Interpretation of the Fundamental Theorem
Well talk about a proof of the fundamental theorem later; for now lets get a
more intuitive interpretation of the theorem. Well use the example of time and
di
The First Fundamental Theorem of Calculus
Our rst example is the one we worked so hard on when we rst introduced
denite integrals:
3
Example: F (x) = x .
3
When we dierentiate F (x) we get f (x) = F (
The Fundamental Theorem and the Mean Value Theorem
Our goal is to use information about F to derive information about F . Our
rst example of this process will be to compare the rst fundamental theorem
Using The Second Fundamental Theorem of Calculus
This is the quiz question which everybody gets wrong until they practice it.
x
d
dt
Example: Evaluate
.
dx 1 t2
This question challenges your ability t
Properties of Integrals
The symbol
originated as a stylized letter S; in French, they call integrals
sums. We know from our discussion of Riemann sums that denite integrals are
just limits of sums. Be
The Fundamental Theorem of Calculus
The fundamental theorem of calculus is probably the most important thing in
this entire course. There will be two versions of it; when we need to abbreviate
well re
Approximations at 0 for ln(1 + x) and (1 + x)r
Next, we compute two linear approximations that are slightly more challenging.
f (x)
f (x)
f (0) f (0)
1
0
1
And here
Heres the table of values: ln(1 + x
Question: Can we use the original formula?
Earlier, we found that:
e3x
7
f (x) =
1 x.
2
1+x
Could we use a dierent method to get a linear approximation of the function
f (x)?
Yes. We could calculate
Example: Cumulative Debt
In this example we see an integral that represents a cumulative sum, rather than
an area.
Let t = time in years
and f (t) = dollars/year; f (t) is a borrowing rate.
Notice the
Approximation of ln e
Heres an example of the power of linear approximation, and of what quadratic
approximation can do for us that linear approximation cannot.
Recall that when we discussed exponenti
Explaining the Formula by Example
As we saw last time, quadratic approximations are a little more complicated
than linear approximation. Use these when the linear approximation is not
enough. For exam
SearchallofeNotes
Search
/3
Findtheintegrals
2
sin + sin tan
sec
0
2
sin + sin tan
sec 2
sin + sin tan
sec
2
2
/3
/3
/3
0
d =
1
2
2
sec
) + C [ cos(0) + C]
3
sin + sin tan
2
2
2
2
d
d = cos(
sin +
Example: Change of Variables
2
Example: 1 (x3 + 2)5 x2 dx
Before, we would have tried to handle this integral by substitution, using
u = x3 + 2. Were going to do the same thing here, taking into accou
Substitution When u Changes Sign
Weve been told that changing variables of integration only works if u(x) is either
always increasing or always decreasing on the interval of integration. Lets see
1
wh
Example: Volume of a Cauldron
In our next, Halloween themed, example well compute the volume of the region
shown below.
y
x
Figure 1: y = x2 rotated around the y -axis.
We could use the method of disk
Example of Estimation
Heres an example in which we use the estimation property of integrals: if
b
b
f (x) g (x) and a < b, then a f (x) dx a g (x) dx.
The example is the same as one weve already seen.
Relative Error
We continue with our example of time dilation in GPS satellite operation. We
started with the following formula from special relativity:
T
Tm =
1
v2
c2
and used a linear approximation
e3x
Example:
1+x
Last lecture we computed the linear approximation for x near 0 of
e3x
= e3x (1 + x)1/2 .
1+x
This lecture well compute a quadratic approximation for this function when x
is near 0.
T
Linear Approximation and the Denition of the Derivative
Another way to understand the formula for linear approximation involves the
denition of the derivative:
f
x0 x
f (x0 ) = lim
Look at this backwa
Alternate Denition of Natural Log
The second fundamental theorem of calculus says that the derivative of an
integral gives you the function back again:
d
dx
x
f (t) dt = f (x).
a
We saw a few examples
Review of the Fundamental Theorem of Calculus
Remember that the First Fundamental Theorem of Calculus (FTC1) said
b
that if F = f , then a f (x) dx = F (b) F (a).
We used this to evaluate denite integ
Area Under the Bell Curve
In addition to exotic but familiar functions like ln x, we can also use definite integrals and Riemann sums to get truly new functions. 2 Example: The solution to y = e-x ; y
Proof of the Second Fundamental Theorem of Calculus
Theorem: (The Second Fundamental Theorem of Calculus) If f is continuous
x
and F (x) = a f (t) dt, then F (x) = f (x).
Proof: Here we use the interp
The Second Fundamental Theorem of Calculus
Were going to start with a continuous function f and dene a complicated
x
function G(x) = a f (t) dt. The variable x which is the input to function G
is actu
Summary of Examples
Now lets look at our results, comparing the function f (x) to the area
under the graph of f between 0 and b.
f (x)
x2
x = x1
1 = x0
b
0
f (x) dx
b
f (x) dx
b3 /3
b2 /2
b = b1 /1
0
Introduction to Curve Sketching
Goal: To draw the graph of f using information about whether f and f are
positive or negative. We want the graph to be qualitatively correct, but not
necessarily to sca
Summation Notation
Youll have noticed working with sums like 12 + 22 + 32 + + (n 1)2 + n2 is
extremely cumbersome; its really too large for us to deal with. Mathematicians
have a shorthand for calcula
Introduction to Related Rates
Next well look at the subject of related rates, which will give us another op
portunity to practice working with several dierent variables and equations.
Example: Police