Calculus for Biologists The Fundamental Theorem of Calculus James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University August 29, 2013
Calculus for Biologists
The Fundamental Theorem of Calculus
James K. Peterson
Department of Biological Sciences and Department of Mathematical SciencesClemson University
August 29, 2013
Calculus for Biologists
Outline
1 Properties of The Riemann Integral
2 Fundamental Theorem Of Calculus
3 The Cauchy Fundamental Theorem Of CalculusExamples
Calculus for Biologists
Abstract
This lecture explains the amazing connection between the idea ofan antiderivative or primitive of a function and the Riemannintegral of that same function on some interval.
Calculus for Biologists
Properties of The Riemann Integral
If you think about how the Riemann integral is defined in terms oflimits of Riemann sums, it is pretty easy to figure out some basicproperties.∫ a
a f (t)dt = 0 as the partitions are all one point and all the
changes in t are 0. Thus,∫ 11 t2dt = 0 as the interval is just
one point.∫ ba f (t)dt =
∫ ca f (t)dt +
∫ bc f (t)dt for any c between a and
b. For example:∫ 5
1t2dt =
∫ 3
1t2dt +
∫ 5
3t2dt.∫ 5
1t2dt =
∫ 2.2
1t2dt +
∫ 5
2.2t2dt
Having the order backwards just changes the sign of theRiemann integral value. So∫ 1
5t2dt = −
∫ 5
1t2dt.∫ −1
8t3dt = −
∫ 8
−1t3dt
Calculus for Biologists
Fundamental Theorem Of Calculus
There is a big connection between the idea of the antiderivativeof a function f and its Riemann integral.
For a positive function f on the finite interval [a, b], we canconstruct the area under the curve functionF (x) =
∫ xa f (t) dt.
Let’s look at the difference in these areas: we assume h ispositive.
F (x + h) − F (x) =
∫ x+h
af (t) dt −
∫ x
af (t) dt
=
∫ x
af (t) dt +
∫ x+h
xf (t) dt
−∫ x
af (t) dt
where we have used standard properties of the Riemannintegral to write the first integral as two pieces.
Calculus for Biologists
Fundamental Theorem Of Calculus
Now subtract to get
F (x + h) − F (x) =
∫ x+h
xf (t) dt
Now divide this difference by the change in x which is h. Wefind
F (x + h) − F (x)
h=
1
h
∫ x+h
xf (t) dt
We show F (x) and F (x + h) for a small positive h in thefigure which follows.
Calculus for Biologists
Fundamental Theorem Of Calculus
F (x) is the area under this curve from a to x .
(a, f (a))(b, f (b))
a bx x + h
F (x) F (x + h)
Figure: The Function F (x)
Calculus for Biologists
Fundamental Theorem Of Calculus
The difference in area,∫ x+hx f (t) dt, is the second shaded area in
the figure you just looked at. We see
If t is any number between x and x + h, the area of therectangle with base h and height f (t) is f (t) × h which isclosely related to the area difference.
Note the difference between this area and F (x + h) − F (x)is really small when h is small.
We know that f is bounded on [a, b] You can easily see that fhas a maximum value for the particular f we draw. Of course,this graph is not what all such bounded functions f look like,but you should be able to get the idea that there is a numberB so that 0 < f (t) ≤ B for all t in [a, b].
Thus, we see
F (x + h) − F (x) ≤∫ x+h
xB dt = B h (1)
Calculus for Biologists
Fundamental Theorem Of Calculus
From this, it follows that
We see our difference lives between 0 and B.
0 ≤ (F (x + h) − F (x)) ≤ B h
And so taking the limit as h gets small, we find
0 ≤ limh→ 0
(F (x + h) − F (x))
≤ limh→ 0
B h = 0.
We conclude that F is continuous at each x in [a, b] as
limh→ 0
(F (x + h) − F (x)) = 0.
It seems that the new function F we construct by integratingthe function f in this manner always builds a new functionthat is continuous.
Calculus for Biologists
Fundamental Theorem Of Calculus
Is F differentiable at x? Let’s do an estimate. We have a lowerand upper bound on the area of the middle slice in our figure.
minx ≤t ≤x+h
f (t) h ≤∫ x+h
xf (t)dt ≤ max
x ≤t ≤x+hf (t) h
Thus, we have the estimate
minx ≤t ≤x+h
f (t) ≤ F (x + h) − F (x)
h≤ max
x ≤t ≤x+hf (t)
Calculus for Biologists
Fundamental Theorem Of Calculus
If f was continuous at x , then we must have
limh→ 0
minx ≤t ≤x+h
f (t) = f (x)
and
limh→ 0
maxx ≤t ≤x+h
f (t) = f (x)
Note the f we draw in our figure is continuous all the time,but the argument we use here only needs continuity at thepoint x! At any rate, we can infer for positive h,
limh→ 0+
F (x + h) − F (x)
h= f (x)
Calculus for Biologists
Fundamental Theorem Of Calculus
You should be able to believe that a similar argument wouldwork for negative values of h: i.e.,
limh→ 0−
F (x + h) − F (x)
h= lim
h→ 0−f (t) = f (x)
This tells us that F ′(p) exists and equals f (x) as long as f iscontinuous at x as
F ′(x+) = limh→ 0+
F (x + h) − F (x)
h= f (x)
F ′(x−) = limh→ 0−
F (x + h) − F (x)
h= f (x)
Calculus for Biologists
Fundamental Theorem Of Calculus
This relationship is called the Fundamental Theorem ofCalculus.
