Single variable calculus deals with two
things: differentiation and integration.
These two concepts are linked by the
fundamental theorem of calculus, seen
already on this slide. You master single
variable calculus if you understand this
theorem and are able to apply it.
A function is a rule which assigns to a real
number of new real number. You see here
an example of a polynomial. If we use the
input x=1 then the output is f(x)=3.
A function can be visualized by
its graph, which is obtained by plotting
the pairs (x,f(x)) in the plane.
A function f is continuous at a point a, if
f(x) converges to f(a) if x converges to a.
The function f is continuous,
if it is continuous at every point.
To the left you see functions which are
continuous. To the right, we see three
basic failures of continuity:
the first is a pole, the second is a jump
discontinuity, the third is an
oscillatory discontinuity.
The exponential function is an
important example of a continuous
function. It takes the value 1 at the
point x=0 and increases
monotonically everywhere. Its inverse, the
natural log function  is only defined
for positive x. The exponential of a sum
is the product of the exponentials. The
logarithm of the product is the sum of
the logarithms.
The sine and cosine function are
continuous functions belonging to the
class of trig functions. They are both
2pi periodic. One can read their values
off from the unit circle. The tangent is
defined as the fraction sine over cosine.
It is periodic with period pi and
discontinuous at the values pi/2 plus
a multiple of Pi. The tangent gives the
slope from an angle x. If that  angle is 90
degrees or - 90 degrees, then the slope is
infinite and not defined.
It is good to to know the values
of the sine and cosine function
for some values. Here are two
triangles: the 45 45 90 triangle and the
30 60 90 triangle.
The first is half of the unit square the
second is half of the equilateral
triangle.
Used quite often are the
double angle formulas. They are especially
handy when doing integrations. The double
angle formulas relate the squares of the
cosine or sine function with a cosine or
sine of the double angles. If you add the
two equations up
you see an important identity. Can you 
see Pythagoras?
An important prototype function
is the sinc function.
It has many applications, for example
in number theory, in Fourier theory
or signal processing. It is also interesting
because there is a mystery at x=0,
where we divide zero over zero. One can verify geometrically that the limit of sin(x) is
always equal to 1 when x converges to 0.
It is therefore possible to assign the
value f(x)=1 to x=0 and have a
continuous function everywhere.
This filling of the gap for this
function is so important that it is
called the 'fundamental theorem of
trigonometry'.
An other important function is the
function of Maria Agnesi. It is
important in statistics, where it is
related to the Cauchy distribution,
a high-risk probability density function.
It is also important because it turns
out to be the derivative of the arctan
function, the inverse of the tangent
function. The function has a horizontal
asymptote. The function on the last
slide was already an example of a rational
function, a quotient of two polynomials.
Here, we see another example of a
rational function; but this function has
poles at x equal to 1 and x equal to -1.
These are points, where the function is not
continuous because the denominator is
zero there.
The function has vertical asymptotes
at those points.
This is the bell curve,
which gives the
Gaussian distribution in statistics. It
has a similar shape than the Maria of
Agnesi function but unlike the Agnesi
function, it has finite variance. The
function again is continuous everywhere
and has horizontal asymptotes at
infinity.
It has been scaled so that the total
area under the curve is 1.
The derivative of a function is defined as a limit:
it is the rate of change of the function
and geometrically given as a slope. You
can approximate the derivative by making
a small step h then look at the rise f(x+h)-f(x)
and divide by the run (x+h)-x which is
h. If the derivative exists at a point a,
then the function is called differentiable at a.
The derivative satisfies a few rules
which need to be practiced. The addition
rule tells that the derivative of a sum is
the sum of the derivatives. Less obvious
are the multiplication -, the quotient - and
the chain rules. These rules allow to
compute the derivatives of complicated
functions obtained from basic functions
You have to know the derivatives on this slide.
They appear so often that you
don't want to have to look them up every
time or have to re-derive the rule.
Note that the log function is the
natural log.
Here are some more derivatives, which are
good to know.
The sign function is the function
which is = 1 for positive x and = -1
for negative x and 0 at 0. The derivative of
the absolute value function |x| is not defined at x=0
Its derivative has a jump discontinuity there.
An observation of Fermat tells that if a differentiable
function has a local maximum at a point a, then the derivative is 0 there.
This follows directly from the
definition. It gives a tool to find
points which are candidates for maxima.
