New York University
May 2, 2011
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Announcements
I Today: 5.4I Wednesday 5/4: 5.5I Monday 5/9: Review andMovie Day!
I Thursday 5/12: FinalExam, 2:00–3:50pm
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ObjectivesI State and explain theFundemental Theoremsof Calculus
I Use the first fundamentaltheorem of calculus tofind deriva ves offunc ons defined asintegrals.
I Compute the averagevalue of an integrablefunc on over a closedinterval.
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Notes
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Notes
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Notes
. 1.
. Sec on 5.4: The Fundamental Theorem. V63.0121.001: Calculus I . May 2, 2011
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OutlineRecall: The Evalua on Theorem a/k/a 2nd FTC
The First Fundamental Theorem of CalculusArea as a Func onStatement and proof of 1FTCBiographies
Differen a on of func ons defined by integrals“Contrived” examplesErfOther applica ons
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The definite integral as a limitDefini onIf f is a func on defined on [a, b], the definite integral of f from a tob is the number ∫ b
af(x) dx = lim
n→∞
n∑i=1
f(ci)∆x
where∆x =b− an
, and for each i, xi = a+ i∆x, and ci is a point in[xi−1, xi].
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The definite integral as a limit
TheoremIf f is con nuous on [a, b] or if f has only finitely many jumpdiscon nui es, then f is integrable on [a, b]; that is, the definite
integral∫ b
af(x) dx exists and is the same for any choice of ci.
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Notes
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Notes
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Notes
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. Sec on 5.4: The Fundamental Theorem. V63.0121.001: Calculus I . May 2, 2011
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Big time Theorem
Theorem (The Second Fundamental Theorem of Calculus)
Suppose f is integrable on [a, b] and f = F′ for another func on F,then ∫ b
af(x) dx = F(b)− F(a).
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The Integral as Net Change
Corollary
If v(t) represents the velocity of a par cle moving rec linearly, then∫ t1
t0v(t) dt = s(t1)− s(t0).
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The Integral as Net Change
Corollary
If MC(x) represents the marginal cost of making x units of a product,then
C(x) = C(0) +∫ x
0MC(q) dq.
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Notes
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Notes
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. Sec on 5.4: The Fundamental Theorem. V63.0121.001: Calculus I . May 2, 2011
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The Integral as Net Change
Corollary
If ρ(x) represents the density of a thin rod at a distance of x from itsend, then the mass of the rod up to x is
m(x) =∫ x
0ρ(s) ds.
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My first table of integrals..∫
[f(x) + g(x)] dx =∫
f(x) dx+∫
g(x) dx∫xn dx =
xn+1
n+ 1+ C (n ̸= −1)∫
ex dx = ex + C∫sin x dx = − cos x+ C∫cos x dx = sin x+ C∫sec2 x dx = tan x+ C∫
sec x tan x dx = sec x+ C∫1
1+ x2dx = arctan x+ C
∫cf(x) dx = c
∫f(x) dx∫
1xdx = ln |x|+ C∫
ax dx =ax
ln a+ C∫
csc2 x dx = − cot x+ C∫csc x cot x dx = − csc x+ C∫
1√1− x2
dx = arcsin x+ C
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OutlineRecall: The Evalua on Theorem a/k/a 2nd FTC
The First Fundamental Theorem of CalculusArea as a Func onStatement and proof of 1FTCBiographies
Differen a on of func ons defined by integrals“Contrived” examplesErfOther applica ons
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Notes
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Notes
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Notes
. 4.
. Sec on 5.4: The Fundamental Theorem. V63.0121.001: Calculus I . May 2, 2011
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Area as a FunctionExample
Let f(t) = t3 and define g(x) =∫ x
0f(t) dt. Find g(x) and g′(x).
Solu on
..0.
x
Dividing the interval [0, x] into n pieces gives∆t =xnand
ti = 0+ i∆t =ixn.
Rn =xn· x
3
n3+
xn· (2x)
3
n3+ · · ·+ x
n· (nx)
3
n3
=x4
n4(13 + 23 + 33 + · · ·+ n3
)=
x4
n4[12n(n+ 1)
]2
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Area as a FunctionExample
Let f(t) = t3 and define g(x) =∫ x
0f(t) dt. Find g(x) and g′(x).
Solu on
..0.
x
Dividing the interval [0, x] into n pieces gives∆t =xnand
ti = 0+ i∆t =ixn.
Rn =x4n2(n+ 1)2
4n4
So g(x) = limx→∞
Rn =x4
4and g′(x) = x3.
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The area function in generalLet f be a func on which is integrable (i.e., con nuous or withfinitely many jump discon nui es) on [a, b]. Define
g(x) =∫ x
af(t) dt.
I The variable is x; t is a “dummy” variable that’s integrated over.I Picture changing x and taking more of less of the region underthe curve.