Our argument works for x equals a or b but we only need tolook at the derivative from one side. So the discussion is a bitsimpler.
Our argument used a positive f but it works just fine if f haspositive and negative spots. Just divide f into it’s postive andnegative pieces and apply these ideas to each piece and thenglue the result together.
We can actually prove this using fairly relaxed assumptions onf for the interval [a, b]. In general, f need only be RiemannIntegrable on [a, b] which allows for jumps in the function.But those arguments are more advanced!
Calculus for Biologists
Fundamental Theorem Of Calculus
Theorem
The Fundemental Theorem of CalculusLet f be Riemann Integrable on [a, b]. Then the function Fdefined on [a, b] by F (x) =
∫ xa f (t) dt satisfies
1 F is continuous on all of [a, b]
2 F is differentiable at each point x in [a, b] where f iscontinuous and F ′(x) = f (x).
Calculus for Biologists
Fundamental Theorem Of Calculus
We can do more!!!
Using the same f as before, suppose G was defined on [a, b]as follows
G (x) =
∫ b
xf (t) dt.
then
F (x) + G (x) =
∫ x
af (t) dt +
∫ b
xf (t) dt
=
∫ b
af (t) dt.
Calculus for Biologists
Fundamental Theorem Of Calculus
So
G (x) =
∫ b
af (t) dt − F (x)
Since the Fundamental Theorem of Calculus tells us F isdifferentiable, we see G (x) must also be differentiable. Itfollows that since the derivative of a constant is 0, we have
G ′(x) = − F ′(x) = −f (x).
Calculus for Biologists
Fundamental Theorem Of Calculus
Let’s state this as a variant of the Fundamental Theorem ofCalculus, the Reversed Fundamental Theorem of Calculus soto speak.
Theorem
Reversed Fundamental Theorem of CalculusLet f be Riemann Integrable on [a, b]. Then the function G
defined on [a, b] by G (x) =∫ bx f (t) dt satisfies
1 G is continuous on all of [a, b]
2 G is differentiable at each point x in [a, b] where f iscontinuous and G ′(x) = −f (x).
Calculus for Biologists
The Cauchy Fundamental Theorem Of Calculus
We can use the Fundamental Theorem of Calculus to learn how toevaluate many Riemann integrals. This is how it works.
If f is continuous, the FToC tells us that F (x) =∫ xa f (t)dt
satisfies F ′(x) = f (x). So F is an antiderivative of f !!
If G is another antiderivative of f , then it also satisfiesG ′(x) = f (x).
We can use this to figure out a way to evaluate Riemannintegrals!
Calculus for Biologists
The Cauchy Fundamental Theorem Of Calculus
Here’s the argument (very cool one too I might add!)
For our f which is continuous on [a, b], let
F (x) =
∫ x
af (t) dt,
So F ′ = f and note F (a) = 0.
Let G be an antiderivative of the function f on [a, b]. Then,by definition, G ′(x) = f (x) and also G is continuous since itis differentiable.
Let H = F − G . Then
H ′(x) =
(F (x) − G (x)
)′= f (x) − f (x) = 0.
The only function whose derivative is 0 is a constant. So forsome constant C ,
H(x) = F (x) − G (x) = C
Calculus for Biologists
The Cauchy Fundamental Theorem Of Calculus
Almost done!
Thus H(a) = H(b) = C as H has the same value everywhere.
But H(a) = F (a) − G (a) and H(b) = F (b) − G (b).
These values are the same, so
F (a) − G (a) = F (b) − G (b).
Rearranging, we have
F (b) =
∫ b
af (t)dt = G (b) − G (a).
This result is huge! It says we can evaluate any Riemannintegral if we can guess an antiderivative.
Calculus for Biologists
The Cauchy Fundamental Theorem Of Calculus
Let’s formalize this as a theorem called the Cauchy FundamentalTheorem of Calculus. All we really need to prove this result isthat f is Riemann integrable on [a, b], which for us is usually trueas our functions f are continuous in general.
Theorem
Cauchy Fundamental Theorem of CalculusLet f be Riemann integral on [a, b] and let G be any
antiderivative of f . Then G (b) − G (a) =∫ ba f (t) dt.
We usually write this as∫ b
af (t)dt = G (t)
∣∣∣∣ba
Calculus for Biologists
The Cauchy Fundamental Theorem Of Calculus
Examples
In the problems that follow it doesn’t matter which antiderivativewe choose as our result above doesn’t care. So we just chooseC = 0 always.
∫ 3
1t3 dt =
t4
4
∣∣∣∣31
=34
4− 14
4=
80
4
∫ 4
−2t3 dt =
t4
4
∣∣∣∣4−2
=44
4− (−2)4
4
=256
4− 16
4=
240
4