Points where the derivative is 0 are
called critical points.
If the second derivative at a critical point is negative,
then the function is concave down there,
and the function has a local maximum.
Similarly, if the second derivative is
positive at a critical point then we get
a local minimum. If the second derivative
is zero at a critical point, then the
point is an inflection point. If this
happens at a critical point we don't
necessarily have a maximum.
an example is f(x) equals x cubed which
has a critical point zero but neither a
maximum not a minimum there.
The integral of a function is defined as a limit
of Riemann sums. Given a continuous function on an interval, the integral exists.
It has the interpretation of a signed
area. The adjective "sign" is used since
the function can also be negative
leading to a negative integral.
The just defined limits of
Riemann sums defines a definite integral.
It is a number if the upper limit of the
interval is a variable x, then we obtain
a new function F(x) called anti-derivative
of f. We see it here in the case when the
interval goes from 0 to X.  If x is
negative, then the result changes sign.
So, we have now
the concepts of area and slope. The area
is related to integral and the slope is
related to the derivative.
How can these tho seemingly completely
unrelated concepts linked?
The answer is given by
the fundamental theorem of calculus. There are 2versions: the first tells the
derivative of the integral is the function itself.
The second tells that the integral of
the derivative is a difference between
the function value f(x) and f(0).
You can check in the second formula that if  x = 0, then both sides are zero.
The chain rule for differentiation can now
be reversd to get the method of
substitution. To use this method, identify
a function g(x) in the formula and call it u.
Then compute du and substitute
both the dx and g(x) if you get a
function of u only
it might lead to the solution. You see here a
typical example: we want to integrate
log(x)/x,  we substitute u equals log x
then compute du=dx/x. We end up
with the integral for the variable u.
Finally we back-substitute to get again
a function of x.
The product rule leads to integration by
parts. To use this, write the function as
a product of two functions and decide
which part to integrate. Do that first,
then subtract a new integral with the
product of the derivative of the first
function with the integral of the second
function.
The method of trig substitution is a special
case of
substitution, where the function is 
an inverse trig function. It is useful if
a trig function allows to get rid of a
square root.
The method of partial fraction is more like
an algebra trick.
It is a method to write a rational
function in terms of functions which we know
how to integrate. Here is a typical
example: in order to find the constants
A and  B, we cross multiply and compare the coefficients.
Here are some integrals to know.
They were obtained using
the fundamental theorem of calculus from
the derivative table. [note that the integral of tan(x)  is - log(sin(x)), it is the derivative which is 1/cos^2(x).
A direct application of integration
is the computation of area of a region
bounded by the sine curve and the x axis.
We have to integrate the sine function
from 0 to Pi. We know the anti-
derivative of sine of x because of the
fundamental theorem of calculus. We see
that the result is -cos(pi)+cos(0)=2. If we compute the same integral
from zero to 2pi, we get 0. How come? Isn't the area not always positive?
Yes, but the integral is a signed area.
The region below the curve is counted
negatively. The positive and negative
areas cancel.
We can now also compute
volumes of solids. Just slice the body
along some convenient axes, then find
the area A(x) at the height x. When summing over all  slices A(x) dx we get the
volume. In the case of a cone,
the area is Pi x^2. It can be
integrated easily.
We see that the volume is pi/3, which
is a third of the volume of the cylinder
of radius 1.
Let's do the same for the sphere.
Now, since the
slices are by Pythagoras the square root of
1 minus x squared, we have to integrate
pi times (1-x^2). So, the result is 2 pi/ 3.
Archimedes already noticed that this is
the volume of the complement of the cone
inside  the cylinder. Archimedes just compared the slicing areas.
The Da Vinci tower with
rotating floors, proposed by architect
David Fisher, illustrates the Cavalieri
principle that if the slice area does
not change, then also the volume does not
change. To compute a series, we have in general
to sum up infinitely many
terms. An important example is the
geometric series, where we saw terms
which decrease in a geometric manner:
each term is a constant "a" times the previous
term. To the left, we see a visualization
of the sum. It is the total area.
Now, if we scale in the horizontally by
factor a, we see that all, except the first
expression have appeared again. If we add
1 to it, then we get the same area.
If S is the sum, this means a S + 1 = S
Solving  this equation for S gives a
closed-form solution S=1/(1-a) for S.