I Ques on: What does f tell you about g?
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Notes
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Notes
. 5.
. Sec on 5.4: The Fundamental Theorem. V63.0121.001: Calculus I . May 2, 2011
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Envisioning the area function
Example
Suppose f(t) is the func ongraphed to the right. Let
g(x) =∫ x
0f(t) dt. What can
you say about g? ...x
..
y
.......f
..2
..4
..6
..8
..10
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Features of g from f
...x
..
y
.... g....f
..2
..4
..6
..8
..10
Interval sign monotonicity monotonicity concavityof f of g of f of g
[0, 2] + ↗ ↗ ⌣
[2, 4.5] + ↗ ↘ ⌢
[4.5, 6] − ↘ ↘ ⌢
[6, 8] − ↘ ↗ ⌣
[8, 10] − ↘ → none
We see that g is behaving a lot like an an deriva ve of f.
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Another Big Time TheoremTheorem (The First Fundamental Theorem of Calculus)
Let f be an integrable func on on [a, b] and define
g(x) =∫ x
af(t) dt.
If f is con nuous at x in (a, b), then g is differen able at x and
g′(x) = f(x).
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Notes
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Notes
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. Sec on 5.4: The Fundamental Theorem. V63.0121.001: Calculus I . May 2, 2011
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Proving the Fundamental TheoremProof.
Let h > 0 be given so that x+ h < b. We have
g(x+ h)− g(x)h
=1h
∫ x+h
xf(t) dt.
LetMh be the maximum value of f on [x, x+ h], and letmh theminimum value of f on [x, x+ h]. From §5.2 we have
mh · h ≤∫ x+h
xf(t) dt ≤ Mh · h
=⇒ mh ≤g(x+ h)− g(x)
h≤ Mh.
As h → 0, bothmh andMh tend to f(x).
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Proving the Fundamental TheoremProof.
From §5.2 we have
mh · h ≤∫ x+h
xf(t) dt ≤ Mh · h
=⇒ mh ≤g(x+ h)− g(x)
h≤ Mh.
As h → 0, bothmh andMh tend to f(x).
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Meet the Mathematician: JamesGregory
I Sco sh, 1638-1675I Astronomer and GeometerI Conceived transcendentalnumbers and found evidencethat π was transcendental
I Proved a geometric version of1FTC as a lemma but didn’t takeit further
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. Sec on 5.4: The Fundamental Theorem. V63.0121.001: Calculus I . May 2, 2011
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Meet the Mathematician: IsaacBarrow
I English, 1630-1677I Professor of Greek, theology,and mathema cs at Cambridge
I Had a famous student
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Meet the Mathematician: IsaacNewton
I English, 1643–1727I Professor at Cambridge(England)
I Philosophiae Naturalis PrincipiaMathema ca published 1687
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Meet the Mathematician: GottfriedLeibniz
I German, 1646–1716I Eminent philosopher as well asmathema cian
I Contemporarily disgraced by thecalculus priority dispute
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Notes
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Notes
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. Sec on 5.4: The Fundamental Theorem. V63.0121.001: Calculus I . May 2, 2011
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Differentiation and Integration asreverse processes
Pu ng together 1FTC and 2FTC, we get a beau ful rela onshipbetween the two fundamental concepts in calculus.Theorem (The Fundamental Theorem(s) of Calculus)
I. If f is a con nuous func on, then
ddx
∫ x
af(t) dt = f(x)
So the deriva ve of the integral is the original func on.
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Differentiation and Integration asreverse processes
Pu ng together 1FTC and 2FTC, we get a beau ful rela onshipbetween the two fundamental concepts in calculus.Theorem (The Fundamental Theorem(s) of Calculus)
II. If f is a differen able func on, then∫ b
af′(x) dx = f(b)− f(a).
So the integral of the deriva ve of is (an evalua on of) theoriginal func on.
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OutlineRecall: The Evalua on Theorem a/k/a 2nd FTC
The First Fundamental Theorem of CalculusArea as a Func onStatement and proof of 1FTCBiographies
Differen a on of func ons defined by integrals“Contrived” examplesErfOther applica ons
.
Notes
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Notes
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Notes
. 9.
. Sec on 5.4: The Fundamental Theorem. V63.0121.001: Calculus I . May 2, 2011
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Differentiation of area functionsExample
Let h(x) =∫ 3x
0t3 dt. What is h′(x)?
Solu on (Using 2FTC)
h(x) =t4
4
∣∣∣∣3x0=
14(3x)4 = 1
4 · 81x4, so h′(x) = 81x3.
Solu on (Using 1FTC)
We can think of h as the composi on g ◦ k, where g(u) =∫ u
0t3 dt
and k(x) = 3x.
Then h′(x) = g′(u) · k′(x), or
h′(x) = g′(k(x)) · k′(x) = (k(x))3 · 3 = (3x)3 · 3 = 81x3.