For an alternating series, the terms to
decrease monotonically and the signs do
do altenate. Such a sum always converges.
You can see it visualized: the first term
overshoots adding the next under shoots,
then we overshoot again etc. The sum is
squeezed between the left lower bound
getting larger and the right upper
bounds which is getting smaller all the
time. In the limit
we have a fixed value. An important
example is the Leibniz series, where the
limit is Pi/4. The computation of
the limit is not always easy. In this
case it can be done by Fourier or power
series techniques.
Not all series do converge. A prototype of the sum 
which is not finite is the harmonic series.
We can make this sum smaller by changing 1/3 to 1/4 and all terms 1/5,1/6 and
1/7 to 1/8 etc. The next seven terms
will be replaced by 1/16. The terms can
now be group to get one half for each group.
An important class of series is formed
by the zeta function. Fix a value s and
look at the sum of 1/n^s.
In the case when s=1, this was the
harmonic series. What happens in general?
It turns out that s=1 is a
threshold. For s larger than 1, the
series converges. For s=2, we 
get a specific value pi^2/ 6 as Euler found
out. Why does the series
converge for s larger than 1? Because we
can relate it with an integral: integrate
the function x^s from 1 to
infinity. This is an indefinite integral
which converges for s larger than than 1.
So, also the series converges. Next we come to power series.
Also Power series are functions. But now, the variable is not 
in the exponent like before but in the base.
Such series have a radius of convergence.
For the absolute value of x smaller than
this radius, the series converges. There
is a formula for the radius of
convergence: it is given as a limsup.
Finally we have the important Taylor series
which allows to write a function as a
series. This works for most of the basic
functions.
It is a magical formula, as we only need
to know all the derivatives of the
function at a point to know the function
nearby.
An example is the exponential function,
where we have a beautiful formula given
as a series involving factorials.
A differential
equation is an equation
for an unknown function
f which involves the function and at least
one derivative of the function. An
important example is the differential
equation f' =a f. It has the
exponential function e^(at)
as a solution. We think now of the
variable as time. Given the initial
value at t=0, we can get the
value at every later t. If the constant a is positive, this differential equation models
exponential growth.
It is the simplest model for population
growth as more members there are in the
group as faster they grow. If it is
negative then we have exponential decay.
in this case as more members there are,
as faster the value decreases.
Here is another
differential equation. It models three fall. Also here, the solution allows us to predict
the future.
In this case we can simply solve for f
by integrating twice and taking care of
the initial position.
A general technique is the separation of variables method. It is a bit magic as we
use the function f like a variable. You see it
work in an example.
Three big heros of calculus are Archimedes,
Newton and Leibniz. Archimedes 
is the father of integral calculus.
Newton and Leibniz share both the credit
for building calculus as we know it.
Fermat, Bolzano and Johann Bernoulli are
three other important
figures in calculus. Fermat got first the
extremization idea. Bolzano is
commemorated for the extreme value
theorem and Bernoulli worked on
differential equations.
Early authors of textbooks were Euler,
Maria of Agnesi and de L'Hopital.
L'hopital's rule is an important method
to find limit. In the case of sin(x)/x for 
example, just differentiate the top and
the bottom to get  1/1 which gives the limit 1.
Calculus allows to predict the path of some 
celestial object.
This can be an asteroid, a rock in space
which is subject to gravitational forces.
Newton has  put this into mathematical
terms.
The most elegant proof that Pi
not be written as a fraction, uses
calculus. This proof is due to Ivan Niven
and uses the fundamental theorem of
calculus and a bit knowledge about
differential equations like the driven
harmonic oscillator.
Movie
Movie
The intermediate value theorem tells that a
continuous function which is negative at
some point a and positive x another
point b must have a zero between a and b.
Not so obvious is the application of the
wobbly table on any surface as uneven
It might be one can always turn the
table so that all four legs are on the
ground.
A Music piece is just a function.
Instead of plotting a function, we
can also play it. Here
you hear an example of a function.
Music
And
here is an example where calculus
is used in number theory. You might have
heard about the Goldbach conjecture. It
tells that every even number larger than
2 is the sum of two primes. Goldbach
wrote this to Euler in a letter of 1742.
Here is a calculus
reformulation: define a function as
a power series where we take all x to
the power p where p is a prime. The
Goldbach conjecture states that all but
the first even derivatives of the square
of this function are nonzero.