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Differentiation of area functionsExample
Let h(x) =∫ 3x
0t3 dt. What is h′(x)?
Solu on (Using 1FTC)
We can think of h as the composi on g ◦ k, where g(u) =∫ u
0t3 dt
and k(x) = 3x. Then h′(x) = g′(u) · k′(x), or
h′(x) = g′(k(x)) · k′(x) = (k(x))3 · 3 = (3x)3 · 3 = 81x3.
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Differentiation of area functions, in generalI by 1FTC
ddx
∫ k(x)
af(t) dt = f(k(x))k′(x)
I by reversing the order of integra on:
ddx
∫ b
h(x)f(t) dt = − d
dx
∫ h(x)
bf(t) dt = −f(h(x))h′(x)
I by combining the two above:
ddx
∫ k(x)
h(x)f(t) dt =
ddx
(∫ k(x)
0f(t) dt+
∫ 0
h(x)f(t) dt
)= f(k(x))k′(x)− f(h(x))h′(x)
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Notes
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Notes
. 10.
. Sec on 5.4: The Fundamental Theorem. V63.0121.001: Calculus I . May 2, 2011
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Another ExampleExample
Let h(x) =∫ sin2 x
0(17t2 + 4t− 4) dt. What is h′(x)?
Solu on
We have
ddx
∫ sin2 x
0(17t2 + 4t− 4) dt =
(17(sin2 x)2 + 4(sin2 x)− 4
)· ddx
sin2 x
=(17 sin4 x+ 4 sin2 x− 4
)· 2 sin x cos x
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A Similar ExampleExample
Let h(x) =∫ sin2 x
3(17t2 + 4t− 4) dt. What is h′(x)?
Solu on
We have
ddx
∫ sin2 x
0(17t2 + 4t− 4) dt =
(17(sin2 x)2 + 4(sin2 x)− 4
)· ddx
sin2 x
=(17 sin4 x+ 4 sin2 x− 4
)· 2 sin x cos x
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CompareQues on
Why isddx
∫ sin2 x
0(17t2 + 4t− 4) dt =
ddx
∫ sin2 x
3(17t2 + 4t− 4) dt?
Or, why doesn’t the lower limit appear in the deriva ve?
Answer
∫ sin2 x
0(17t2+4t−4) dt =
∫ 3
0(17t2+4t−4) dt+
∫ sin2 x
3(17t2+4t−4) dt
So the two func ons differ by a constant.
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Notes
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. 11.
. Sec on 5.4: The Fundamental Theorem. V63.0121.001: Calculus I . May 2, 2011
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The Full NastyExample
Find the deriva ve of F(x) =∫ ex
x3sin4 t dt.
Solu on
ddx
∫ ex
x3sin4 t dt = sin4(ex) · ex − sin4(x3) · 3x2
No ce here it’s much easier than finding an an deriva ve for sin4.
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Why use 1FTC?Ques on
Why would we use 1FTC to find the deriva ve of an integral? Itseems like confusion for its own sake.
Answer
I Some func ons are difficult or impossible to integrate inelementary terms.
I Some func ons are naturally defined in terms of otherintegrals.
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Erferf(x) =
2√π
∫ x
0e−t2 dt
I erf measures area the bell curve.I We can’t find erf(x), explicitly,but we do know its deriva ve:
erf′(x) =2√πe−x2.
...x
.
erf(x)
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. 12.
. Sec on 5.4: The Fundamental Theorem. V63.0121.001: Calculus I . May 2, 2011
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Example of erfExample
Findddx
erf(x2).
Solu on
By the chain rule we have
ddx
erf(x2) = erf′(x2)ddx
x2 =2√πe−(x2)22x =
4√πxe−x4.
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Other functions defined by integralsI The future value of an asset:
FV(t) =∫ ∞
tπ(s)e−rs ds
where π(s) is the profitability at me s and r is the discountrate.
I The consumer surplus of a good:
CS(q∗) =∫ q∗
0(f(q)− p∗) dq
where f(q) is the demand func on and p∗ and q∗ theequilibrium price and quan ty.
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Surplus by picture
..quan ty (q)
.
price (p)
..
demand f(q).
market revenue
.
supply
.
equilibrium
..q∗
..
p∗
.
consumer surplus
.
producer surplus
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Notes
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. 13.
. Sec on 5.4: The Fundamental Theorem. V63.0121.001: Calculus I . May 2, 2011
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SummaryI Func ons defined as integrals can be differen ated using thefirst FTC:
ddx
∫ x
af(t) dt = f(x)
I The two FTCs link the two major processes in calculus:differen a on and integra on∫
F′(x) dx = F(x) + C
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Notes
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. 14.
. Sec on 5.4: The Fundamental Theorem. V63.0121.001: Calculus I . May 2, 2011